# Optimal Blind Focusing on Perturbation-Inducing Targets in Sub-Unitary Complex Media

Jérôme Sol, Luc Le Magoarou, Philipp del Hougne

► To cite this version:

Jérôme Sol, Luc Le Magoarou, Philipp del Hougne. Optimal Blind Focusing on Perturbation-Inducing Targets in Sub-Unitary Complex Media. Laser and Photonics Reviews, 2024, 2024, pp.2400619. 10.1002/lpor.202400619 . hal-04716664

HAL Id: hal-04716664

https://hal.science/hal-04716664v1

Submitted on 10 Nov 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

# Optimal Blind Focusing on Perturbation-Inducing Targets in Sub-Unitary Complex Media

Jérôme Sol, Luc Le Magoarou, and Philipp del Hougne*

The scattering of waves in a complex medium is perturbed by polarizability changes or displacements of embedded targets. These perturbations can serve as perfectly non-invasive guidestars for focusing on the targets. Here, a fundamental but to-date overlooked difference between these two perturbation types is theoretically derived: the change of the scattering matrix is of rank one [two] for a polarizability change [displacement] of a point-like target; optimal strategies to perfectly focus on the target in both cases are identified accordingly. In particular, in the latter case, optimal focusing requires at least two target displacements. Furthermore, for the case of dynamic complex media additionally featuring parasitic perturbers, a non-invasive scheme to achieve optimal time-averaged power delivery to a perturbation-inducing target is established. In all cases, no assumptions about the unitarity of the system's scattering matrix or the perturbation strength are necessary. All results are experimentally demonstrated in the microwave regime using a strongly sub-unitary lossy chaotic cavity as a complex medium. The experiments highlight that the target's "structural scattering" is irrelevant [must be negligible] in the case of target polarizability changes [displacements]. The presented results are expected to find applications in communications, cybersecurity, wireless bioelectronics, flow-cytometry, and self-propelled nano-swimmers.

## 1. Introduction

Coherently focusing waves on a target embedded inside a complex medium is a wave control primitive underpinning applications spanning from communications via sensing to wireless power transfer, across the entire spectrum of wave phenomena and scales.[1,2] In this paper, we define “optimal focusing” as delivering as much power as possible to a point-like target. Given the vector **h** containing the transmission coefficients between the

coherently controlled sources and the target, provably optimal focusing on the target by phase conjugation is possible.[3,4] The difficulty usually lies in determining **h** given an unknown complex medium. Various guidestar-based focusing techniques were explored but they involve external invasive manipulation of the system to establish the guidestar.[5–11] Virtual guidestars still rely on exposing the system to external fields (distinct from the fields associated with the wave phenomenon intended to focus on the target) and sometimes suffer from limited resolution.[12–17] The holy grail would be to use a natural perturbation of the system originating from the target itself (e.g., a change of the target’s polarizability or a target displacement) as a perfectly non-invasive guidestar. The crux lies in retrieving **h** (or a vector **h'** = **ch** that is collinear with **h**, where **c** is an arbitrary complex-valued scalar) only based on how the system scatters known wavefronts for various perturbation states of the target.

Three techniques for focusing on perturbation-inducing targets inside complex media are reported in the literature. Let **S** be the system's scattering matrix that relates any incoming wavefront **x** and outgoing wavefront **y**: **y** = **Sx**; the target-induced perturbation alters **S**: **S** → **S** + **ΔS**. First, Ref. [18] proposed to focus on a fault in a cable network using the first singular vector of **ΔS**[18–23]; the appearance of the fault can be understood as a target polarizability change. Second, Refs. [24, 25] proposed the TRACK method that considers the change of the scattered field due to the target perturbation while the probing field remains the same: **h'**<sub>TRACK</sub> = **ΔSx**. Third, Refs. [26, 27] define **h'**<sub>GWS</sub> as the conjugate of the first eigenvector of the generalized Wigner–Smith (GWS) operator **Q**<sub>q</sub> = −∫**S**<sup>−1</sup>(∂**S**/∂**q**)<sup>†</sup> that is evaluated based on the derivative of **S** with respect to the perturbed parameter **q**. TRACK and GWS are agnostic to the perturbation's nature (polarizability change vs displacement). In all cases, some experimental evidence of focusing was presented but without an experimental proof of having achieved “optimal” focusing (i.e., without benchmarking against an experimentally measured phase-conjugation result). Based on our results presented in this paper, it appears inconceivable that the focusing reported in Refs. [24–27] was close to optimal for the displacement-induced perturbations. Moreover, the GWS approach has so far not been analyzed based on a

J. Sol, L. Le Magoarou, P. del Hougne
Univ Rennes
INSA Rennes
CNRS
IETR - UMR 6164, Rennes F-35000, France
E-mail: philipp.del-hougne@univ-rennes.fr

The ORCID identification number(s) for the author(s) of this article can be found under https://doi.org/10.1002/lpor.202400619

© 2024 The Author(s). Laser & Photonics Reviews published by Wiley-VCH GmbH. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited and is not used for commercial purposes.

DOI: 10.1002/lpor.202400619
system model, its relation to the other two methods predating it has not been worked out, and its assumptions that the perturbation strength is infinitesimal and that S is unitary limit its scope of applicability; in most imaginable application scenarios, S is strongly sub-unitary due to significant absorption and/or leakage. A rigorous theoretical understanding of scattering in sub-unitary unknown complex media involving perturbation-inducing targets and existing techniques for focusing on such targets is to date missing. Moreover, as we elaborate below, no existing technique enables non-invasive optimal focusing on a target by leveraging the target's displacements, and no ideas for non-invasive optimal time-averaged power delivery to a perturbation-inducing target in a dynamic complex medium have been put forth to date.

In this paper, we fill these gaps. Based on a recent physics-compliant model of tunable complex media,[28-30] we show that there is a qualitative difference between various types of target-induced perturbations. If the target alters its polarizability, ΔS is of rank one and optimal focusing simply requires a singular value decomposition (SVD) of ΔS. In the case of a target displacement, ΔS is of rank two and we show that based on the SVDs of at least two realizations of ΔS originating from at least two target displacements, optimal focusing is possible. Moreover, we show that the SVD approach can be extended to optimal non-invasive time-averaged power delivery to targets in dynamic complex media where ΔS contains contributions from parasitic perturbations and the perturbation-inducing target. In all cases, no assumption about unitarity or the perturbation strength is required. We corroborate all findings experimentally in the microwave domain using a chaotic cavity as a complex medium. Along the way, we clarify the important role of the target's "structural scattering cross section".

In the following, the superscripts T, * and † and denote the transpose, the conjugate and the transpose conjugate, respectively; ă = a/||a||₂; A<sub>BC</sub> denotes the block of the matrix A comprising rows [columns] whose indices are in the set B [C]. I<sub>a</sub> denotes the a × a identity matrix.

To prevent possible confusion, let us note and point out the
following:

1) “Non-invasive” refers to the fact that the target is not directly accessed. In particular, no scattering channel is attached to the target such that the ground truth transmission vector **h** from the sources to the target cannot be measured directly.

2) In our proof-of-principle experiments, we can actually attach a scattering channel to the target and hence determine the ground truth transmission vector **h** from the sources to the target. However, we only make use of the possibility to attach a scattering channel to the target in order to determine the ground truth on **h** when we assess the success of our proposed methods.

3) Given the ground truth, we can rigorously assess the fidelity with which our non-invasive focusing methods are capable of retrieving a vector **h′** that is collinear with **h** (as opposed to just showing a maximum on a field map without benchmarking it against the ground truth optimum). Specifically, we quantify the efficiency of our “non-invasive” focusing methods based on **h′** (which result in an intensity |**h**†**h′**|² at the

target) compared to optimal focusing by phase conjugation using the “invasively” obtained ground truth knowledge of **h** (which result in an intensity |**h**†**h**'|² at the target[4]) by defining the metric η = |**h**†**h**'|²/|**h**†**h**'|². Our metric η cannot exceed unity, and η should not be confused with the completely differently defined “enhancement factor” in the wavefront shaping literature.[2]

4) In our experiments, the target is the port of an inaccessible antenna; this setting directly maps into many important technological applications in areas such as wireless bioelectronics and cybersecurity.

5) In our experiments, the sources and detectors are embedded inside the complex medium, whereas in many optical wavefront shaping experiments, sources and detectors are in front of and behind the complex medium, respectively. First, this fact does not compromise the non-invasiveness of our methods: implanting sources and detectors somewhere in the complex medium is not equivalent to directly accessing the target embedded somewhere else in the complex medium. Second, the formal definition of the scattering matrix does not impose any requirements on the spatial locations of the ports relative to the complex medium. The usual optical wavefront shaping setup is, in fact, a special case in which light scattering by the sources and detectors is neglected. In contrast, our setup is more general because the sources and detectors do significantly scatter waves.

6) The literature on wavefront shaping in complex media often-times does not pay attention to whether the considered complex medium involves recurrent scattering (such as our experiments in the present work) or whether the considered complex medium is limited to forward scattering (such as Refs. [31, 32]). Incidentally, recurrent scattering arises not only in chaotic cavities like that used in our experiments but it can also arise in multiply scattering optical samples, and similar coupled-dipole approaches have been used to simulate both cases.[28,33] In the context of the present work concerned with the impact of a perturbation inside the complex medium on the latter's transfer function, the distinction between recurrent scattering and forward scattering becomes significant: correctly modeling recurrent scattering is more challenging because there is an infinite number of multi-bounce paths encountering the target (see Section 2 below and Ref. [34]). Our proof-of-principle experiments based on a complex medium involving recurrent scattering hence provide a more challenging and generic test of our methods than experiments based on a complex medium limited to forward scattering could.

7) Our analysis assumes that the entire system (the complex medium and the target therein) is static during any given measurement, and this assumption is satisfied in our proof-of-principle experiments. In practical applications, if the measurement is sufficiently rapid so that the system does not notably decorrelate over the course of the measurement, this assumption will be effectively valid.

This paper is organized as follows. In Section 2, we introduce our system model based on a coupled-dipole formalism. We go on to consider theoretically and demonstrate experimentally the *optimal non-invasive* focusing on a perturbation-inducing
target embedded inside an otherwise static complex medium, first, for a target polarizability change (Section 3) and, second, for a target displacement (Section 4). Then, we consider theoretically and demonstrate experimentally the optimal non-invasive time-averaged power delivery to a target that periodically changes its polarizability and is embedded inside a dynamic complex medium featuring parasitic time-varying perturbations not originating from the target. Finally, we close with a brief discussion in Section 6 and conclusion in Section 7. Technical details about our experiments, the derivations of Equations (1) and (2), as well as the mathematical background of Section 5 are provided in the Methods [Section 8].

## 2. System Model

To start, we briefly summarize the key features of our system model for tunable complex media that was recently validated experimentally in the context of “smart” radio environments.[29] Applied to the problem of interest in this paper, the ports of both the $N_A$ antennas (acting as sources and/or detectors) and the target (itself an antenna so that we can measure the ground-truth transmission coefficients) are modeled as point-like dipoles. This approximation, as we elaborate in the following, is very accurate within the generic realm of microwave systems, and hence our results are of direct technological relevance in important applications like wireless bioelectronics and cybersecurity. In optics, significant instances of electrically small targets like gold nanobeads[23] exist, too. A brief comment about extended targets is included in the Discussion in Section 6.

In our microwave experiments, an electromagnetic antenna is a device that couples guided waves (incident via the single-mode coaxial cable that connects the antenna to a measurement device) to waves propagating in free space, and vice versa. The calibration plane is typically at the antenna port, i.e., at the connection between the coaxial cable and the antenna, such that the cable does not impact the measurement outcome. The “dipole” we are concerned with arises due to the charge separation between the two conductors of the coaxial cable in the calibration plane, i.e., at the port, implying that the dipole is indeed electrically very small (hence approximately point-like). To be clear, this definition is independent of the antenna design and we are hence not referring to the entire antenna as the “dipole.” An antenna usually scatters waves even when it is short-circuited (implying zero dipole moment). This so-called “structural scattering”[35–37] can be significant even for electrically small antennas and plays a pivotal role below.

In our system model (illustrated in Figure 1 and detailed further in Note S1, Supporting Information),[38–40] the ith dipole is characterized by its polarizability $\alpha_i$ and coupled to the jth dipole via the background Green’s function $G_{ji}$ that takes into account the overwhelmingly complicated scattering in the unknown complex medium (excluding the dipoles representing the ports of the antennas acting as sources/detectors and target but including the structural scattering of the antennas acting as sources/detectors and target).[29,30] In general, $G_{ii} \neq 0$ in a complex medium due to self-interactions. Let $\mathcal{A}$ and $\mathcal{T}$ be the sets of dipole indices representing the sources/detectors and the target, respectively. We define an interaction matrix $\mathbf{W}$ whose ith diagonal entry is $\alpha_i^{-1} - G_{ii}$ while its $(i,j)$th off-diagonal entry is $-G_{ij}$. Our system

<figure><img src="image_3.png" /><figcaption>**Figure 1.** System model and sketch. All antenna ports and the target port are modeled as dipoles characterized by their polarizabilities. In this sketch and in our experiments, there are seven antenna ports and one target port. The interaction matrix is the sum of a local term (diagonal matrix containing the dipoles' inverse polarizabilities) and a non-local term (fully populated symmetric matrix containing the background Green's functions between the dipoles' locations). Importantly, the structural scattering of the antennas is part of the background scattering. Upon inverting the interaction matrix, the top left block yields the scattering matrix $\mathbf{S}$ while the top right block $\mathbf{h}$ (a column vector) is the sought-after vector of transmission coefficients from the antenna ports to the target port. The model makes no assumption about unitarity.</figcaption></figure>

model postulates that the scattering matrix defined by the antennas is proportional to the $\mathcal{A}\mathcal{A}$ block of the inverse interaction matrix: $\mathbf{S} = [\mathbf{W}^{-1}]_{\mathcal{A}\mathcal{A}}$, where the hat indicates that the quantity has absorbed multiplicative and additive factors that do not depend on the target polarizability (see Note S1, Supporting Information for details). Note that the inversion of the interaction matrix compactly captures the infinite number of multi-bounce paths between the dipoles, each of which itself involves an infinite number of multi-bounce paths that ricochet within the complex medium without encountering any of the dipoles along the way; by casting the inversion into nested infinite series, a clear physical picture in terms of multi-bounce paths can be established.[34]

In the following, we assume reciprocity ($\mathbf{S} = \mathbf{S}^T$, $\mathbf{W} = \mathbf{W}^T$) but not unitarity ($\mathbf{SS}^\dagger \neq \mathbf{I}_{N_A}$). (A brief comment about non-reciprocity is provided in the Discussion in Section 6.) The sought-after transmission vector is $\mathbf{h} = [\mathbf{W}^{-1}]_{\mathcal{A}\mathcal{T}}$. As mentioned earlier, knowing some vector $\mathbf{h}'$ that is collinear with $\mathbf{h}$ is sufficient for our goal of blind optimal focusing on the target, where “blind” refers to the fact that the spatial coordinates of the target are unknown.

## 3. Target Polarizability Change

The first target-induced perturbation we examine is a change of the target’s inverse polarizability: $\alpha_T^{-1} \rightarrow \alpha_T^{-1} + \Delta\alpha_T^{-1}$. The inverse of the new interaction matrix $\mathbf{W}_p$ after the target’s change of polarizability can be related to that of the previous interaction matrix $\mathbf{W}$ via the Woodbury identity.[41,42] Straightforward algebraic manipulations yield

$$ \Delta \mathbf{S}_p = [\mathbf{W}_p^{-1}]_{\mathcal{A}\mathcal{A}} - [\mathbf{W}^{-1}]_{\mathcal{A}\mathcal{A}} = -khh^T, \quad (1) $$
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Blind optimal focusing on polarizability-changing target (load-impedance-modulated target antenna) inside unknown chaotic microwave cavity for five target antenna types. The top left inset shows the experimental setup involving seven antennas and a target antenna inside a chaotic cavity (see Methods [Section 8] for details). The top right inset shows the considered five distinct types of target antennas, ranging from electrically large to electrically small. The bottom row shows the frequency-averaged focusing efficiencies as a function of SNR for the three considered methods (SVD, GWS, TRACK) as well as the use of random wavefronts (RAND) for comparison.</figcaption></figure>

where *k* is a complex-valued scalar; details on the derivation of Equation (1) are provided in the Methods [Section 8]. Given Equation (1), it is clear that $\Delta\mathbf{S}_p$ is a rank-one matrix whose first (and only) left singular vector is collinear with **h**, implying the ability to achieve blind optimal focusing.[43–45] $\Delta\mathbf{S}_p = \mathbf{U}_p\Sigma_p\mathbf{V}_p^\dagger$ is the SVD, where $\Sigma_p$ is a diagonal matrix containing the singular values (in descending order); the ith column of $\mathbf{U}_p$ (resp. $\mathbf{V}_p$) is the ith left (resp. right) singular vector $\mathbf{u}_{p_i}$ (resp. $\mathbf{v}_{p_i}$) of $\Delta\mathbf{S}_p$. The probably optimal “SVD approach” yields $\mathbf{h}'_p = \mathbf{u}_{p_i}$. Our derivation did not make any assumptions regarding the unitarity of **S** or the magnitude of the polarizability change. TRACK is equivalent to right-multiplying $\Delta\mathbf{S}_p$ with some non-zero wavefront **x**; GWS left multiplies $\Delta\mathbf{S}_p$ (taken as approximation of $\partial\mathbf{S}/\partial\mathbf{\alpha}_T$) with $-j\mathbf{S}^{-1}$ and extracts the first (and only) eigenvector. Hence, TRACK and GWS in principle also identify vectors that are collinear with **h** (see Notes S2 and S3, Supporting Information), enabling optimal focusing. The assumptions in Refs. [26, 27] that $\mathbf{S}^\dagger\mathbf{S} = \mathbf{I}_{N_A}$ and $\Delta\mathbf{\alpha}_T^{-1} \to 0$ are unnecessary for optimal focusing.

In our microwave experiments, a change of polarizability takes the form of changing the termination of the target antenna's port: instead of connecting it to a coaxial cable, it can be, for instance, short-circuited or open-circuited. Antennas that self-modulate their load impedance arise in many practical contexts: “reconfigurable intelligent surfaces” in next-generation wireless networks,[30,46,47] backscatter communication schemes,[48,49] spy equipment,[50] cyber-attacks based on electromagnetic information leakage,[51] biomedical ingestible devices.[52] While Ref. [18] performed SVD-based focusing without discussing its optimality, Refs. [45, 53] recently observed the optimality of GWS-based focusing on load-impedance-modulated antennas in chaotic cavities without explaining how that was possible despite obvious violations of fundamental GWS assumptions (sub-unitary sys-

tem, large change in load impedance). This is now clear. Two additional questions remain: i) Does the target antenna design play any role? (Ref. [45] assumes the target antenna should be electrically small.) ii) Are the three approaches (SVD,[18] TRACK,[24,25] GWS[27]) fully equivalent in the presence of noise?

We conducted experiments with five distinct target antennas of drastically different sizes in the chaotic microwave cavity seen in Figure 2 (59 × 60 × 58cm³; composite quality factor: Q = 758; modal overlap: N = 4). $N_A = 7$ antennas were coupled to the system. Our results in Figure 2 shows that for high signal-to-noise ratios (SNRs), the antenna design does not play any role: with the SVD approach, $\eta_p = |\mathbf{h}^\dagger\mathbf{h}'_p|^2 / |\mathbf{h}^\dagger\mathbf{h}|^2 = 100\%$ for all considered antenna types. This observation makes sense because the target antenna's size impacts its structural scattering cross section but this is fixed and an indistinguishable part of the background scattering. In other words, the load impedance modulation is always a local perturbation because of the target antenna's electrically very small port, irrespective of the structural scattering of the target antenna itself. However, $||\Delta\mathbf{S}_p||_F$ depends on the target antenna type because the latter impacts the coupling strength (background Green's functions) between the target dipole and the $N_A$ source/detector dipoles. Hence, for lower SNRs the performance degrades more rapidly in case V than in cases I–IV. We also observe that TRACK and GWS fall slight short of the SVD performance. This also makes sense because of their avoidable vulnerabilities (see Notes S2 and S3, Supporting Information), and additionally TRACK is fundamentally disadvantaged in this comparison because it relies on $N_A$ fewer measurements. Below SNRs of roughly 0 dB, all methods fail (their performance equals the baseline of taking a normalized random vector as $\mathbf{h}'$, yielding $\langle\eta_p\rangle_f = 1/N_A = 1/7$).
To recapitulate our key contributions from this section:

1) We derived from first principles why the first left singular vector of $\Delta S_p$ is collinear with $h$ in an arbitrarily complex (potentially sub-unitary) medium for an arbitrarily large change of the target's polarizability. In particular, our derivation rigorously takes into account the infinite number of multi-bounce paths contributing to $h$. Our theoretical development also establishes the to-date not documented equivalence of all three considered methods (SVD,[18] TRACK,[24,25] GWS[27]), up to some avoidable vulnerabilities of GWS and TRACK detailed in Notes S2 and S3 (Supporting Information). Previously, GWS was suggested to differ from TRACK which precedes GWS,[26] its link to SVD had not been clearly stated in Ref. [26], it had not been derived from first principles using a concrete system model, and only unitary systems and infinitesimal target polarizability changes were considered.

2) We explained and experimentally demonstrated that the target antenna size is irrelevant for focusing on the target antenna by leveraging modulations of its port's load impedance (except for the sensitivity to noise during the measurements). The role of the target antenna's structural scattering was previously not discussed nor understood, resulting in the incorrect assumption that the target antenna should be electrically small.[45]

This section has hence not introduced a new focusing method but it has rigorously explained why previously used methods for non-invasive focusing on a polarizability-changing target embedded inside a complex medium (SVD,[18] TRACK,[24,25] GWS[27,45,53]) work, even beyond their assumed limitations in the case of GWS (unitarity, infinitesimal polarizability change, electrically small target antenna), and that they are essentially all equivalent.

## 4. Target Displacement

We now proceed to examine a second (more challenging) target-induced perturbation: target displacements. Whereas TRACK and GWS do not distinguish between different types of perturbations, our model-based analysis reveals an important difference compared to the previously examined change of target polarizability. The interaction matrix $W_m$ after the target displacement and the previous interaction matrix $W$ differ regarding an entire column and an entire row because all background Green's functions involving the target (denoted by $\hat{g}$, $\hat{g}^T$ and $\hat{g}$ in Figure 1) change. Hence, $\Delta S_m$ (change of $S$ due to target displacement) must be of rank two whereas $\Delta S_p$ (change of $S$ due to target polarizability change) was of rank one. The Woodbury identity yields an exact expression for $\Delta S_m$[42]:

$$ \Delta S_m = (K_{11} + K_{12}d_T)\mathbf{e}\mathbf{h}^T + (K_{21} + K_{22}d_T)\mathbf{h}\mathbf{h}^T + K_{12}\mathbf{e}\mathbf{e}^T + K_{22}\mathbf{h}\mathbf{e}^T, \quad (2) $$

where $\mathbf{e} = \mathbf{Sd}$, $\mathbf{d} = [\mathbf{W}_m]_{AT} - [\mathbf{W}]_{AT}$, and $d_T$, $K_{11}$, $K_{12}$, $K_{21}$, and $K_{22}$ are complex-valued scalars; details on the derivation of Equation (2) are provided in the Methods [Section 8]. Inspection of Equation (2) reveals that the first two left singular vectors $\mathbf{u}_{m1}$ and

$\mathbf{u}_{m2}$ of $\Delta S_m$ span a space containing $h$ and $e$. Hence, $\mathbf{u}_{m1}$ is a linear combination of $h$ and $e$ that is, in general, not collinear with $h$. Consequently, perfect focusing on a target subject to a single displacement is impossible based on $\Delta S_m$. As seen in Figure 3D,E, SVD, TRACK, and GWS still achieve some focusing on average but fall significantly short of the optimum.

Fortunately, blind optimal focusing on a target subject to multiple displacements is nonetheless possible. To that end, note that every target displacement yields a distinct $e$. Hence, given at least three measurements $S_A$, $S_B$ and $S_C$ corresponding to three arbitrary distinct positions A, B and C of the target, we evaluate the two rank-two matrices $\Delta S_m^{AC} = S_C - S_A$ and $\Delta S_m^{BC} = S_C - S_B$ (assuming, without loss of generality, that we wish to focus on the target while it is at position C). Then, we extract $h'_m$ as the intersection of the spaces spanned by the two first singular vectors of $\Delta S_m^{AC}$ and $\Delta S_m^{BC}$. To find $h'_m$, we define the matrix $H = [\mathbf{u}_1^{AC}, \mathbf{u}_2^{AC}, \mathbf{u}_1^{BC}, \mathbf{u}_2^{BC}]$ and define $h'_m$ as the first left singular vector of $H$. $h'_m$ is collinear with $h_C$ and hence optimal for focusing on the target while it is at position C. Our method requires $N_A \ge 3$ to ensure that the SVD of $H$ can separate its three constituent vectors (for $N_A = 2$, the efficiency is only that of a random wavefront,[54] as seen in the inset of Figure 3D).

To experimentally demonstrate blind optimal focusing on a target subject to displacements, the target's structural scattering must be minimal; otherwise, it would be as if parts of the system were displaced jointly with the target dipole such that based on realizations of $\Delta S_m$ the contribution from the target dipole could not be isolated. We heuristically found that the antenna shown in Figure 3B has minimal structural scattering. Displacing this target along the indicated trajectory involving straight, curvy and random segments, we evaluated the efficiency $\eta_m = |\mathbf{h}_C^+ h'_m|^2 / |\mathbf{h}_C^+ \tilde{h}_C|^2$ of focusing on C for all possible choices of A and B and all frequency points. This implies that we considered a wide range of step sizes of the two displacements, ranging from sub-wavelength displacements to multiple-wavelengths-large displacements. As seen in Figure 3D,E, our method achieves perfect focusing for SNRs above 40 dB at all considered frequency points, irrespective of the relative distances between A, B and C. In contrast, the benchmarks SVD, TRACK[24,25] and GWS[26] only reach around 60% efficiency on average and slightly deteriorate as the distance between A and C increases.

To recapitulate our key contributions from this section:

1) We proved from first principles that optimal focusing on a target based on a single target displacement is impossible because $\Delta S_m$ is fundamentally of rank two. Hence, since TRACK and GWS only consider a single target displacement, by construction they cannot achieve optimal focusing on the target. (It is stated in the first paragraph on page two of Ref. [26] that GWS allows one to “maximally focus” on a target subject to a displacement. Our results contradict this statement unless Ref. [26] tacitly assumes an unusual definition of “maximal focusing” that is not related to how much power is delivered to the target.)

2) We devised and experimentally validated the first method for optimal non-invasive focusing on a target subject to displacements, requiring at least two target displacements. Our experiments in a 3D chaotic cavity allowed us to directly measure
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Blind optimal focusing on target subject to displacements inside an unknown chaotic microwave cavity. A–C) The photographs show the utilized target antenna found to have minimal structural scattering and how it was inserted via small holes in the top cover into our chaotic cavity. The target is displaced along a complex trajectory. D) Focusing efficiency averaged over frequencies and choices of position triplets {A,B,C} for $N_A = 7$ as a function of SNR, as well as for SNR = 70 dB as a function of $N_A$ in the inset, for the four considered methods (SVD, GWS, TRACK, OUR method) as well as the use of random wavefronts (RAND) for comparison. E) Frequency-averaged focusing efficiencies as a function of the average distance between the three positions {A,B,C} for $N_A = 7$ and SNR = 70 dB.</figcaption></figure>

the ground truth to validate our method (field scans in 2D systems as reported in Refs. [26, 27] can reveal some focusing but do not prove its optimality without benchmarking).

3) The derivation of our provably optimal method from first principles clarifies that no assumption about the size of the target displacements nor the system's unitarity is necessary. Our experimental results confirm these insights.

4) We explained the importance of minimal structural scattering of the target antenna for optimal focusing on the target antenna based on perturbations originating from its displacements.

## 5. Dynamic Complex Media

So far, we (and prior literature) only considered non-invasive focusing in complex media that are perfectly static except for the target-induced perturbation. Practical applications often involve dynamic complex media[32,55,56] in which parasitic perturbations not originating from the target distort $\Delta S$ and thereby the discussed estimation of $\mathbf{h}'$. Specifically, we now discuss parasitic perturbations occurring on time scales faster than that at which the previously outlined techniques can be performed – otherwise the latter could simply be applied repeatedly in a loop. Hence, the goal is to find a wavefront $\hat{\mathbf{x}}$ that delivers the largest possible power to the target on average (over realizations of the parasitic perturbations). The time-averaged delivered power is $|\langle\mathbf{h}^T(t)\hat{\mathbf{x}}\rangle|^2 = |\langle\mathbf{h}^T(t)\rangle|\hat{\mathbf{x}}|^2$, revealing that the optimal choice of $\hat{\mathbf{x}}$ is collinear with $\bar{\mathbf{h}} = \langle\mathbf{h}(t)\rangle_i^*$. However, $\bar{\mathbf{h}}$ cannot be extracted directly from $\Delta S$: for a target polarizability change, any given $\Delta S_p(t) = S(t) - S(t - \Delta t)$ has $\min(1 + m, N_A)$ non-zero singular values if $m \le N_S$ of the $N_S$ parasitic perturbers (also described as dipoles in our system model) changed their polarizabilities between time instants $t - \Delta t$ and $t$. The left singular vectors of $\Delta S_p(t)$ are hence linear combinations of $\mathbf{h}(t)$ and the $m$ transmission vectors $l_v(t)$ from the sources/detectors to the $m$ parasitic perturbers that were active in the considered time interval (see Note S4, Supporting Information). $\mathbf{h}$ and $l_v$ are composed of a static and a dynamic component: $\mathbf{h}(t) = \bar{\mathbf{h}} + \hat{\mathbf{h}}(t)$ and $l_v(t) = \bar{l}_v + \hat{l}_v(t)$.

Let us assume that the target periodically changes its polarizability at a known rate $1/\Delta t$ that is not correlated with the perturbers. This assumption is certainly practical within important application contexts such as non-invasively focusing on a target antenna inside an ingestible capsule that periodically modulates its load impedance, e.g., to achieve wireless power transfer, wireless communications or sensing. Then, to extract $\bar{\mathbf{h}}$, we stack in a matrix $\mathbf{F}$ the first $w$ left singular vectors of $\gamma$ realizations of $\Delta S_p(t)$ measured at time instants $t$ separated by $\Delta t$. We define our estimate $\bar{\mathbf{h}}'$ as the first left singular vector of $\mathbf{F}$. If $N_A \ge 2 + N_S$ so that we can choose $w \ge 1 + N_S$ and $\gamma \to \infty$, we expect that $\bar{\mathbf{h}}$ is collinear with $\bar{\mathbf{h}}$. To understand why, it is sufficient to note that $\bar{\mathbf{h}}$ is the most present vector in $\mathbf{F}$: every realization of $\hat{\mathbf{h}}(t)$ and $\hat{l}_v(t)$ is uncorrelated with the others, and any given $\bar{l}_v$ appears statistically only in every second realization of $\Delta S_p$. Additional mathematical background is provided in the Methods [Section 8]. If the sources-target coupling is stronger than the sources-perturber coupling, smaller values of $w$ can be chosen since $\mathbf{h}(t)$ will be present in the first few left singular vectors of $\Delta S_p(t)$, and fewer realizations are necessary for $\bar{\mathbf{h}}'$ to converge toward being collinear with $\bar{\mathbf{h}}$. Hence the target antenna type is expected to matter.

**Remark:** Simply averaging over the columns of $\mathbf{F}$ is not a viable method to obtain an estimate of $\bar{\mathbf{h}}$ because each version of $\bar{\mathbf{h}}$ included in $\mathbf{F}$ has a random complex-valued pre-factor. Additional mathematical background is provided in the Methods [Section 8].

To systematically explore our approach, we emulate the parasitic perturbations with a 15-element programmable metasurface (PM, see **Figure 4E**). Within the PM's operating band (2.39 – 2.51 GHz), each of its meta-atoms can be modeled as a dipole whose polarizability can be individually switched between two values.[29] By sweeping $N_S \le 15$ meta-atoms through ran-
<figure><img src="image_6.png" /><figcaption>**Figure 4.** Blind non-invasive optimal time-averaged power delivery to polarizability-changing target in a dynamic complex environment. $N_S \le 15$ 1-bit programmable meta-atoms working around 2.45 GHz emulate parasitic perturbers; the setup is shown in (E). A) Time-averaged power delivery efficiency as a function of frequency and $w$ for target antenna type III and $N_S = 15$. B) Standard deviation of a transmission coefficient to the target antenna across random configurations of the programmable meta-atoms as a function of frequency. C,D) Time-averaged power delivery efficiency averaged solely over frequencies for which the meta-atoms impact the field significantly [highlighted in red in (A,B)] as a function of $N_S$ and $w$ for target antenna types III (C) and V (D). F–I) Maximum achieved time-averaged power delivery efficiency (F,G) and normalized parasitic perturbation strength (H,I) for each considered value of $N_S$ for target antenna types III (F,H) and V (G,I). All results are for $\gamma = 10^3$ and SNR = 50 dB.</figcaption></figure>

dom configurations while keeping the remaining ones static, a dynamic environment with a given number of (identical) parasitic perturbers can be emulated.[57]

Because target antenna types I–IV yielded almost identical results in Figure 2, we limit our study in this section to target antenna types III and V. For target antenna type III considered in Figure 4A,C,F,H, $||\Delta\mathbf{S}_p||_F$ only weakly depends on $N_S$ such that $\mathbf{h}(t)$ is present in the first few left singular vectors of $\Delta\mathbf{S}_p(t)$. Hence, even though $N_A = 7$ forces us to use $w < 1 + N_S$ for $N_S > 5$, we achieve $\langle\tilde{\eta}_p\rangle_f = \langle|\tilde{h}^\dagger\tilde{h}^\dagger|^2/|\tilde{h}^\dagger\tilde{h}^\dagger|^2\rangle_f = 93\%$ for $N_S = 15$ (averaging only across 2.39–2.51 GHz where the meta-atoms strongly influence the field, see Figure 4A,B). For target antenna type V considered in Figure 4D,G,I, $||\Delta\mathbf{S}_p||_F$ significantly depends on $N_S$; for larger values of $N_S$, $\mathbf{h}(t)$ is thus not guaranteed to be entirely present in the first few left singular vectors of $\Delta\mathbf{S}_p$ that are stacked in F. Hence, $\tilde{h}^\dagger$ is not perfectly collinear with $\tilde{h}$. Nonetheless, we achieve $\langle\tilde{\eta}_p\rangle_f = 79\%$ for $N_S = 15$. Overall, the proposed method is seen to be very effective at approaching optimal time-averaged power delivery to a polarization-changing target in a dynamic complex medium. Additional results from a systematic analysis of how $\gamma$, SNR and the target antenna type impact $\langle\tilde{\eta}_p\rangle_f$ are summarized in Note S6 (Supporting Information).

To recapitulate our key contributions from this section:

1) We devised and experimentally validated the first method for non-invasive optimal time-averaged power delivery to

a polarizability-changing target inside a dynamic complex medium. An important requirement are periodic target polarizability modulations not correlated with the parasitic perturbers' activity. This requirement is certainly practical in applications like wireless bioelectronics.

2) We explained and experimentally evidenced the impact of the coupling strength between sources and target compared to that between sources and parasitic perturbers, justifying why the target antenna type influences the results.

3) Our results reveal that not only the norm but also the rank of a change of the scattering matrix due to parasitic perturbers in a dynamic complex medium matters. Prior works on dynamic complex media did not pay attention to the rank.[32,55,56] Our method of emulating a complex medium with a programmable metasurface conveniently allowed us to systematically study this effect.

## 6. Discussion

In case only an off-diagonal block of $S$ can be measured, optimal focusing is still possible with the techniques outlined in this paper except that instead of the left singular vector(s) one should use the conjugate(s) of the right singular vector(s). Theoretical details and experimental validation are provided in Note S5 (Supporting Information). Such transmission-only scenarios arise in many wireless communication systems and optical wavefront shaping experiments. Similarly, if $S$ can be measured but the system is
non-reciprocal, instead of the left singular vector(s) one should use the conjugate(s) of the right singular vector(s) (in reciprocal systems, they are equivalent to the left singular vectors).

One may wonder to what extent the reported techniques for optimal non-invasive focusing on perturbation-inducing targets inside generic sub-unitary complex media could enable purposeful micromanipulations of the target (e.g., applying a force or torque to the target) or optimal sensing of information about the target's perturbation state. To the best of our knowledge, there is in general no direct relation between focusing and micromanipulation; the link to GWS-based micromanipulation made in Refs. [26, 27] appears to pivotally hinge on the unitarity of the complex medium. However, the wavefront that optimally focuses on a point-like target is also optimal to distinguish between two load impedance states of the target.[31]

Finally, regarding our assumption of a point-like target, first, we note that our experiments have revealed that this assumption is very reasonable within the generic realm of microwave systems where the coaxial connectors we used are widespread; the electrically very small cross section of the coaxial cables ultimately ensures the validity of the point approximation. Indeed, our retrieved vectors $h'$ were perfectly collinear with the ground-truth $h$, as long as the measurement noise was weak. Nonetheless, despite instances of electrically small targets like gold nanobeads[23] in optics, especially in optics one might also be confronted with electrically large targets. However, for extended targets, already the definition of focusing is non-trivial ("energy deposition" is often interpreted as maximizing the energy integrated across the target,[58] without paying attention to the spatial distribution of the energy across the target). Moreover, if a non-invasive scheme for optimal energy deposition on an extended target could be conceived, independently verifying its effectiveness in a proof-of-principle experiment by accessing the ground truth would be rather challenging. So far, optimal energy deposition on extended targets has only been achieved invasively in a 2D waveguide.[58] Even without aiming for a non-invasive method, the path toward energy deposition on extended targets in realistic 3D volumetric settings still appears long. Hence, extended targets are clearly beyond the scope of the present work.

## 7. Conclusion

To summarize, our model-based analysis revealed a fundamental difference between perturbations of complex media due to polarizability changes versus displacements of an embedded point-like target, allowing us to theoretically and experimentally achieve non-invasive blind optimal focusing on the target in both cases. In the case of a target polarizability change, we derived from first principles that three existing methods in the literature (SVD,[18] TRACK,[24,25] GWS[27]) are essentially equivalent, and that fundamental assumptions of the GWS approach (unitarity, infinitesimal change) are unnecessary for optimal focusing. We also demonstrated that the size of the target antenna whose load impedance is modulated does not matter: the resulting change of polarizability is always point-like since the target antenna port is always electrically very small. More importantly, we evidenced theoretically and experimentally the requirement for at least two target displacements for optimal non-invasive focusing on a target based on perturbations originating from its displacements;

this requirement had been overlooked in the literature such that prior to the present paper optimal non-invasive focusing on a target based on perturbations induced by its displacements was not achieved. Furthermore, whereas all previous works studying non-invasive focusing based on target-induced perturbations assumed that the complex medium is static besides the target-induced perturbation, we additionally ventured into the thus far uncharted territory of dynamic complex media that feature parasitic perturbations in addition to the target-induced perturbations. Specifically, we experimentally demonstrated non-invasive blind optimal time-averaged power delivery to a target inside a dynamic complex medium. The requirement that the target's polarizability changes are not correlated with the perturbers was easily satisfied by a target that periodically modulates its polarizability.

Looking forward, we expect our techniques for focusing on polarizability-changing targets to find practical microwave applications in communications, cybersecurity and wireless bioelectronics. Meanwhile, we expect our technique for focusing on targets subject to displacements to find applications in optics where targets of interest include gold nanobeads,[23] self-propelled nano-swimmers converting received energy into motion,[59] and cells in flow-cytometry.[25] Moreover, we ultimately envision that particles with tunable polarizability can act as "virtual ports" to the complex medium, enabling, for instance, the non-invasive retrieval of the transmission coefficient between two tunable particles,[40] promising non-invasive protocols for sensing, imaging, communications and wireless power transfer in complex media.

## 8. Methods

*Experimental setup and procedure:* As seen in Figures 2 and 4, the complex medium is an irregularly shaped metallic enclosure of dimensions 59 × 60 × 58 cm³. Note that its top wall was removed to show its interior in Figures 2 and 4. The holes seen in Figure 3 for the experiment on target displacement were drilled into the top wall.

Seven identical antennas (ANT-24G-HL90-SMA; identical to target antenna type IV) were coupled to the system in order to input and output waves. As seen in Figures 2 and 4, the seven antennas were regularly spaced, although their spatial arrangement has no influence on the presented results given the chaoticity of the system. Based on the average impulse response envelope's decay time, the composite quality factor of the system is found to be $Q = 758$, implying a modal overlap of $N' = 4$.

The target antenna in Figure 3 is a metallic screw that is electrically connected to the inner conductor of an SMA connector. No electric contact between the screw and the outer conductor exists.

All measurements were made with an 8-port vector network analyzer (VNA, Keysight M9005A) with 201 frequency points linearly spaced between 2 and 3 GHz. The intermediate-frequency bandwidth was 1 kHz and the emitted power was 13 dB. The cables connecting the VNA to the seven antennas were included in the calibration such that the calibration plane was where the antennas are connected to the cables. The cable connected to the target antenna in Figures 2 and 4 is composed of two segments, one inside the cavity and one outside the cavity. Only the outside segment is included in the calibration such that the load impedance of the antenna can be changed without opening the cavity. This
implies that the cable segment inside the cavity is effectively part of the target antenna in that case. The cable connected to the target antenna in Figure 4 is included in the calibration up to the plane where it is connected to the antenna.

$||S^†S||_2$ varies roughly between 0.3 and 0.7 in the experiments, evidencing that the system is clearly sub-unitarity. Most of the energy decay is of irreversible nature, originating from homogeneous absorption due to Ohmic losses on the cavity walls.

The 1-bit-programmable metasurface design is the same as in Ref. [57]. The operating bandwidth lies in the range 2.39 – 2.51 GHz, as seen in Figure 3. The configuration is digitally imposed with an Arduino microcontroller.

**Derivation of Equation (1):** After a target polarizability change, the updated inverse interaction matrix $\mathbf{W}_p^{-1}$ is related to the previous inverse interaction matrix $\mathbf{W}^{-1}$ according to the Woodbury identity[41,42] via the following relation:

$$ \mathbf{W}_p^{-1} = (\mathbf{W} + \text{UCV})^{-1} = \mathbf{W}^{-1} - \mathbf{W}^{-1}\mathbf{U}(\mathbf{C}^{-1} + \mathbf{V}\mathbf{W}^{-1}\mathbf{U})^{-1}\mathbf{V}\mathbf{W}^{-1}. \quad (3) $$

Herein,

$$ \mathbf{C} = \Delta\alpha_T^{-1} \quad (4) $$

$$ \mathbf{U} = [\mathbf{0}_{N_A} \ 1]^T = \mathbf{V}^T, \quad (5) $$

where $\Delta\alpha_T^{-1}$ is a complex-valued scalar and $\mathbf{0}_{N_A}$ denotes a $N_A$-element vector whose entries are zero. As pointed out in Ref. [42], U and V act as “ selectors ” here, implying that

$$ \mathbf{W}^{-1}\mathbf{U} = \begin{bmatrix} \mathbf{h} \\ a \end{bmatrix} \quad (6) $$

$$ \mathbf{V}\mathbf{W}^{-1}\mathbf{U} = [\mathbf{W}^{-1}]_{\tau\tau} = a \quad (7) $$

$$ \mathbf{V}\mathbf{W}^{-1} = [\mathbf{h}^T \ a]. \quad (8) $$

Substituting these results into Equation (3), we find

$$ \mathbf{W}_p^{-1} = \mathbf{W}^{-1} - \begin{bmatrix} \mathbf{h} \\ a \end{bmatrix} \left( \frac{1}{\Delta\alpha_T^{-1}} + a \right)^{-1} \begin{bmatrix} \mathbf{h}^T \ a \end{bmatrix}, \quad (9) $$

which, upon defining the complex-valued scalar $k = ((1/\Delta\alpha_T^{-1}) + a)^{-1}$, yields:

$$ \mathbf{W}_p^{-1} = \mathbf{W}^{-1} - \begin{bmatrix} \mathbf{h} \\ a \end{bmatrix} k \begin{bmatrix} \mathbf{h}^T \ a \end{bmatrix}, \quad (10) $$

from which Equation (1) directly follows.

**Derivation of Equation (2):** After a target displacement, the updated inverse interaction matrix $\mathbf{W}_m^{-1}$ is related to the previous inverse interaction matrix $\mathbf{W}^{-1}$ according to the Woodbury identity[41,42] via the following relation:

$$ \mathbf{W}_m^{-1} = (\mathbf{W} + \text{AI}_2\mathbf{B})^{-1} = \mathbf{W}^{-1} - \mathbf{W}^{-1}\mathbf{A}(\mathbf{I}_2 + \mathbf{B}\mathbf{W}^{-1}\mathbf{A})^{-1}\mathbf{B}\mathbf{W}^{-1}. \quad (11) $$

Herein,

$$ \mathbf{A} = \begin{bmatrix} \mathbf{d} \ \mathbf{0}_{N_A} \\ 0 \ 1 \end{bmatrix} \quad (12) $$

$$ \mathbf{B} = \begin{bmatrix} \mathbf{0}_{N_A}^T \ 1 \\ \mathbf{d}^T \ d_T \end{bmatrix}, \quad (13) $$

where $\mathbf{0}_{N_A}$ denotes a zero vector of dimensions $N_A \times 1$, $\mathbf{d} = [\mathbf{W}_m]_{A\tau} - [\mathbf{W}]_{A\tau}$ and $d_T = [\mathbf{W}_m]_{\tau\tau} - [\mathbf{W}]_{\tau\tau}$. Note that $d_T$ represents the change of the target’s self-interaction due to its displacement; meanwhile, the target’s polarizability remains unchanged. To avoid that $d_T$ is incorrectly taken into account twice, it only appears in the definition of **B**. Indeed, it can be verified that the definitions of **A** and **B** yield the correct form of $\mathbf{W}_m - \mathbf{W}$:

$$ \mathbf{W}_m - \mathbf{W} = \text{AI}_2\mathbf{B} = \begin{bmatrix} \mathbf{d} \ \mathbf{0}_{N_A} \\ 0 \ 1 \end{bmatrix} \begin{bmatrix} \mathbf{0}_{N_A}^T \ 1 \\ \mathbf{d}^T \ d_T \end{bmatrix} = \begin{bmatrix} \mathbf{0}_{N_A \times N_A} \ \mathbf{d} \\ \mathbf{d}^T \ d_T \end{bmatrix}, \quad (14) $$

where $\mathbf{0}_{N_A \times N_A}$ denotes a zero matrix of dimensions $N_A \times N_A$.

We begin by evaluating the following terms that appear in Equation (11):

$$ \mathbf{W}^{-1}\mathbf{A} = \begin{bmatrix} \mathbf{S} \ \mathbf{h} \\ \mathbf{h}^T \ a \end{bmatrix} \begin{bmatrix} \mathbf{d} \ \mathbf{0}_{N_A} \\ 0 \ 1 \end{bmatrix} = \begin{bmatrix} \mathbf{Sd} \ \mathbf{h} \\ \mathbf{h}^T \ d a \end{bmatrix} \quad (15) $$

$$ \mathbf{B}\mathbf{W}^{-1} = \begin{bmatrix} \mathbf{0}_{N_A}^T \ 1 \\ \mathbf{d}^T \ d_T \end{bmatrix} \begin{bmatrix} \mathbf{S} \ \mathbf{h} \\ \mathbf{h}^T \ a \end{bmatrix} = \begin{bmatrix} \mathbf{h}^T \ & \mathbf{a} \\ \mathbf{d}^T\mathbf{S} + d_T\mathbf{h}^T \ & \mathbf{d}^T\mathbf{h} + d_T\mathbf{a} \end{bmatrix}. \quad (16) $$

Moreover, we define

$$ \mathbf{K} = \begin{bmatrix} K_{11} & K_{12} \\ K_{21} & K_{22} \end{bmatrix} = (\mathbf{I}_2 + \mathbf{B}\mathbf{W}^{-1}\mathbf{A})^{-1}. \quad (17) $$

For the purposes of the present analysis, there is no need to work out an exact expressions for **K** in terms of **S**, **h**, **a**, **d** and $d_T$. Next, we evaluate the second term on the right-hand side of Equation (11):

$$ \begin{aligned} \mathbf{W}^{-1}\mathbf{A}\mathbf{K}\mathbf{B}\mathbf{W}^{-1} &= \begin{bmatrix} \mathbf{S}\mathbf{d}\mathbf{h} \\ \mathbf{h}^T\mathbf{d}\mathbf{a} \end{bmatrix} \begin{bmatrix} K_{11} & K_{12} \\ K_{21} & K_{22} \end{bmatrix} \begin{bmatrix} \mathbf{h}^T & \mathbf{a} \\ \mathbf{d}^T\mathbf{S} + d_T\mathbf{h}^T & \mathbf{d}^T\mathbf{h} + d_T\mathbf{a} \end{bmatrix} \\ &= \begin{bmatrix} K_{11}\mathbf{S}\mathbf{d} + K_{21}\mathbf{h} & K_{12}\mathbf{S}\mathbf{d} + K_{22}\mathbf{h} \\ K_{11}\mathbf{h}^T\mathbf{d} + K_{21}\mathbf{a} & K_{12}\mathbf{h}^T\mathbf{d} + K_{22}\mathbf{a} \end{bmatrix} \begin{bmatrix} \mathbf{h}^T & \mathbf{a} \\ \mathbf{d}^T\mathbf{S} + d_T\mathbf{h}^T & \mathbf{d}^T\mathbf{h} + d_T\mathbf{a} \end{bmatrix}. \end{aligned} \quad (18) $$

Before continuing, it should be noted that only the top left block is of interest because $\Delta\mathbf{S}_m = [\mathbf{W}_m^{-1}]_{AA} - \mathbf{S}$. We obtain

$$ \Delta\mathbf{S}_m = (K_{11}\mathbf{S}\mathbf{d} + K_{21}\mathbf{h})\mathbf{h}^T + (K_{12}\mathbf{S}\mathbf{d} + K_{22}\mathbf{h})(\mathbf{d}^T\mathbf{S} + d_T\mathbf{h}^T). \quad (19) $$

Let us now define $\mathbf{e} = \mathbf{S}\mathbf{d}$ and $\mathbf{f} = \mathbf{d}^T\mathbf{S}$. Since reciprocity imposes $\mathbf{S} = \mathbf{S}^T$, we have $\mathbf{e} = \mathbf{f}^T$ such that Equation (2) directly follows from Equation (19).

**Theoretical background for Section 5:** We model the perturbers as dipoles that change their polarizability randomly, in line with the experimental implementation based on a PM.[57] For brevity, we refer to perturbers that changed their state in a considered time interval as perturbers that were “active” in the considered
time interval. As stated in Section 5, any given realization of $\Delta S_p(t)$ has min(1 + m, $N_A$) non-zero singular values, where m is the number of active perturbers. The corresponding singular vectors of $\Delta S_p(t)$ are linear combinations of the transmission vector from the sources/detectors to the target at the considered instant in time, $\mathbf{h}(t) = \bar{\mathbf{h}} + \check{\mathbf{h}}(t)$, as well as the transmission vectors from the sources/detectors to each of the active perturbers at the considered instant in time, $\mathbf{I}_v(t) = \bar{\mathbf{I}}_v + \check{\mathbf{I}}_v(t) \forall v \in Q(t)$, where $Q(t)$ is a set containing the indices of the perturbers that were active in the considered time interval. $\check{\mathbf{h}}(t)$ depends on the configuration of all perturbers at the considered time instant, and $\bar{\mathbf{I}}_v(t)$ depends on the configuration of all perturbers except for the vth one at the considered time instant.

The first step of the proposed method consists of selecting the first w left singular vectors of every realization of $\Delta S_p(t)$. In the following, we refer to the matrix containing these first w left singular vectors of the ith realization of $\Delta S_p$ as $\mathbf{G}_i \in \mathbb{C}^{N_A \times w}$. The w orthogonal vectors contained in $\mathbf{G}_i$ constitute a basis of the span of the w most dominant left singular vectors of the ith realization of $\Delta S_p$; this span is denoted by $\mathcal{D}_i$. The orthogonal projector[60] onto $\mathcal{D}_i$ is given by

$$ \mathbf{P}_{\mathcal{D}_i} = \mathbf{G}_i (\mathbf{G}_i^\dagger \mathbf{G}_i)^{-1} \mathbf{G}_i^\dagger = \mathbf{G}_i \mathbf{G}_i^\dagger. \quad (20) $$

where $\mathbf{G}_i^\dagger \mathbf{G}_i = \mathbf{I}_w$ is used which follows from the fact that $\mathbf{G}_i$ is composed of normalized and mutually orthogonal singular vectors.

The hypotheses underlying the proposed method are that

i) each $\mathcal{D}_i$ contains a noisy version of the sought-after $\bar{\mathbf{h}}$, namely $\mathbf{h}(t) = \bar{\mathbf{h}} + \check{\mathbf{h}}(t)$.

ii) no other vector or noisy forms thereof are present in each $\mathcal{D}_i$.

The second step of the proposed method consists of stacking the first w left singular vectors from $\gamma$ realizations, yielding a matrix $\mathbf{F} \in \mathbb{C}^{N_A \times w}$:

$$ \mathbf{F} = [\mathbf{G}_1, \mathbf{G}_2, \dots, \mathbf{G}_\gamma]. \quad (21) $$

The third step of the proposed method yields the estimate $\bar{\mathbf{h}}'$ of $\bar{\mathbf{h}}$ by computing the left singular vector of $\mathbf{F}$ associated with the largest singular value. This procedure can be expressed as

$$ \begin{aligned} \bar{\mathbf{h}}' &= \underset{\|\mathbf{u}\|_2=1}{\operatorname{argmax}} (\mathbf{u}^\dagger \mathbf{F} \mathbf{F}^\dagger \mathbf{u}) \\ &= \underset{\|\mathbf{u}\|_2=1}{\operatorname{argmax}} \left( \sum_{i=1}^{\gamma} \mathbf{u}^\dagger \mathbf{G}_i \mathbf{G}_i^\dagger \mathbf{u} \right) = \underset{\|\mathbf{u}\|_2=1}{\operatorname{argmax}} \left( \sum_{i=1}^{\gamma} \| \mathbf{P}_{\mathcal{D}_i} \mathbf{u} \|_2^2 \right), \end{aligned} \quad (22) $$

where the definition of the first left singular vector is used for the first step, Equation (21) for the second step and Equation (20) for the third step.[61] Maximizing $\sum_{i=1}^{\gamma} \| \mathbf{P}_{\mathcal{D}_i} \mathbf{u} \|_2^2$ subject to the constraint $\|\mathbf{u}\|_2 = 1$ amounts to finding the unit vector $\mathbf{u}$ whose sum of the squared norms of projections onto the $\gamma$ spans $\mathcal{D}_i$ is the largest. Given the two above-listed hypotheses, the maximizing vector approaches $\bar{\mathbf{h}}$ for a sufficiently large number $\gamma$ of realizations.

As mentioned in Section 5, simply taking the sum of the columns of $\mathbf{F}$ does not yield an estimate of $\bar{\mathbf{h}}$: even though each $\mathcal{D}_i$ contains a noisy version of $\bar{\mathbf{h}}$, the vectors of a particular basis (the columns of $\mathbf{G}_i$) are not necessarily positively correlated with $\bar{\mathbf{h}}$ because there is an orientation ambiguity.

## Supporting Information

Supporting Information is available from the Wiley Online Library or from the author.

## Conflict of Interest

The authors declare no conflict of interest.

## Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## Keywords

complex medium, moving target, non-invasive focusing, physical model, wavefront shaping

Received: April 30, 2024
Revised: July 10, 2024
Published online:

[1] A. P. Mosk, A. Lagendijk, G. Lerosey, M. Fink, Nat. Photonics 2012, 6, 283.

[2] S. Rotter, S. Gigan, Rev. Mod. Phys. 2017, 89, 015005.

[3] T. Lo, IEEE Trans. Commun. 1999, 47, 1458.

[4] X. Cheng, A. Z. Genack, Opt. Lett. 2014, 39, 6324.

[5] I. M. Vellekoop, E. Van Putten, A. Lagendijk, A. Mosk, Opt. Express 2008, 16, 67.

[6] C.-L. Hsieh, Y. Pu, R. Grange, D. Psaltis, Opt. Express 2010, 18, 12283.

[7] I. M. Vellekoop, M. Cui, C. Yang, Appl. Phys. Lett. 2012, 101, 8.

[8] X. Tao, J. Crest, S. Kotadia, O. Azucena, D. C. Chen, W. Sullivan, J. Kubby, Opt. Express 2012, 20, 15969.

[9] O. Katz, E. Small, Y. Guan, Y. Silberberg, Optica 2014, 1, 170.

[10] R. Horstmeyer, H. Ruan, C. Yang, Nat. Photonics 2015, 9, 563.

[11] P. del Hougne, M. Fink, G. Lerosey, Phys. Rev. Applied 2017, 8, 061001.

[12] B. Larrat, M. Pernot, G. Montaldo, M. Fink, M. Taner, IEEE Trans. Ultrason. Ferroelectr. Freq. 2010, 57, 1734.

[13] X. Xu, H. Liu, L. V. Wang, Nat. Photonics 2011, 5, 154.

[14] B. Judkewitz, Y. M. Wang, R. Horstmeyer, A. Mathy, C. Yang, Nat. Photonics 2013, 7, 300.

[15] T. Chaigne, O. Katz, A. C. Boccara, M. Fink, E. Bossy, S. Gigan, Nat. Photonics 2014, 8, 58.

[16] D. B. Conkey, A. M. Caravaca-Aguirre, J. D. Dove, H. Ju, T. W. Murray, R. Piestun, Nat. Commun. 2015, 6, 7902.

[17] H. Ruan, T. Haber, Y. Liu, J. Brake, J. Kim, J. M. Berlin, C. Yang, Optica 2017, 4, 1337.

[18] L. Abboud, A. Cozza, L. Pichon, IEEE Trans. Veh. Technol. 2013, 62, 1010.

[19] Ref. [18] built on the DORT method for focusing on a static and strongly reflecting target in a weakly scattering medium.[20–23]
[20] C. Prada, M. Fink, *Wave Motion* **1994**, *20*, 151.

[21] C. Prada, S. Manneville, D. Spoliansky, M. Fink, *J. Acoust. Soc. Am.* **1996**, *99*, 2067.

[22] M. Davy, J.-G. Minonzio, J. de Rosny, C. Prada, M. Fink, *Prog. Electromagn. Res.* **2009**, *98*, 333.

[23] S. M. Popoff, A. Aubry, G. Lerosey, M. Fink, A.-C. Boccara, S. Gigan, *Phys. Rev. Lett.* **2011**, *107*, 263901.

[24] C. Ma, X. Xu, Y. Liu, L. V. Wang, *Nat. Photonics* **2014**, *8*, 931.

[25] E. H. Zhou, H. Ruan, C. Yang, B. Judkewitz, *Optica* **2014**, *1*, 227.

[26] P. Ambichl, A. Brandstötter, J. Böhm, M. Kühmayer, U. Kuhl, S. Rotter, *Phys. Rev. Lett.* **2017**, *119*, 033903.

[27] M. Horodynski, M. Kühmayer, A. Brandstötter, K. Pichler, Y. V. Fyodorov, U. Kuhl, S. Rotter, *Nat. Photonics* **2020**, *14*, 149.

[28] R. Faqiri, C. Saigre-Tardif, G. C. Alexandropoulos, N. Shlezinger, M. F. Imani, P. del Hougne, *IEEE Trans. Wirel. Commun.* **2022**, *22*, 580.

[29] J. Sol, H. Prod'homme, L. Le Magoarou, P. del Hougne, *Nat. Commun.* **2024**, *15*, 2841.

[30] P. del Hougne, arXiv:2311.11651 **2024**.

[31] D. Bouchet, L. M. Rachbauer, S. Rotter, A. P. Mosk, E. Bossy, *Phys. Rev. Lett.* **2021**, *127*, 253902.

[32] C. K. Mididoddi, C. Sharp, P. del Hougne, S. A. Horsley, D. B. Phillips, arXiv:2301.04461 **2023**.

[33] A. Cazé, R. Pierrat, R. Carminati, *Phys. Rev. Lett.* **2013**, *111*, 053901.

[34] A. Rabault, L. Le Magoarou, J. Sol, G. C. Alexandropoulos, N. Shlezinger, H. V. Poor, P. del Hougne, *IEEE Trans. Wirel. Commun.* **2024**, *23*, 10001.

[35] D. King, *Proc. IRE 1949*, *37*, 770.

[36] W. K. Kahn, H. Kurss, *IEEE Trans. Antennas Propag.* **1965**, *13*, 671.

[37] R. C. Hansen, *Proc. IEEE 1989*, *77*, 659.

[38] Our results on target polarizability changes could be derived equivalently (but more arduously) with a related model formulation in terms of multi-port network theory;[39,40] however, this multi-port network formulation does not lend itself to the analysis of target displacements.

[39] J. Tapie, H. Prod'homme, M. F. Imani, P. del Hougne, *IEEE J. Sel. Areas Commun.* **2024**, *42*, 2026.

[40] P. del Hougne, arXiv:2403.08074 **2024**.

[41] W. W. Hager, *SIAM Rev.* **1989**, *31*, 221.

[42] H. Prod'homme, P. del Hougne, *IEEE Commun. Lett.* **2023**, *27*, 3375.

[43] The importance of $\Delta S$ being of rank one to achieve optimal focusing with the first left singular vector of $\Delta S$ has also been recognized in Refs. [44, 45].

[44] P. del Hougne, R. Sobry, O. Legrand, F. Mortessagne, U. Kuhl, M. Davy, *Laser Photonics Rev.* **2021**, *15*, 2000335.

[45] K. B. Yeo, C. Leconte, P. del Hougne, P. Besnier, M. Davy, *IEEE Access* **2022**, *10*, 91119.

[46] L. Subrt, P. Pechac, *IET Commun.* **2012**, *6*, 1004.

[47] C. Liaskos, S. Nie, A. I. Tsioliaridou, A. Pitsillides, S. Ioannidis, I. F. Akyildiz, *IEEE Commun. Mag.* **2018**, *56*, 162.

[48] V. Liu, A. Parks, V. Talla, S. Gollakota, D. Wetherall, J. R. Smith, *ACM SIGCOMM Comput. Commun. Rev.* **2013**, *43*, 39.

[49] H. Zhao, Y. Shuang, M. Wei, T. J. Cui, P. del Hougne, L. Li, *Nat. Commun.* **2020**, *11*, 3926.

[50] G. Brooker, J. Gomez, *IEEE Aerosp. Electron. Syst. Mag.* **2013**, *28*, 4.

[51] S. Kaji, D. Fujimoto, M. Kinugawa, Y. Hayashi, *IEEE Trans. Electromagn. Compat.* **2023**, *65*, 655.

[52] S. Mandal, R. Sarpeshkar, *IEEE Trans. Biomed. Circuits Syst.* **2008**, *2*, 301.

[53] P. del Hougne, K. B. Yeo, P. Besnier, M. Davy, *Phys. Rev. Lett.* **2021**, *126*, 193903.

[54] For $N_A = 2$, both the ambient space $\mathbb{C}^2$ as well as the spans of $\Delta S_m^{AC}$ and $\Delta S_m^{BC}$ are two-dimensional and hence equal: $\mathbb{C}^2 = \text{span}(\Delta S_m^{AC}) = \text{span}(\Delta S_m^{BC})$. Finding the intersection of the spans of $\Delta S_m^{AC}$ and $\Delta S_m^{BC}$ thus amounts to choosing a random vector in $\mathbb{C}^2$.

[55] P. del Hougne, *Phys. Rev. Research* **2020**, *2*, 043224.

[56] C. Saigre-Tardif, P. del Hougne, *IEEE Wirel. Commun.* **2023**, *30*, 24.

[57] I. Ahmed, M. Davy, H. Prod'homme, P. Besnier, P. del Hougne, *IEEE Trans. Antennas Propag* **2024**.

[58] N. Bender, A. Yamilov, A. Goetschy, H. Yılmaz, C. W. Hsu, H. Cao, *Nat. Phys.* **2022**, *18*, 309.

[59] C. Bechinger, R. Di Leonardo, H. Löwen, C. Reichhardt, G. Volpe, G. Volpe, *Rev. Mod. Phys.* **2016**, *88*, 045006.

[60] The vector $\mathbf{q}$ within the span $D_i$ that is closest (in terms of the Euclidean distance) to some vector $\mathbf{b}$ is $\mathbf{q} = P_{D_i}\mathbf{b}$.

[61] The general orthogonal projector property $P_{D_i} = P_{D_i}^{\dagger} = P_{D_i}^2$ implies that $\mathbf{u}^{\dagger}P_{D_i}\mathbf{u} = \mathbf{u}^{\dagger}P_{D_i}^{\dagger}P_{D_i}\mathbf{u} = \|P_{D_i}\mathbf{u}\|_2^2$. Moreover, $\|P_{D_i}\mathbf{u}\|_2^2 \le \|\mathbf{u}\|_2^2$ with equality if and only if $\mathbf{u} \in D_i$, in which case the squared norm of the projection is maximized.