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Disorder-Induced Coherence Enables Control of Wave Transport

This study demonstrates that disorder-induced coherence enables precise control of wave transport by revealing how modal alignment evolves with depth to sustain high transmission through constructive interference in specific eigenchannels, while suppressing transmission in others via destructive interference, thereby allowing the measurement of transmission levels far below conventional noise floors.

Original authors: Israel Kurtz, Yiming Huang, Zhou Shi, Azriel Z Genack

Published 2026-05-19

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Israel Kurtz, Yiming Huang, Zhou Shi, Azriel Z Genack

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

Imagine you are trying to send a message through a crowded, chaotic room full of people who keep bumping into each other. In physics, this is like sending a wave (such as light or microwaves) through a "disordered medium"—a material filled with random obstacles like glass beads or rocks. Usually, we think that because the waves bounce around so randomly, the signal gets scrambled and lost. You'd expect the message to come out as a jumbled mess or disappear entirely.

However, this paper reveals a surprising secret: Even in total chaos, the waves can secretly organize themselves.

Here is the breakdown of what the researchers found, using simple analogies:

1. The "Crowded Room" and the "Secret Paths"

Think of the disordered material as a hallway filled with random furniture. When you shout (send a wave) into one end, the sound bounces off the furniture in every direction.

The Old View: Scientists used to think that because the sound bounces randomly, you can't control where it goes. It's just noise.
The New Discovery: The researchers found that if you look closely at the "transmission matrix" (a mathematical map of how the wave enters and exits), there are special "super-highways" hidden inside the chaos.
- The VIP Lane: Some paths allow the wave to zip through almost perfectly, as if the furniture wasn't even there.
- The Dead End: Other paths are designed so the wave cancels itself out completely, resulting in zero transmission.

2. The Magic of "Modal Alignment" (The Orchestra Analogy)

How does a wave get through a crowded room without getting lost? The paper introduces a concept called Modal Alignment.

Imagine a group of 62 musicians (the different parts of the wave) trying to play a song together.

In a "Low-Transmission" path (The Dead End): The musicians are playing the same notes, but they are all out of sync. One plays loud, the next plays soft, and their timing is slightly off. When you add them all up, the sounds cancel each other out. It's like a choir where everyone is singing a different note at a different time; the result is silence. The paper shows that even though the musicians are playing loudly, the timing is so perfectly wrong that the sound vanishes.
In a "High-Transmission" path (The VIP Lane): The musicians are playing the exact same note at the exact same time. They are perfectly in sync. Even though they are bouncing off walls and furniture, their waves line up perfectly to push the signal through.

The paper proves that as the wave travels deeper into the material, these "musicians" (wave components) naturally fall into step. In the best paths, they align perfectly to build a strong signal. In the worst paths, they align perfectly to destroy the signal.

3. Seeing the Invisible

One of the most impressive findings is about measuring the "Dead End" paths.

Usually, if a signal is too weak, it gets lost in the background static (noise floor) of your equipment.
The researchers found that even when the signal is one billion times weaker than the strongest signal, they could still measure it.
Why? Because the signal isn't weak because the musicians are playing quietly. They are playing loudly, but they are canceling each other out. By measuring the individual "musicians" (the components of the wave) before they cancel out, the researchers could calculate the result even when the final result was nearly invisible.

4. The Journey Through the Material

The paper also looked at what happens inside the material, not just at the entrance and exit.

They tracked the energy and speed of the waves as they moved through the "crowded room."
They found that in the "VIP lanes," the waves speed up and organize as they go deeper.
In the "Dead End" lanes, the waves slow down and get confused, eventually canceling themselves out right before they exit.

The Bottom Line

This paper shows that disorder doesn't just mean randomness. It means there is a hidden, complex order where waves can either cooperate to pass through easily or cooperate to cancel themselves out completely.

By understanding how these waves "align" (or misalign) as they travel, scientists can now see and control waves in materials that were previously thought to be too chaotic to manage. This explains why some waves get through a messy wall while others bounce back, and it allows us to measure signals that are so faint they were previously thought to be impossible to detect.

Technical Summary: Disorder-Induced Coherence Enables Control of Wave Transport

Problem Statement
When classical waves traverse a disordered medium, multiple scattering typically transforms the wave into a stationary speckle pattern, seemingly rendering transmission uncontrollable. While the transmission matrix provides a framework for understanding wave transport—characterizing transmission eigenchannels that span from complete to vanishing transmission—the specific mechanism governing the evolution of coherence within these channels as waves propagate through the sample remains unclear. Although the transmission matrix theory is well-established for mesoscopic electronic transport, direct measurement of phase coherence is often impossible in electronic systems. In contrast, classical waves allow for direct measurement of the transmission matrix, yet the manner in which modal contributions align to produce the vast dynamic range of transmission eigenvalues (spanning orders of magnitude) has not been fully examined. Specifically, it is unknown how wave interference evolves with depth to produce constructive or destructive interference across different eigenchannels.

Methodology
The authors employ a dual approach combining experimental microwave measurements and numerical simulations to resolve the internal dynamics of wave transport:

Experimental Setup: The transmission matrix of a disordered medium was measured in a copper waveguide ( $N=62$ propagating modes) of length $L=40$ cm, filled with randomly packed alumina spheres. A vector network analyzer was used to measure field transmission coefficients between scanned wire antennas at input and output grids. Rotating the tube generated new random configurations. The study focused on the dimensionless conductance $g=3.2$ .
Transmission Matrix Analysis: The transmission matrix $t$ was decomposed via Singular Value Decomposition (SVD) to identify transmission eigenchannels and their eigenvalues ( $\tau_n$ ). The study utilized the factorized form of the determinant of the transmission matrix to analyze poles and zeros in the complex frequency plane.
Modal Decomposition: To analyze coherence, the flux amplitude within each eigenchannel was decomposed into a superposition of the $N$ propagating waveguide modes of the empty waveguide. This allowed the tracking of modal amplitudes and phases.
Numerical Simulations: Complementary simulations using recursive Green's function methods were performed on quasi-one-dimensional disordered waveguides. These simulations resolved the internal field into forward- and backward-propagating components, enabling the calculation of the flux matrix $t(z)$ at any depth $z$ . This allowed for the extraction of flux, energy density, and velocity for each eigenchannel throughout the sample.

Key Contributions and Results

Modal Alignment as the Governing Factor: The paper identifies "modal alignment" as the decisive factor determining transmission eigenvalues. The flux amplitude at any depth is determined by three factors: the incident modal profile, the coupling of incident modes to internal sample modes, and the collective alignment of complex modal contributions.
- High-Transmission Channels: In the highest-transmission eigenchannel, modal contributions align nearly perfectly (constructive interference) as sample length increases. The degree of modal alignment ( $\kappa$ ) can exceed 70% of its maximum possible value, allowing transmission to remain high despite extensive multiple scattering.
- Low-Transmission Channels: In the lowest-transmission eigenchannel, transmission is suppressed not because individual modal amplitudes are small, but because they undergo near-perfect destructive interference. The individual contributions remain appreciable (only ~3 times smaller than in high-transmission channels), but their coherent sum is vanishingly small.
Dynamic Range and Noise Floor: The study demonstrates that transmission in the lowest eigenchannel can be measured far below the conventional noise floor (more than nine orders of magnitude below the highest eigenvalue) when the frequency is tuned near a transmission zero. This is possible because the small transmission results from the destructive interference of measurable modal flux amplitudes, rather than the absence of signal.
Evolution with Depth: Simulations reveal that modal alignment evolves with depth. For eigenchannels with transmission above the ensemble average ( $g/N$ ), modal alignment increases with depth, reflecting enhanced constructive interference. Conversely, for channels below the average, alignment decreases, reflecting increasing destructive interference. This evolution is most pronounced for the fastest modes.
Phase Dynamics at Zeros and Poles: The paper analyzes the phase behavior near transmission zeros. It finds that the sharp phase jumps (approaching $180^\circ$ ) observed in the lowest-transmission eigenchannel as frequency crosses a zero arise primarily from discontinuous phase shifts in the incident eigenchannel coefficients ( $v_{sn}$ ), rather than in the transmission matrix elements ( $t_{ms}$ ) themselves. This contrasts with phase singularities in speckle patterns where the constituent phasors evolve continuously.
Flux, Energy Density, and Velocity: The simulations establish a consistent relationship between flux ( $f$ ), energy density ( $u$ ), and eigenchannel velocity ( $v$ ) via $f = uv$. The net flux in each eigenchannel remains constant throughout the sample, while the average energy density decreases linearly with depth. Notably, the eigenchannel velocity exhibits a minimum within the sample before returning to input values at the output, a behavior necessary to maintain constant flux despite the linear decrease in energy density.

Significance and Claims
The paper claims to reveal how modal alignment evolves throughout disordered media, providing a physical basis for the contrasting characteristics of transmission eigenchannels. By demonstrating that transmission is controlled by the interference of appreciable modal contributions rather than their magnitude, the work establishes a framework for understanding and potentially controlling wave transport in complex systems.

The authors state that these findings underlie applications in communications, sensing, imaging, and energy delivery. Specifically, the ability to measure transmission far below the noise floor via the analysis of modal interference could enable the detection of minute changes in a sample, such as the merging of transmission zeros. The work highlights an "unexpected internal organization of coherence" within disordered media, suggesting that coherence self-organizes over length scales much larger than the scattering mean free path.

1. The "Crowded Room" and the "Secret Paths"

2. The Magic of "Modal Alignment" (The Orchestra Analogy)

3. Seeing the Invisible

4. The Journey Through the Material

The Bottom Line

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