🔬 optics

Measurement and Optimal Targeting of a Hidden Scatterer in a Complex Environment Utilizing Fisher Information

This paper experimentally demonstrates that the principal eigenvector of a Fisher information operator, derived from the scattering matrix of a complex non-Hermitian system, enables the quantitative measurement of hidden parameter changes, the implementation of a discrete feedback loop for optimal targeting, and energy focusing, even in the presence of absorption and broken time-reversal symmetry.

Original authors: Nadav Shaibe, Jared Erb, Steven M. Anlage

Published 2026-08-21

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Nadav Shaibe, Jared Erb, Steven M. Anlage

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the chaotic world of waves—whether they are sound bouncing through a crowded room, light filtering through a forest canopy, or radio signals navigating a city of skyscrapers—predicting exactly where a signal will go is often impossible. When waves encounter a complex environment filled with obstacles, they scatter in countless directions, interfering with one another to create a tangled mess. For decades, scientists have treated these messy systems as "black boxes," measuring only what goes in and what comes out, without being able to see what happens inside. This limitation has made it difficult to locate hidden objects or to focus energy on a specific spot within such disorder. However, a new approach developed by researchers at the University of Maryland offers a way to peer into this chaos without needing to see inside. By treating the relationship between incoming and outgoing waves as a source of statistical clues, they have found a method to not only detect tiny, hidden changes within a system but also to steer energy precisely to where those changes are occurring, even when the system is absorbing energy or behaving in ways that defy simple reversal.

The team, led by Nadav Shaibe, Jared Erb, and Steven Anlage, set out to solve a problem that feels like trying to find a single moving part inside a sealed, vibrating machine. They worked with a specialized microwave chamber, a metal box shaped like a quarter of a bow-tie, which acts as a chaotic playground for electromagnetic waves. Inside this box, they placed a special surface made of tiny electronic components that could be tuned by changing the voltage applied to them. This surface acted as the "hidden" element, a variable that could be adjusted to slightly alter how the waves bounced around the room. The researchers wanted to know: if they changed the voltage on this surface by a tiny amount, could they figure out exactly how much it changed just by listening to the waves coming out of the box?

To do this, they used a mathematical tool called the Fisher information operator. Think of this tool as a way to sort through all the possible ways to send a signal into the box and identify which specific pattern of waves is most sensitive to the hidden change. The researchers calculated this operator by observing how the system's behavior shifted when they tweaked the voltage. They found that the operator produced a set of special wave patterns, ranked by how much information they carried about the hidden change. The most powerful pattern, the one carrying the most information, was the "principal eigenvector." When they used this specific pattern to send waves into the box, they could measure the voltage change with remarkable precision. In their tests, they could detect shifts in voltage as small as a few thousandths of a volt, a level of sensitivity that was impossible to achieve with any other wave pattern.

This ability to measure the invisible led to a second, more active application: a "discrete feedback loop." Imagine trying to keep a spinning top perfectly balanced while someone keeps nudging it. The researchers demonstrated that by repeatedly measuring the hidden voltage change and then applying an opposite adjustment to cancel it out, they could keep the system locked in a stable state. They ran this process for over a hundred cycles, successfully counteracting random jitters in the voltage. In some configurations, they were able to keep the voltage stable within a fraction of a volt, proving that this method could actively stabilize complex systems against drift and noise. This is a significant step toward controlling environments that are usually too chaotic to manage.

The researchers then asked a different question: if they knew which wave pattern was most sensitive to the hidden object, could that same pattern also be used to focus energy directly onto that object? In complex media, focusing energy is notoriously difficult because the waves scatter everywhere. However, the team discovered that the very same wave pattern that revealed the hidden voltage also acted as a perfect key to concentrate energy on the surface where the change was happening. To prove this, they used a circular microwave chamber and placed the tunable surface at different locations. When they sent in the special wave pattern, the surface absorbed so much concentrated power that it began to behave in a non-linear way, a sign that the energy was intensely focused. When they used any other wave pattern, the surface remained calm and linear. This showed that the method could target a specific spot inside a chaotic room without needing to know the exact layout of the room or having a beacon to guide the signal.

One of the most compelling aspects of this work is that it does not rely on the system being perfectly reversible or lossless. Many traditional methods for focusing waves or imaging through disorder require the system to behave symmetrically, meaning the waves can theoretically travel backward in time to retrace their path. This new approach works even when the system absorbs energy or when the waves behave differently depending on their direction of travel. The researchers tested this by introducing components that broke these symmetries, and the method still succeeded. This suggests that the technique could be applied in real-world scenarios where materials absorb energy or where the environment is not perfectly uniform, such as in biological tissue or lossy communication networks.

The study also highlighted the limits of the method. The researchers noted that to use this technique, one must be able to toggle the hidden parameter between at least two states to observe how the system changes. They also pointed out that while the method is excellent at detecting a change, it cannot always distinguish between different types of changes if multiple things are shifting at once. If the system is altered by two different factors simultaneously, the method might attribute the change to the wrong cause. The authors suggest that future work involving machine learning could help untangle these complex scenarios. Furthermore, while they could identify the best wave pattern, they did not yet have a theoretical way to predict exactly how much information a system would provide before running the experiment; they had to rely on empirical testing to see if the signal was strong enough.

Ultimately, this research provides a new way to interrogate the hidden details of complex systems. By treating the scattering of waves as a rich source of data rather than just noise, the team showed that it is possible to measure microscopic changes and focus energy with high precision. Their work bridges the gap between abstract statistical theory and practical engineering, offering a tool that works in the messy, imperfect conditions of the real world. Whether for stabilizing sensitive instruments, locating hidden objects, or transferring power wirelessly through cluttered environments, the ability to find the "right" wave pattern to ask the right question represents a quiet but powerful advance in how we interact with the complex world around us.

Technical Summary: Measurement and Optimal Targeting of a Hidden Scatterer in a Complex Environment Utilizing Fisher Information

Problem Statement
Complex, non-Hermitian scattering systems characterized by high degrees of multiple scattering and interference are often treated as "black boxes," described solely by the relationship between incoming and outgoing waves via a scattering matrix ( $S$ ). While $S$ effectively describes the macroscopic behavior of such systems (e.g., in microwave cavities, quantum billiards, and optics), it reveals little about the microscopic processes or hidden internal parameters. Existing methods to estimate internal field amplitudes or transport constraints often rely on specific assumptions or require time-reversal symmetry. The authors address the challenge of quantitatively measuring a hidden, varying parameter $x$ within such a complex system using only scattering matrix information, and further, of optimally focusing energy onto the location of that parameter without requiring the target to emit signals or the system to be lossless or reciprocal.

Methodology
The authors utilize the Fisher Information Operator (FIO), denoted as $F_x$ , defined as:
$F_x = \left(\frac{\partial S}{\partial x}\right)^\dagger \frac{\partial S}{\partial x}$
where $S$ is the scattering matrix and $x$ is the parameter of interest. $F_x$ is a Hermitian matrix whose eigenvectors $|E_m\rangle$ are ranked by their associated eigenvalues $\lambda_m$ . The eigenvalues quantify the amount of information about $x$ returned by the outgoing wave $|E_{out,m}\rangle = S|E_m\rangle$ .

The methodology involves two primary experimental approaches:

Parameter Estimation: The principal eigenvector $|E_1\rangle$ (associated with the largest eigenvalue $\lambda_1$ ) is used as the optimal input excitation $|E_{in}\rangle$ . The change in the parameter $\Delta x$ is estimated using a minimum variance unbiased estimator derived from the linear expansion of the outgoing wave.
Discrete Feedback Loop: A protocol is implemented where the system state is repeatedly measured, $\Delta x$ is estimated, and a counter-perturbation is applied to stabilize the parameter at a fixed benchmark value.
Energy Focusing: The hypothesis that $|E_1\rangle$ maximally deposits energy at the location of the parameter $x$ is tested. Since direct internal field measurement is not feasible, the authors exploit the non-linear behavior of a varactor-loaded metasurface. When energy is sufficiently concentrated on the metasurface, the system exhibits non-linearity, observable as a deviation in the output-to-input power ratio ( $P_{out}/P_{in}$ ).

Experiments were conducted using quasi-two-dimensional microwave billiards (quarter-bowtie and circular geometries) equipped with tunable metasurfaces and multi-port network analyzers. The system was tested under conditions of time-reversal symmetry breaking (due to absorption and non-reciprocity).

Key Results

Quantitative Parameter Estimation: The principal eigenvector $|E_1\rangle$ consistently provided accurate quantitative estimates of small changes in the hidden parameter (voltage applied to a metasurface), whereas other eigenvectors or random excitations failed to do so, particularly when the ratio of the first to second eigenvalue ( $\lambda_1/\lambda_2$ ) was large.
Stabilization: The "discrete feedback loop" successfully stabilized the system against random perturbations. In a 4-port system, the voltage drift was reduced to the order of $10^{-4}$ V. The authors note that stability degrades if the excitation source itself drifts (e.g., in simultaneous multi-port injection with external generators), highlighting the importance of source stability.
Energy Focusing: Injecting $|E_1\rangle$ drove the metasurface into a non-linear regime, evidenced by a change in the $P_{out}/P_{in}$ ratio, while other eigenvectors (e.g., $|E_2\rangle$ ) maintained linear behavior. This confirms that $|E_1\rangle$ optimally focuses energy on the perturbation location. This effect was observed even when the metasurface was moved to different locations within the billiard.
Robustness: The method functioned effectively in systems with broken time-reversal symmetry, including lossy and non-reciprocal environments, where traditional time-reversal or phase-conjugation methods might fail.

Significance and Claims
The paper claims to demonstrate a non-invasive, non-iterative method for both interrogating complex scattering systems and targeting specific internal locations.

General Applicability: Unlike "guidestar" techniques that require the target to emit signals, this method only requires the target to perturb the scattering matrix. It applies to any tunable or toggleable parameter (location, electromagnetic properties, etc.).
Advantage over Existing Methods: The FIO approach is distinct from the Wigner-Smith Operator (WSO) and Generalized Wigner-Smith Operator (GWSO). While GWSO is equivalent to FIO for unitary systems, the FIO remains Hermitian with real, ordered eigenvalues in lossy (sub-unitary) systems, whereas GWSO does not.
Wireless Power Transfer (WPT): The ability to focus energy on a specific target while minimizing amplitudes elsewhere via the principal eigenvector is presented as a potential approach for WPT in complex media.
Limitations: The authors modestly acknowledge limitations, including the requirement for some control over the parameter $x$ to calculate derivatives, the inability to discriminate between multiple simultaneous sources of change (leading to inaccurate estimates if multiple parameters vary), and the lack of a theoretical threshold for the eigenvalue $\lambda_1$ required to guarantee successful retrieval of information. They suggest that machine learning could be a future avenue for multi-parameter discrimination.

The work establishes the Fisher Information Operator as a practical tool for wavefront shaping and parameter estimation in complex, disordered environments without relying on time-reversal invariance.

More like this