# Matrix approach for optimal spatio-temporal coherent control of wave
scattering

Clément Ferise  Univ Rennes, CNRS, IETR - UMR 6164, F-35000 Rennes, France
Philipp del Hougne  Univ Rennes, CNRS, IETR - UMR 6164, F-35000 Rennes, France
Matthieu Davy  Univ Rennes, CNRS, IETR - UMR 6164, F-35000 Rennes, France
<EMAIL_ADDRESS>
###### Abstract

We present and experimentally verify a matrix approach for determining how to
optimally sculpt an input wavefront both in space and time for any desired
wave-control functionality, irrespective of the complexity of the wave
scattering. We leverage a singular value decomposition of the transport matrix
that fully captures how both the spatial and temporal degrees of freedom
available to shape the input wavefront impact the output wavefront’s spatial
and temporal form. In our experiments in the microwave domain, we use our
formalism to successfully tackle three iconic wave-control tasks in a
disordered cavity: (i) reflectionless transient excitation (“virtual perfect
absorption”), (ii) optimal energy deposition, and (iii) scattering-invariant
time-varying states.

††preprint: APS/123-QED

##  I Introduction

The ability to sculpt the wavefront incident on a scattering system allows for
a judicious control of the outgoing wavefront’s spatial and/or temporal shape,
thanks to the linearity of the wave equation Rotter and Gigan (2017); Cao  _et
al._ (2022). The crux lies in identifying the optimal input wavefront for a
desired wave-control functionality. While iterative techniques optimize in
situ the incident wavefront until the output converges to the desired one by
relying on a feedback Vellekoop and Mosk (2007); Vellekoop  _et al._ (2010);
Ma  _et al._ (2014); Horstmeyer  _et al._ (2015), matrix-based techniques
achieve provably optimal wave control using tools from linear algebra Tanter
_et al._ (2000); Vellekoop and Mosk (2008); Rotter and Gigan (2017). Different
wave-engineering domains (acoustics, optics, etc.) have explored matrix
approaches for optimal coherent wave control under various constraints (e.g.,
only spatial control over the input, only aiming at focusing in space and
time). Here we present a general approach based on a multi-spectral transport
matrix that can optimally leverage both spatial and temporal degrees of
freedom (DoFs) on input and output sides.

Figure 1: (a) Photographic image of our quasi-2D disordered cavity
($L=W=205.74$ mm, $h=10.16$ mm) with the top plate removed to show the inside.
One coax-to-waveguide transition is attached at each side of the cavity,
yielding a two-channel scattering system. The inset illustrates our spatio-
temporal matrix formalism defined in Eq. (S7). (b) Spectrum of the two
eigenvalues of $\left[S(\nu)\right]^{\dagger}S(\nu)$. (c) Spectrum of the real
part of the trace of the Wigner-Smith operator $Q(\nu)$.(d) Singular value
spectrum of $H$ (see Eq. (S7)).

For monochromatic waves, optimal wave-control functionalities are based on
measuring the scattering matrix that fully encodes the linear relation between
spatial input and spatial output DoFs. In optics, the transmission matrix was
initially measured using deformable mirrors capable of spatially shaping the
input wavefront to achieve maximal focusing of light traversing a multiple
scattering medium Popoff  _et al._ (2010). More complicated functionalities
beyond focusing are then achieved using a singular value decomposition (SVD)
of the transport matrix. This includes selective focusing inside the
scattering medium Popoff  _et al._ (2011), perfect absorption of incident
radiations Chong  _et al._ (2010); Wan  _et al._ (2011), speckle engineering
Devaud  _et al._ (2021) and access to open and closed transmission
eigenchannels Vellekoop and Mosk (2008); Kim  _et al._ (2012); Gérardin  _et
al._ (2014); Sarma  _et al._ (2016) in diffusive media, e.g., for optimal
energy deposition Jeong  _et al._ (2018); Bender  _et al._ (2022). New
possibilities have recently emerged by applying an SVD or an
eigendecomposition to other operators constructed from the transport matrix.
This has led to proposals for schemes aimed at optimal energy storage Durand
_et al._ (2019); del Hougne  _et al._ (2021a), optimal micro-manipulation of
targets embedded in scattering environments Horodynski  _et al._ (2019) and
ultimate sensitivity of the output wavefield to perturbations Bouchet  _et
al._ (2021) – provided that the system’s transport matrix can be rapidly
characterized under low-noise conditions.

Manipulating spatially the incident wavefront also enables a degree of control
on output signals in the time domain for incident pulses Aulbach  _et al._
(2011); Katz  _et al._ (2011). Focusing and maximal energy deposition at
arbitrary times are based on measuring the multispectral TM Andreoli  _et al._
(2015); Mounaix  _et al._ (2016a) or the time-gated TM Choi  _et al._ (2013);
Mounaix  _et al._ (2016b); Jeong  _et al._ (2018); Devaud  _et al._ (2022).
More recently, spatio-temporal control over the output has also been achieved
by engineering the scattering system as opposed to sculpting the input
wavefront del Hougne  _et al._ (2016); F. Imani  _et al._ (2022), but this
approach is conceptually very different because the parametrization of the
transport matrix via tunable scattering parameters is in general non-linear
Rabault  _et al._ (2023).

While these approaches are still limited to spatial incident DoFs, existing
optimal spatio-temporal control makes use of both spatial and temporal DoFs on
the input side Mosk  _et al._ (2012). For the specific functionality of
maximal focusing on an output channel, time reversal (TR) is an established
technique pioneered in acoustics that captures the spatio-temporal input-
output relation of a (usually complex) scattering system Fink (1992); Derode
_et al._ (1995). TR consists in emitting phase-conjugated broadband
transmission spectra, and can be refined with iterative techniques Lemoult
_et al._ (2009). In reverberating media, the spatio-temporal compression of
long impulse responses shows that a control on output spatial DoFs is achieved
using input temporal DoFs only for focusing Derode  _et al._ (1995); Draeger
and Fink (1997); Lerosey  _et al._ (2007) and perturbation sensing Bouchet and
Bossy (2022). Corresponding results have also been achieved in the optical
domain for a single-channel McCabe  _et al._ (2011) and multi-channel Mounaix
_et al._ (2020) systems. However, the TR technique does not offer any insights
into how to achieve more elaborate wave control.

In this paper, we introduce an SVD-based approach enabling optimal spatio-
temporal wavefront shaping for arbitrary desired wave-control functionalities.
We illustrate the capabilities of our formalism by applying it (i) to achieve
reflectionless transient excitation (“virtual perfect absorption”, VPA), (ii)
to optimally deliver energy at a targeted time, and (iii) to excite time-
varying scattering-invariant states. Thereby, we generalize the VPA concept
from the single-mode Baranov  _et al._ (2017) to the multi-resonance regime,
as well as scattering-invariant states from the harmonic Pai  _et al._ (2021)
to the time-varying regime. Moreover, our formalism constitutes a unifying
framework to understand to-date seemingly unrelated wave-control concepts.

##  II Matrix Formalism

For monochromatic waves with frequency $\nu$, the transport matrix
$\mathcal{H}(\nu)$ describes the linear relation between the $N$ incoming
channels $x(\nu)$ and $M$ outgoing channels $y(\nu)$ as
$y(\nu)=\mathcal{H}(\nu)x(\nu)$. If all connected channels are consider for
both input and output, $\mathcal{H}(\nu)$ is the system’s full scattering
matrix $S(\nu)$.

To extend this monochromatic input-output relation to the time domain on both
input and output sides, we consider that the $N$ time signals $x_{n}(t_{m})$
are sampled on $N_{tx}$ points within a time interval $\Delta t_{x}=[0\
t_{c}]$. Their spectra $x_{n}(\nu)$ are computed on $N_{\nu}$ frequencies
using the discrete-Fourier-transform (DFT) operator,
$x_{n}(\nu)=D_{x}x_{n}(t)$, where
$D_{x_{km}}=e^{-i2\pi\nu_{k}t_{m}}/\sqrt{N_{tx}}$. To obtain a single multi-
channel spatio-temporal linear relation, we first concatenate these vectors
into the single vectors $x_{t}$ and $x_{\nu}$ of length $NN_{tx}$ and
$NN_{\nu}$. The elements of $x_{t}$ are $x_{t_{j}}=x_{n}(t_{m})$ with
$j=(m-1)N+n$. The linear relation between vectors $x_{t}$ and $x_{\nu}$ is
described by the sparse multi-channel DFT operator $\tilde{D}_{x}$ of
dimensions $NN_{\nu}\times NN_{tx}$ (see SM), $x_{\nu}=\tilde{D}_{x}x_{t}$.
For $N_{tx}=N_{\nu}$ and $t_{c}=t_{\mathrm{max}}=1/\delta\nu$, both $D_{x}$
and $\tilde{D}_{x}$ are unitary operators. However, in the more general case
of $t_{c}<t_{\mathrm{max}}$, they are non-square matrices verifying
$D_{x}^{\dagger}D_{x}=\mathbb{1}_{N_{tx}}$ and
$\tilde{D}_{x}^{\dagger}\tilde{D}_{x}=\mathbb{1}_{NN_{tx}}$ but
$D_{x}D_{x}^{\dagger}\neq\mathbb{1}_{N_{\nu}}$ and
$\tilde{D}_{x}\tilde{D}_{x}^{\dagger}\neq\mathbb{1}_{NN_{\nu}}$. We apply the
same procedure to the $M$ outgoing signals $y_{n}(t)$ measured within the time
window $\Delta t_{y}$.

Finally, we introduce the $NN_{\nu}\times NN_{\nu}$ multi-spectral block-
diagonal matrix $S_{D}$ with the $k$th diagonal block given by $S(\nu_{k})$,
yielding the following linear relation,

$y_{t}=\tilde{D}_{y}^{\dagger}S_{D}\tilde{D}_{x}x_{t}={H}x_{t},$ (1)

where we have defined the operator
${H}=\tilde{D}_{y}^{\dagger}S_{D}\tilde{D}_{x}$. This formulation relates the
spatial and temporal DoFs of a coherent input wavefront to the spatial and
temporal DoFs of the corresponding output wavefront.

Using Eq. (S7), the spatio-temporal input wavefront $x$ giving any arbitrary
multi-channel time-varying output signal $y_{T}$ can be estimated using the
pseudo-inverse ${H}^{+}$ of ${H}$ as $x={H}^{+}y_{T}$. The case of the well-
established TR technique Fink (1992) can also be interpreted in light of Eq.
(S7). In a time-reversal experiment, the goal is to maximize the signal on the
$k$th output channel at time $t=t_{i}$ by injecting a spatio-temporally
sculpted wavefront in the time interval $\Delta t_{x}=[0\ t_{i}]$. The desired
output signal is $y_{n}(t)=\delta(k-n)\delta(t-t_{i})$. Simple algebra shows
that for a pseudo-inverse ${H}^{+}={H}^{\dagger}$, the corresponding input
wavefront is $x_{n}(t)=\tilde{D}_{x}S_{nk}^{*}$, corresponding as expected to
the time-reversed impulse responses between the $N$ input ports and the $k$th
output port. TR is hence a special case of our more general technique for
optimal spatio-temporal wave control.

Of particular interest is the SVD of ${H}$: ${H}=U\sqrt{\Lambda}V^{\dagger}$.
The diagonal matrix $\Lambda$ directly yields the integral of output
intensities over the time interval $\Delta t_{y}$ and all ports
$\lambda_{n}=||{H}V_{n}||^{2}=\int_{\Delta t_{y}}[\Sigma_{n}|y_{n}(t)|^{2}]dt$
upon injection through all ports within the time interval $\Delta t_{x}$ of
the spatio-temporal states $x_{n}(t)$. The incident signals $x_{n}(t)$
composing the vector $V_{n}$ are normalized such as $\int_{\Delta
t_{x}}[\Sigma_{n}|x_{n}(t)|^{2}]dt=1$. If our transport matrix $\mathcal{H}$
is taken to be $S$, the highest and lowest values of $\lambda_{n}$ determine
the complete range of achievable output reflection, and hence offer a simple
method to identify the provably optimal input states for maximal or minimal
output reflections. If instead $\mathcal{H}$ is a transmission matrix, then
$\lambda_{n}$ corresponds to transmitted rather than reflected intensity
instead.

Figure 2: Experimentally measured output signals upon coherent spatio-temporal
in situ injection of various right singular vectors of $H$. For each
considered state, we display the injected (blue) and outcoming (red) time-
varying signals on the two channels (middle and right columns), as well as the
corresponding signal spectra averaged over both channels (left column). The
first and second rows correspond to two representative choices of the smallest
near-zero singular values that yield qualitatively distinct results.

##  III Experimental Setup

We consider the quasi-2D electromagnetic cavity shown in Fig. 1(a) that
contains $15$ randomly placed metallic cylinders. This disordered system is
coupled via matched coax-to-waveguide transitions to two channels at the left
and right sides. Using a vector network analyzer, we can conveniently measure
the $2\times 2$ scattering matrix between 8 and 9 GHz (frequency step:
$\delta\nu=1.5625$ MHz). In order to inject spatio-temporally sculpted signals
in situ, we generate baseband signals between 0 and 1 GHz on an FPGA board at
a sampling rate of 4 GS/s and multiply them with a local oscillator at $f=8$
GHz (see SM for additional details). We coherently record the spatio-temporal
output signals with an ultra-wideband multi-channel oscilloscope using a
sampling rate of 40 GS/s.

For the particular choice of $\Delta t_{x}=\Delta t_{y}=[0~{}t_{c}]$ with
$t_{c}=1/(2\delta\nu)$, we display in Fig. 1(d) the singular value (SV)
spectrum of $H$ for our disordered system shown in Fig. 1(a). The largest SV
$\lambda_{1}=0.84$ remains below unity due to the inevitable presence of
absorption in our experimental system. The presence of absorption is also
evidenced by the unitarity deficit of $S$ that is seen in Fig. 1(b). This
unitarity deficit is frequency dependent and it is notable in Fig. 1(b) that
for some frequencies the smaller one of the two eigenvalues of $S^{\dagger}S$
even gets close to zero, indicating that a time-harmonic state yielding a
rather small reflected intensity exists. For the transient regime, we observe
in Fig. 1(d) that multiple of the smallest SVs of ${H}$ are extremely close to
zero, suggesting that in the transient regime (i.e., within the interval
$\Delta t_{y}$), the reflected signal intensity can be suppressed very
efficiently if the corresponding spatio-temporal states are injected.

##  IV Reflectionless Transient Excitation.

We begin by considering reflectionless transient excitation (RTE) of a
scattering system. As mentioned above and seen in Fig. 1(d), multiple
$\lambda_{n}$ are extremely close to zero for $\Delta t_{x}=\Delta
t_{y}=[0~{}t_{c}]$. We observe that two qualitatively different types of such
RTEs exist in our system, and we display one experimentally measured example
of each in the first and second rows of Fig. 2.

The first type, in the first row of Fig. 2, is an instance of a concept
previously introduced as “coherent virtual absorption”(CVA) Baranov  _et al._
(2017). CVA is a promising concept for energy storage Trainiti  _et al._
(2019), sensing and transient application of optical forces Lepeshov and
Krasnok (2020) where an isolated complex-valued zero of the scattering
response is engaged in the transient regime to suppress reflection within the
excitation interval. Any scattering response can be decomposed into its
constituent scattering singularities, namely poles $\tilde{\nu}_{n}$ and zeros
$z_{n}$ Grigoriev  _et al._ (2013a, b); Krasnok  _et al._ (2019). If a zero is
real-valued, reflectionless time-harmonic excitation of the scattering
structure is possible Chong  _et al._ (2010); Pichler  _et al._ (2019);
Sweeney  _et al._ (2020). Within a limited frequency interval, a disordered
system like ours usually does not have a real-valued zero unless the
scattering system is purposefully perturbed to impose one F. Imani  _et al._
(2020); Frazier  _et al._ (2020); del Hougne  _et al._ (2021b, c); Sol  _et
al._ (2022, 2023). However, given only complex-valued zeros, reflectionless
excitation can still be achieved, albeit only in the transient regime for a
limited temporal interval, by shaping the incident signal in space and time.
When the incident signal is interrupted, the energy accumulated within the
medium leaks out through the channels.

Previous experimental observations of RTE Trainiti  _et al._ (2019); Delage
_et al._ (2022, 2023) considered systems with isolated resonances (similar to
the first row in Fig. 2). In the simple picture of isolated resonances
experiencing uniform absorption within the medium, poles $\tilde{\nu}_{n}$ and
zeros $z_{n}$ are found at the same real frequency $\nu_{n}$ but acquire
different imaginary parts, $\tilde{\nu}_{n}=\nu_{n}-i(\gamma_{n}+\gamma_{a})$
and $z_{n}=\nu_{n}+i(\gamma_{n}-\gamma_{a})$ ($\gamma_{n}$ is the resonance
width and $\gamma_{a}$ is the absorption strength). The incident signal
corresponding to a zero $z_{n}$ is a monochromatic signal at frequency
$\mathrm{Re}[z_{n}]$ modulated with an exponentially increasing envelope at
the rate $\mathrm{Im}[z_{n}]$ Baranov  _et al._ (2017). Identifying poles and
zeros however requires cumbersome protocols that becomes highly challenging in
the regime of overlapping resonances Trainiti  _et al._ (2019); Delage  _et
al._ (2022, 2023); Chen and Anlage (2022); Mandelshtam and Taylor (1997); Kuhl
_et al._ (2008); Davy and Genack (2018). In contrast, our technique solely
requires the SVD of the transport matrix $H$ and directly yields the non-
intuitive spatio-temporal input signals. For overlapping resonances, these
signals involve multiple zeros as seen in the second row of Fig. 2.

How many VPA states exist for a given transport matrix $H$? Theoretically, we
expect this number to correspond to the number of complex zeros in the upper
half of the complex frequency plane. We show in the SM that a VPA state indeed
disappears as the corresponding zero crosses the real axis and acquires a
negative imaginary part. The total number of poles and zeros can be estimated
using Weyl’s law Weyl (1911); Arendt  _et al._ (2009); Pierrat  _et al._
(2014). The average density of states $\rho(\omega)\sim A\omega/(2\pi
c_{0}^{2})$, where $A$ is the area of the scattering region, gives a
theoretical estimate of 25 modes between 8 and 9 GHz. This number can also be
roughly confirmed by counting the peaks in Fig. 1(b). However, counting the
number of zeros with positive imaginary part is more challenging. To do so, we
study in Fig. 1(c) the number of positive peaks of the real part of the
Wigner-Smith time delay $\tau_{W}(\nu)=[\mathrm{Tr}(Q(\nu))]$. Here,
$Q(\nu)=-i\left[S(\nu)\right]^{-1}[\partial S(\nu)/\partial\nu]$ is the
Wigner-Smith operator Wigner (1955); Smith (1960). The real part of the
eigenvalues of $Q$ is assumed to be proportional to a dwell time Fan and Kahn
(2005); Rotter  _et al._ (2011); Davy  _et al._ (2015); Böhm  _et al._ (2018);
Durand  _et al._ (2019); del Hougne  _et al._ (2021a); Huang  _et al._ (2022)
which is diverging as a zero crosses the real axis Asano  _et al._ (2016); del
Hougne  _et al._ (2021b); Chen  _et al._ (2021a, b). A positive (negative)
peak of $\mathrm{Re}[\tau_{W}]$ is a clear signature of a zero with positive
(negative) imaginary part. The number of positive peaks (16) in
$\mathrm{Re}[\tau_{W}(\nu)]$ is in good agreement with the number of
$\lambda_{n}$ below $\lambda=10^{-5}$ (15). This number increases with the
number $N$ of ports connected to the cavity as the channel decay rate scales
as $\langle\gamma_{n}\rangle=N\gamma_{0}+\gamma_{a}$, where $\gamma_{0}$ is
the single-channel decay rate. In our system, if we increase $N$ from 2 to 6,
$\gamma_{n}\gg\gamma_{a}$ so that all the zeros have positive imaginary parts
and the number of VPA states is equal to 25 (see SM).

##  V Optimal Energy Delivery

Figure 3: (a,b) Optimal deposition of energy corresponding to the largest
singular value of 0.84 for $\Delta t_{x}=\Delta
t_{y}=[0~{}0.5t_{\mathrm{max}}]$ with $t_{\mathrm{max}}=0.64$ $\mu$s. (c,d)
Optimal energy deposition at two chosen output times: (c)
$t=0.5t_{\mathrm{max}}$, (d) $t=0.52t_{\mathrm{max}}$ with
$t_{\mathrm{max}}=0.80$ $\mu$s. The last two rows show the spatially averaged
time-varying input (blue) and output (red) intensities upon injecting states
for maximal reflection corresponding to the first singular value of $H$. The
insets are a zoom around the focal time.  Figure 4: (a) Temporal transmitted
intensity through the cavity for an incident pulse at
$t=0.5t_{\mathrm{max}}=0.128~{}\mu$s transmitted on a single channel. (b-d)
Optimal energy deposition at $t_{c}=0.5t_{\mathrm{max}}$ (b),
$t_{c}=0.55t_{\mathrm{max}}$ (c) and $t_{c}=0.6t_{\mathrm{max}}$ (d) with
$t_{\mathrm{max}}=0.26$ $\mu$s.

We now turn our attention to the optimal energy deposition, which consists in
delivering the maximum possible amount of energy within the output time
interval $\Delta t_{y}$ given spatio-temporal coherent control in the input
time interval $\Delta t_{x}$. In Fig. 3(a,b), we experimentally observe the
input and output signals upon injecting the state corresponding to the largest
singular value of ${H}$ for the intervals considered in Fig. 1(d). During the
transient excitation, a very large fraction ($\lambda_{1}=0.84$) of the
injected energy exits the system within the same temporal interval ($\Delta
t_{x}=\Delta t_{y}$). The remainder of the injected energy is absorbed in the
system. Maximal-reflection states corresponding to other temporal intervals
are presented in the SM.

By adjusting the definition of the output time interval $\Delta t_{y}$, the
same approach can also yield optimal energy delivery at arbitrary times after
the transient excitation interval. Two examples in the limit of a short output
interval $\Delta t_{y}=\delta(t-t_{c})$ are shown in Fig. 3(c,d). In all
cases, a sharp increase of total reflection at the selected time is observed.
Of course, the duration of the focused output signals is not arbitrarily short
but is instead determined by the bandwidth of the spatio-temporal input
signal.

The number of transmitting channels is here limited ($N=2$) by our electronic
hardware. To further demonstrate the interest of our approach, we consider a
second 2D cavity of length $L=0.5$ m and width $W=0.25$ m with two arrays of
$N=8$ coax-to-waveguide transitions attached on the left and right interfaces
(see Ref.Davy  _et al._ (2021) for a description of the experimental setup).
We measure the transmission matrix (TM) in the diffusive regime between
between 10 and 15 GHz with a frequency step of 3.9 MHz. For a short incident
pulse delivered at $t=0.5t_{\mathrm{max}}$ through a single channel, the
temporal profile of the output intensity decays exponentially with a decay
time equal to $\tau=0.0186t_{\mathrm{max}}$ ($t_{\mathrm{max}}=0.26$ $\mu$s),
see Fig. 4(a). Using the linearity of the wave equation, we reconstruct the
transmitted intensity averaged over outgoing channels for an optimal energy
delivery at time $\Delta t_{y}=\delta(t-t_{c})$ with
$t_{c}=0.5t_{\mathrm{max}}$, $t_{c}=0.55t_{\mathrm{max}}$ and
$t_{c}=0.6t_{\mathrm{max}}$ (see Fig. 4(b-d)). In each case the optimal
incoming signal is assumed to be injected within the interval $\Delta
t_{y}=[0-0.5t_{\mathrm{max}}]$. A clear enhancement of the transmitted signals
at selected temporal interval is now observed even at late times.

Our spatio-temporal matrix formulation takes full advantage of both spectral
and spatial DoFs to achieve optimal total transmission at selected times. We
show in Fig. 4 that the transmitted intensity at the focal time scales
linearly with the number of incoming antennas $N$ and the bandwidth of
incident signals $\Delta\nu$. This is in agreement with theoretical
predictions for time-reversal experiments in diffusive or chaotic systems,
which demonstrates that the focused intensity increases linearly with the
number of DoFs $N\times N_{f}$. Here $N_{f}$ is the ratio between the
bandwidth $\Delta\nu$ and the spectral correlation length $\delta\nu\sim
1/\tau$. For $N=8$ and a bandwidth of 5 GHz, we estimate that the total DoFs
is equal to 190, which is of the same order of magnitude as the number of
spatially controllable optical modes in Ref. Devaud  _et al._ (2022).

Figure 5: Enhancement of transmitted intensity with respect to the bandwidth
(a) and number of antennas (b) used for optimal transmitted energy. The
markers represent experimental data for different selected times $t_{c}$
between $t_{c}=0.5t_{\mathrm{max}}$ and $t_{c}=0.7t_{\mathrm{max}}$ with
$t_{\mathrm{max}}=0.26$ $\mu$s. The dashed lines are linear fits of the data.
The intensity is normalized by its result found for $N=8$, a bandwidth of 5
GHz and $t_{c}=0.5t_{\mathrm{max}}$

##  VI Scattering-Invariant Time-Varying States.

Finally, we explore the possibility of exciting states that have the same
spatial and temporal input and output patterns, generalizing the recent
concept of time-harmonic (monochromatic) scattering-invariant states that have
the same spatial input and output patterns Pai  _et al._ (2021). To that end,
we solve the eigenvalue problem

${H}x_{n}=\alpha_{n}x_{n}$ (2)

and extract the input signals giving optimal similarity to the output signals
within an arbitrary interval. States with smallest $|\alpha_{n}|$ obviously
correspond to low-reflection transient excitations. Of more interest are
states with maximal $|\alpha_{n}|$. Our approach guarantees that the
similarity between input and output states is also maximal. We have injected
in the smaller cavity with $N=2$ two such scattering-invariant time-varying
states with large eigenvalue $|\alpha_{n}|$, $|\alpha|=0.71$ and
$|\alpha|=0.58$ see Fig. 6. The input and output temporal intervals are
$\Delta t_{x}=\Delta t_{y}=[0~{}0.5t_{\mathrm{max}}]$. Their spectra are
peaked on a single or a few frequencies with a bandwidth inversely
proportional to the temporal interval. In the time domain, both signals are
increasing exponentially. While RTE states are associated with seemingly large
delay times, the scattering-invariant time-varying states require quasi-non-
dispersive transport and correspond to short delay times. They are therefore
peaked within frequency ranges in which the reflection coefficient presents a
flattened shape [see Fig. 6(a,c)].

Figure 6: Spectral (a,c) and temporal (b,d) representation of two scattering-
invariant time-varying states for a temporal interval of $[0\ 3.18]$ $\mu$s
(i.e. $[0\ t_{c}]$ with $t_{c}=2/\delta\nu$). The input (blue) and output
(red) signals are shown for the first channel (and look similar for the second
channel). The eigenvalues are $|\alpha|=0.71$ in (b) and $|\alpha|=0.58$ in
(d). The black curve in (a,c) is the reflection $R(\nu)$ averaged over the
input ports. Figure 7: Temporal intensity profile of an invariant time-varying
state in transmission through the large cavity for an incident interval
$[0~{}25]$ns and an output interval $[8~{}0.33]$ns. The blue line is the
incident signal, the red line is the outgoing signal and the green dashed line
is the output signal shifted by $-\Delta\tau=-8$ns. The inset is a zoom of the
field on a small interval represented by the dashed black lines in the figure.

We then consider scattering-invariant time-varying states in transmission
through the larger cavity with $N=8$ antennas on each side. The length of the
input interval is fixed to $0.3t_{\mathrm{max}}$ and the output temporal
interval is now shifted by $d\tau$, $\Delta t_{y}=\Delta t_{x}+\Delta\tau$. We
find a maximal similarity coefficient $|\alpha|=0.77$ for
$\Delta\tau=0.08t_{\mathrm{max}}$. The corresponding length
$c_{0}\Delta\tau=2.04$ m is close to the mean path length of the cavity Davy
_et al._ (2021). The average temporal envelopes of input and output signals
are shown in Fig. 7. The output signal shifted by $-\Delta\tau$ is very
similar to the injected signal. Since input and output speckle patterns are
correlated over a temporal interval, these states could be useful for
broadband imaging through disordered systems.

##  VII Conclusion

We have presented a matrix formalism enabling optimal spatio-temporal coherent
control of waves in arbitrarily complex scattering systems by taking full
advantage of both spatial and temporal DoFs. Our work presents a unifying
perspective on many contemporary wave-control techniques (time reversal,
wavefront shaping, virtual perfect absorption, scattering-invariant modes) and
even generalizes them. Beyond providing fundamental insights, our technique
may readily find applications in the microwave and acoustics regimes where the
necessary hardware for coherent spatio-temporal wavefront generation is
available.

### Acknowledgements

We acknowledge Clara Moy for assistance with the work presented in Sec. IV.
This publication was supported by the European Union through the European
Regional Development Fund (ERDF), by the French region of Brittany and Rennes
Métropole through the CPER Project SOPHIE/STIC & Ondes. M. D. acknowledges the
Institut Universitaire de France. C. F. acknowledges funding from the French
“Ministère de la Défense, Direction Générale de l’Armement”.

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Supplemental Material for
“Matrix approach for optimal spatio-temporal coherent control of wave
scattering”

Clément Ferise,1 Philipp del Hougne,1 and Matthieu Davy1

1Univ Rennes, CNRS, IETR - UMR 6164, F-35000 Rennes, France
(Dated: )

##  I Theory

We provide here details on the multichannel temporal operator given in the
main text. We consider $N$ time signals $x_{n}(t_{m})$ sampled on $N_{tx}$
points between $0$ and $t_{c}$. The Fourier transform on $N_{\nu}$ frequencies

$S(\nu_{k})=\frac{1}{\sqrt{N_{tx}}}\displaystyle\sum_{m=0}^{N_{tx}-1}s(t_{m})e^{-2i\pi\nu_{k}t_{m}}$
(S1)

can be expressed in terms of the discrete Fourier transform (DFT) operator,
$x_{n}(\nu)=D_{x}x_{n}(t)$ with elements

$[D_{x}]_{km}=\frac{1}{\sqrt{N_{tx}}}e^{-i2\pi\nu_{k}t_{m}}.$ (S2)

The dimensions of $D_{x}$ is $N_{\nu}\times N_{tx}$. When we set $\delta
t=1/(N_{\nu}d\nu)$ and $t_{c}=t_{\mathrm{max}}$ with
$t_{\mathrm{max}}=1/\delta\nu$, the operator $D_{x}$ is unitary, meaning that
$D_{x}D_{x}^{\dagger}=\mathbb{1}_{N_{\nu}}$. In the more general case of
control over smaller time invervals $t_{c}<t_{\mathrm{max}}$
($N_{tx}<N_{\nu}$), the DFT operator is non-square. It however still verifies
${D}_{x}^{\dagger}{D}_{x}=\mathbb{1}$ but
${D}_{x}{D}_{x}^{\dagger}\neq\mathbb{1}$.

We then seek for a single multichannel linear relation for the $N$ input and
output channels. We first concatenate the spectral and temporal matrices into
the single vectors $x_{t}$ and $x_{\nu}$ of length $NN_{tx}$ and $NN_{\nu}$,
given below for $N=2$:

$x_{t}=\begin{pmatrix}x_{1}(t_{1})\\\ x_{2}(t_{1})\\\ x_{1}(t_{2})\\\
x_{2}(t_{2})\\\ \vdots\\\ x_{1}(t_{N_{tx}})\\\
x_{2}(t_{N_{tx}})\end{pmatrix}~{}\textnormal{and}~{}x_{f}=\begin{pmatrix}x_{1}(f_{1})\\\
x_{2}(f_{1})\\\ x_{1}(f_{2})\\\ x_{2}(f_{2})\\\ \vdots\\\
x_{1}(f_{N_{\nu}})\\\ x_{2}(f_{N_{\nu}})\end{pmatrix}$ (S3)

The elements of $x_{t}$ are therefore $x_{t_{j}}=x_{n}(t_{m})$ with
$j=(m-1)N+n$. Using this formulation, we can find a linear relation between
$x_{t}$ and $x_{\nu}$. We introduce the sparse multichannel DFT operator
$\tilde{D}_{x}$ of dimensions $NN_{\nu}\times NN_{tx}$ expressed as (still for
$N=2$):

$\tilde{D}_{x}=\begin{pmatrix}(D_{x})_{00}&0&\cdots&(D_{x})_{0N_{tx}}&0\\\
0&(D_{x})_{00}&\cdots&0&(D_{x})_{0N_{tx}}\\\
(D_{x})_{10}&0&\cdots&(D_{x})_{1N_{tx}}&0\\\
0&(D_{x})_{10}&\cdots&0&(D_{x})_{1N_{tx}}\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\
(D_{x})_{N_{\nu}0}&0&\cdots&(D_{x})_{N_{\nu}N_{tx}}&0\\\
0&(D_{x})_{N_{\nu}0}&\cdots&0&(D_{x})_{N_{\nu}N_{tx}}\end{pmatrix}$ (S4)

to get

$x_{\nu}=\tilde{D}_{x}x_{t}$ (S5)

One can easily verify that $\tilde{D}_{x}$ has the same properties as $D_{x}$:
it is unitary for $t_{c}=t_{\mathrm{max}}$ and that
$\tilde{D}_{x}^{\dagger}\tilde{D}_{x}=\mathbb{1}$ for any temporal interval.
However $\tilde{D}_{x}\tilde{D}_{x}^{\dagger}\neq\mathbb{1}$ for
$t_{c}<t_{\mathrm{max}}$.

We apply the same procedure to the $M$ outgoing signals $y_{n}(t)$ measured
within the time window $\Delta t_{y}$. Finally, we define the block diagonal
matrix $S_{D}$ with

$S_{D}=\begin{pmatrix}S(\nu_{1})&0&0&\cdots&0\\\ 0&S(\nu_{2})&0&\cdots&0\\\
\vdots&0&\ddots&0&\vdots\\\ \vdots&\vdots&0&\ddots&\vdots\\\
0&0&\cdots&0&S(\nu_{N_{\nu}})\end{pmatrix}$ (S6)

We finally get $\tilde{D}_{y}y_{t}=S_{D}\tilde{D}_{x}x_{t}$ and

$y_{t}=\tilde{D}_{y}^{\dagger}S_{D}\tilde{D}_{x}x_{t}.$ (S7)

which is Eq. (1) of the main text.

##  II Absorption and virtual perfect absorption

Figure S1: (a) Reflection $|S(\nu)|^{2}$ for a single resonance at
$\nu_{n}=8.5$ GHz. The scattering matrix is described by the Breit-Wigner
formula given in Eq. (S8). The channel decay rate is $\gamma_{n}=8$ MHz and
$\gamma_{a}=\gamma_{n}/2$. (b-d) Input and output signals for the minimal
subspace of $H$ corresponding to an excitation between $0$ and
$t_{c}=t_{\mathrm{max}}/2$ ($t_{\mathrm{max}}/2=1/\delta\nu$) for
$\gamma_{a}=\gamma_{n}/2$ (b), $\gamma_{a}=\gamma_{n}$ (c) and
$\gamma_{a}=2\gamma_{n}$ (c)  Figure S2: Variation of the last eigenvalue of
$H\dagger H$ as a function of the absorption rate $\gamma_{a}/\gamma_{n}$. VPA
states with suppressed reflection are obtained only as
$\gamma_{a}\ll\gamma_{n}$.

In this section, we demonstrate numerically that the VPA states disappear when
a zero of the scattering matrix $S$ crosses the real axis. We consider in the
following a single channel $N=1$ system but an extension to multichannel
systems is straightforward. For a single resonance, the scattering matrix is
expressed by a Breit-Wigner formula,

$S(\nu)=1-\frac{i\gamma_{n}}{\nu-\nu_{n}+i(\gamma_{n}+\gamma_{a})}$ (S8)

Here $\gamma_{n}$ represents the coupling of the channel to the system and
$\gamma_{a}$ is the absorption linewidth. The reflection coefficient
$|S(\nu)|^{2}$ is shown in Fig. S1(a) for $\gamma_{a}=\gamma_{n}/2$. The
resonance is clearly visible at $\nu_{n}=8.5$ GHz.

The pole of $S$ is $\tilde{\nu}_{n}=\nu_{n}-i(\gamma_{n}+\gamma_{a})$ and the
zero of $S$ is $z_{n}=\nu_{n}+i(\gamma_{n}-\gamma_{a})$. When
$\gamma_{n}>\gamma_{a}$, the imaginary part of $z_{n}$ is positive and the
zero is located in the upper complex plane. This zero can then be accessed by
sending a monchromatic signal at frequency $\nu_{n}$ modulated by an
exponentially growing envelope at rate $\gamma_{n}-\gamma_{a}$. This input
temporal signal $x(t)$ for an excitation between $0$ and
$t_{c}=t_{\mathrm{max}}/2$ is directly extracted from the last subspace of the
corresponding $H=\tilde{D}_{y}^{\dagger}S_{D}\tilde{D}_{x}$ matrix. The
outgoing energy in such state of virtual perfect absorption is suppressed
between $0$ and $t_{c}$. Once the excitation is turned off, the energy stored
within the cavity that has not been absorbed is released at rate $\gamma_{n}$.
The envelopes of input and output signals are illustrated in Fig.S1(b) for
$\gamma_{a}=\gamma_{n}/2$.

However, as $\gamma_{a}$ increases, zero reflection between $0$ and $t_{c}$
becomes impossible for an excitation only within this range. For instance, for
$\gamma_{a}=\gamma_{n}$, the zero of $S$ is real, $z_{n}=\nu_{n}$, and the
excitation $x(t)$ must be fully monochromatic to be perfectly absorbed with no
reflection. This is the condition of coherent perfect absorption found when
the absorption decay rate balances the excitation rate Chong  _et al._ (2010).
The reflection between $0$ and $t_{c}$ for any non-monochromatic signal cannot
be suppressed. In Fig.S1(c), we observe that the minimal reflection state of
the $H$ matrix is still increasing in time but the reflection is not zero. For
$\gamma_{a}>\gamma_{n}$, the state with minimal reflection presents a
pronounced peak at $t=t_{c}/2$ [see Fig.S1(d)].

In Fig. S2, we show the variation of the minimal reflection value found from
the singular value decomposition of $H$. It clearly shows that the reflection
can indeed be completely suppressed for $\gamma_{a}\ll\gamma_{n}$ but this
property is lost as $\gamma_{a}$ approaches $\gamma_{n}$ as expected. For
$\gamma_{a}=2\gamma_{n}$, the smallest reflection is even quite strong
($\mathrm{min}[\lambda_{n}]=0.13$).

##  III Details on the experimental setup

In this section we present in details our experimental setup. A sketch is
shown on Fig S3. A $4\times 4$ multi-channel square cavity (side $L=205.74$ mm
and height $H=10.16$ mm) comprising only two coupled ports to the system is
used. Uncoupled ports are obstructed by a metallic plate.The measurement of
the $2\times 2$ scattering matrix between $8$ and $9$ GHz is operated with a
Vector Network Analyser (VNA) on the two coupled ports of the cavity via coax-
to-waveguide transitions with one circulator (Aerocomm $J80.160$) connected to
it. We use circulators on each port to inject and measure the signals with a
$4$-ports VNA.

By applying the SVD-based approach and matrix formalism presented in the main
text, we obtain baseband temporal signals with frequencies between $0$ and $1$
GHz. These signals are first up-converted in the frequency range between $0.2$
and $1.2$ GHz via a Fourier transform and an inverse Fourier transform
successively. The two pseudo-baseband signals ($s_{1}$ and $s_{2}$ on Fig S3)
are finally uploaded on a FPGA board (Xilinx ZYNQ-ZCU111). They are
analogically up-converted in the frequency range $[8~{}9]$ GHz using two wide
band frequency mixers (Mini-Circuits ZX05-153LH-S+) connected to a local
oscillator at $7.8$ GHz. The same local oscillator is connected to the
frequency mixers via a 4 ways power divider (AMD-GROUP PD4-2-18-10) to
synchronize the signals. The RF output of the frequency mixers is connected to
the two circulators matched with the two coax-to-waveguide transitions coupled
to the cavity. To measure the signals reflected from the cavity, the third
channel of each circulator is connected to an ultra-wideband oscilloscope with
a sampling rate of $40$GSamples/s (SDA 816Zi-B).

Figure S3: Detailed sketch of the experimental setup. The inset explicitly
shows the role of the circulators.

##  IV Impact of the size of the input time interval

Figure S4: (a) Spectrum of the first (blue) and last (red) singular value of H
as a function of the variation of $t_{c}$. (b-i) Temporal representation from
numerical simulations upon coherent spatio-temporal injection of the first
(b,d,f,h) and the last (c,e,g,i) right singular vectors of $H$ for four
different input time intervals: (b,c) $t_{c1}=0.004t_{max}$, (d,e)
$t_{c2}=0.05t_{max}$, (f,g) $t_{c3}=0.35t_{max}$ and $t_{c4}=0.9t_{max}$.

In Fig. 2 on the main text we have presented our new approach by displaying
optimal energy deposition and its opposite: reflectionless transient
excitation for $\Delta t_{x}=\Delta t_{y}=[0~{}t_{c}]$ with
$t_{c}=t_{\mathrm{max}}/2$ and for $N=2$ channels coupled to the scattering
system. In this section we numerically explore the impact of the time interval
on reflection coefficients.

We start by varying the interval length $t_{c}$. We display in Fig. S4(a) the
spectrum of the first (blue) and the last (red) singular value (SV) of $H$ as
a function of $t_{c}$. The time axis is normalized by the maximum time which
is inversely proportional to the frequency step:
$t_{\mathrm{max}}=1/\delta\nu$. These states correspond respectively to the
maximum and minimum possible reflection within the output interval $\Delta
t_{y}$. The first SV is zero for $t_{c}=0$, as expected, and then
exponentially increases to rapidly reach a plateau at $\lambda_{1}\approx
0.84$ shortly after $t_{c}=t_{c2}=0.05t_{\mathrm{max}}$. Thus, for $t>t_{c2}$,
the interval length does not impact the reflection coefficient. On the other
hand, the last SV is minimum for $t_{c2}<t<0.8t_{\mathrm{max}}$ with an
average value $\langle\lambda_{N_{\Lambda}}\rangle\approx 2\times 10^{-11}$.
For very short and large values of $t_{c}$ and $t_{c}$, it is however not
possible to shape the incident wavefront temporally to modulate the reflection
coefficient. This implies that VPA states presented in the main text do not
exist when $t_{c}$ is close to $0$ or $t_{\mathrm{max}}$.

The temporal responses for four time intervals
($t_{c1}=0.004t_{\mathrm{max}}$, $t_{c2}=0.05t_{\mathrm{max}}$,
$t_{c3}=0.35t_{\mathrm{max}}$ and $t_{c4}=0.9t_{\mathrm{max}}$) are shown in
the four last rows of Fig. S4. These signals are reconstructed numerically
from measurements of the scattering matrix. On the first column, input (blue)
and output (red) signals upon injecting the state corresponding to the maximum
possible amount of energy are shown for the four time intervals considered.
RTE’s input and output signals, i.e. signals corresponding to minimal
reflection, are displayed in the same way on the second column. In order to
better observe the signals on the first line ($t_{c}=0.004t_{\mathrm{max}}$),
the time axis is adjusted to $[0\ 0.03]\mu s$.

##  V Number of reflectionless transient states

As explained in the main text, the number of reflectionless states found from
the spatio-temporal operator $H$ is theoretically equal to the number of zeros
located in the upper complex plane. The total number of zero within the
frequency range $[f_{1}=\omega_{1}/(2\pi);f_{2}=\omega_{2}/(2\pi)]$ is equal
to the number of poles of the scattering matrix and can be estimated from
Weyl’s law : $N_{\omega}=\int_{\omega_{1}}^{\omega_{2}}\rho(\omega)d\omega$,
where $\rho(\omega)\sim A\omega/(2\pi c_{0}^{2})$ is the density of states.
Here $A$ is the area of the cavity.

In this section, we provide further evidence of the relation between the
number of reflectionless transient states and the number of zeros with
positive imaginary parts. To do so, we vary the number of channels coupled to
the cavity, from $N=2$ (corresponding to measurements presented in the main
text) to $N=8$. In each case, we measure the $N\times N$ scattering matrix and
obtain $\tau_{W}(\nu)$.

The real part of the Wigner time delay is represented on the first column of
Fig. S5 for different number of coupled channels $N$, $N=2$ (a), $N=4$ (c) and
$N=6$ (e). For each case, the corresponding spectrum of the last $40$ singular
values of $H$ is shown on the second column.

For $N=2$, the Wigner time delay comprises positive and negative peaks. We
find $18$ singular values of $H$ bellow $\lambda=10^{-5}$. The small
difference from the number presented in the main text is due to the slight
changes in the system. This time, all the channels are matched with coax-to-
waveguide transitions. For each measurements, $N$ ports are connected to the
Vector Network Analyser, the others being in open-circuit states. This pushes
the reflective boundaries away from the center of the cavity and so virtually
increases its size. For $N=4$, the Wigner time delay comprises only one
negative peak and $25$ singular values below $\lambda=10^{-5}$ are found. For
$N=6$, the peaks of $\tau_{W}(\nu)$ are all positive so that we expect that
the number of reflectionless states to be equal to the number of resonances
(or poles) with the bandwidth. We find $25$ singular values of $H$ smaller
than $\lambda=10^{-5}$ which is in good agreement with Weyl’s estimate.

Figure S5: (a,c,e) Spectra of the real part of the trace of the Wigner-Smith
operator $Q(\nu)=-i\left[S(\nu)\right]^{-1}[\partial S(\nu)/\partial\nu]$ and
(b,d,f) spectra of last singular values of $H$ for $N=2$ (a,b), $N=4$ (c,d)
and $N=6$ (e,f).

## References

  * Chong  _et al._ (2010) Y. Chong, L. Ge, H. Cao, and A. D. Stone, Phys. Rev. Lett. 105, 053901 (2010).