🔬 condensed matter

Wave Transport in Fourier Quasicrystals Revealed by Water Waves

Using surface water waves, researchers experimentally demonstrated that Fourier quasicrystals enable precise control over wave transport by creating distinct regimes of transparency and scattering dictated by their unique, sparse reciprocal-space structure.

Original authors: Angélique Campaniello, Lior Alon, Rémi Carminati, Emmanuel Fort, Marcel Filoche

Published 2026-08-17

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CC BY 4.0

Original authors: Angélique Campaniello, Lior Alon, Rémi Carminati, Emmanuel Fort, Marcel Filoche

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 the world of waves as a grand, invisible dance floor. In this ballroom, waves—whether they are ripples on a pond, beams of light, or sound traveling through air—carry energy and information. Usually, when these waves hit a wall of obstacles, they scatter chaotically, like a crowd of people bumping into each other in a dark room. But scientists have long been fascinated by "crystals," which are like perfectly organized dance floors where obstacles are arranged in repeating patterns. When waves hit these crystals, they don't just bounce randomly; they perform a precise, predictable dance called diffraction, creating sharp, bright spots of light or sound.

Then, there are "quasicrystals." Think of these as the cool, rebellious cousins of regular crystals. They have long-range order and symmetry, but they never repeat their pattern exactly. For decades, scientists thought all quasicrystals were a bit messy in their organization, with diffraction spots so dense they formed a solid, continuous cloud. This meant waves hitting them would scatter in a wide, confusing variety of directions. But recently, a new, stranger type of quasicrystal was discovered in the math books: the "Fourier quasicrystal." Unlike its cousins, this one is incredibly picky. It leaves huge empty spaces in its "dance floor" where no diffraction spots exist at all. This raises a thrilling question: if you send a wave through such a sparse, picky structure, will it pass right through like a ghost, or will it get stuck in a specific, weird pattern? Understanding this helps us design better materials for controlling light, sound, and even quantum particles, potentially leading to new ways to hide objects from detection or guide energy with perfect precision.

In a recent experiment, a team of researchers decided to test this mathematical idea using something very tangible: water waves in a tank. They built a "Fourier quasicrystal" not out of atoms or light, but out of 287 small plastic cylinders floating on the surface of water. These cylinders were arranged in a very specific, non-repeating pattern that looked random but followed strict mathematical rules. The team then sent ripples across the tank at different speeds (frequencies) to see how the water waves would behave when they hit this floating forest of cylinders.

What they found was a fascinating story of three distinct "moods" for the waves, depending on how fast they were moving. When the waves were slow (low frequency), they acted like ghosts. The water ripples flowed right through the cylinders without any trouble, barely noticing they were there. The researchers call this "transparency." It happened because the structure was so sparse that the slow waves couldn't find any "doorways" to bounce off of.

As they sped up the waves (medium frequency), the mood changed. Suddenly, the waves started to scatter, but not randomly. They were deflected in a very specific direction, like a beam of light hitting a prism. The researchers saw this as "selective scattering." The waves were finding just one or two specific "doorways" in the structure's pattern that matched their speed, causing them to bounce off at a precise angle.

Finally, when they cranked the waves up to high speeds (high frequency), the water went wild. The waves bounced off in every direction, creating a messy, speckled pattern that looked like static on an old TV screen. This was "strong scattering." At these speeds, the waves could access so many different "doorways" in the pattern that they scattered everywhere, behaving almost like they were hitting a chaotic, disordered mess.

The most exciting part of the paper is how the researchers proved why this happened. They didn't just guess; they used the water waves themselves to "map" the invisible structure of the quasicrystal. By analyzing the ripples, they reconstructed a map of the structure's "reciprocal space"—a fancy way of describing the map of all the possible directions the waves could bounce. Their map showed exactly what the math predicted: a huge empty zone in the middle (explaining the transparency) and isolated, scattered points further out (explaining the selective and strong scattering).

The team showed that even though their water tank was finite and the water absorbed some energy (making the waves fade out), the results matched the theory perfectly. They proved that Fourier quasicrystals are a unique type of material where you can control how waves travel just by arranging the "doorways" in the pattern. While this experiment was done with water, the authors suggest this discovery opens the door to using similar patterns to control sound, light, and other waves in the real world, offering a new toolkit for engineers who want to design materials that can guide waves with incredible precision.

Technical Summary: Wave Transport in Fourier Quasicrystals Revealed by Water Waves

Problem Statement
Conventional quasicrystals are characterized by long-range order without translational symmetry, typically exhibiting a dense set of Bragg peaks in their diffraction spectrum (reciprocal space). This dense structure has been associated with unconventional wave phenomena such as localization and disorder-enhanced transport. However, a distinct class of structures known as Fourier quasicrystals (FQCs) has recently been identified. Unlike conventional quasicrystals, FQCs possess a structure factor consisting of isolated Bragg peaks scattered across a discrete, nonperiodic set, leaving extended regions of reciprocal space devoid of peaks. Theoretical predictions suggest this "sparse" reciprocal-space structure should constrain momentum transfers for scattered waves, leading to specific transport regimes: transparency at low wavenumbers, selective scattering at intermediate wavenumbers, and strong scattering at high wavenumbers. Prior to this work, these predictions had not been experimentally verified, particularly in the context of finite samples and absorbing media.

Methodology
The authors experimentally tested these predictions using surface water waves scattering off a two-dimensional Fourier quasicrystal.

Sample Construction: The scattering medium consisted of $N=287$ cylindrical scatterers (diameter $d=1$ cm) arranged in a nonperiodic point pattern $(x, z)$ generated by a specific mathematical construction involving trigonometric systems and irrational vectors ( $\omega_1, \omega_2$ ). The resulting pattern is nonperiodic but possesses a diffraction spectrum of isolated Bragg peaks.
Experimental Setup: Experiments were conducted in a water tank ( $150 \times 60$ cm) with a scattering region of $45 \times 60$ cm. Monochromatic surface waves were generated by a horizontally oscillating paddle in the frequency range of $3.5 $–$ 9$ Hz.
Measurement and Analysis: The complex surface elevation (amplitude and phase) was reconstructed using Fourier checkerboard demodulation. A lock-in analysis extracted the fields at the driving frequency. To analyze transport, spatial Fourier transforms of the measured wavefields were computed (using Hann windows to reduce spectral leakage).
Structure Factor Reconstruction: The authors developed a method to reconstruct the structure factor $S(q)$ directly from the measured wavefields. By processing spatial Fourier transforms at various excitation frequencies and incident orientations, they selected annular regions corresponding to elastic scattering shells. These were recentered by subtracting the incident wavevector to express data in terms of the scattering vector $q$ . Stacking these recentered annuli allowed for the reconstruction of a reciprocal-space disk, which was then combined to form a full reciprocal-space map.

Key Results
The experiments revealed three distinct transport regimes as the incident wavevector increased, consistent with the sparse reciprocal-space organization of the FQC:

Transparency (Low Frequency): At low frequencies (e.g., 4.1 Hz), the phase pattern remained undistorted across the medium, and the Fourier spectrum was dominated solely by the incident wavevector. This indicates that the incident plane wave propagated without scattering, consistent with the depleted region of the structure factor around $q=0$ .
Selective Scattering (Intermediate Frequency): At intermediate frequencies (e.g., 5.4 Hz), the phase fronts became tilted and redirected. The Fourier spectrum showed the emergence of specific additional peaks, analogous to diffraction by a crystal. This corresponds to scattering occurring only at specific, discrete reciprocal-space vectors where Bragg peaks exist.
Strong Scattering (High Frequency): At higher frequencies (e.g., 6.2 Hz), the phase field became strongly distorted with a speckle-like structure, and the Fourier spectrum exhibited numerous peaks forming a quasi-isotropic pattern. This reflects multi-channel scattering as more Fourier components became accessible.

The reconstructed structure factor from the experimental data showed very good agreement with the theoretical prediction for an infinite FQC, accurately recovering the positions of dominant Fourier peaks up to $q \simeq 4$ cm $^{-1}$ . Minor discrepancies were attributed to the finite size of the sample, viscous attenuation at higher frequencies, and slight experimental misalignments.

Significance and Claims
The paper establishes Fourier quasicrystals as a physical platform where wave transport can be controlled through the organization of diffraction peaks in reciprocal space. The authors claim this is the first experimental test of the predicted transport regimes in FQCs, demonstrating that these regimes emerge even in finite samples and absorbing media.

The significance lies in the unique scattering behavior of FQCs, which differs qualitatively from conventional quasicrystals:

They exhibit "stealthy hyperuniformity," suppressing scattering near $q=0$ .
Unlike the dense spectrum of conventional quasicrystals, FQCs allow scattering only through a discrete set of reciprocal-space peaks.
This allows for the engineering of the radial distribution of peaks, thereby controlling the wavevector dependence of scattering.

The authors conclude that water-wave Fourier quasicrystals serve as a versatile, tabletop platform for reciprocal-space engineering of wave transport, with potential implications extending to acoustics and photonics. The work confirms that the transition from transparency to directional scattering, and finally to strong multichannel scattering, is a specific feature of FQCs resulting from their distinct reciprocal-space organization.

Technical Summary: Wave Transport in Fourier Quasicrystals Revealed by Water Waves

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