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Nondiffracting Supertoroidal Pulses: Optical "Kármán vortex streets"

This paper reports the discovery of nondiffracting supertoroidal pulses (ND-STPs), which are propagation-robust, skyrmionic electromagnetic waves featuring a unique field structure analogous to von Kármán vortex streets in fluid dynamics, making them promising candidates for directed energy channels in telecommunications.

Original authors: Yijie Shen, Nikitas Papasimakis, Nikolay I. Zheludev

Published 2026-07-28

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CC BY 4.0

Original authors: Yijie Shen, Nikitas Papasimakis, Nikolay I. Zheludev

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 light not just as a beam that travels in a straight line, but as a complex, living sculpture that can twist, spin, and hold its shape. For decades, scientists have been fascinated by the "topology" of light—its shape and how it's knotted in space. Think of it like the difference between a simple rubber band and a pretzel; both are loops, but the pretzel has a more intricate structure that can do things a simple band can't. This field of study helps us understand how light interacts with matter, how we can carry information, and even how to see things smaller than ever before. Usually, when you shine a flashlight, the beam spreads out and gets blurry as it travels; this is called diffraction. But what if you could create a beam of light that refuses to spread out, a "non-diffracting" beam that stays perfectly focused forever? That's the holy grail of optical engineering, and it's the playground where this new research takes place.

Now, meet the "Supertoroidal Pulse." Imagine a pulse of light that doesn't just move forward but also spins and twists in a doughnut-like shape, carrying a unique kind of energy flow that can even loop backward for a moment. Scientists recently discovered these pulses, but they had a problem: like most things in nature, they tended to spread out and lose their cool shape over long distances. In this new study, researchers from the University of Southampton and Nanyang Technological University have figured out how to tune these pulses so they stop spreading out entirely. They call these new creations "Nondiffracting Supertoroidal Pulses" (or ND-STPs for short).

Here is the magic trick: by adjusting a specific mathematical "knob" in the way the light is generated, the team simulated pulses that travel through empty space without ever getting wider. It's as if you threw a ball that, instead of falling to the ground or slowing down, kept rolling in a perfect, unchanging circle forever. But the most surprising part is what these pulses look like inside. The researchers found that the swirling patterns of energy within these pulses look exactly like a "Kármán vortex street." You might have seen this in nature if you've ever watched smoke curling off a chimney or water swirling behind a rock in a river. It's a pattern of alternating, spinning vortices that creates a "street" of turbulence. In the world of fluids, this is what makes telephone lines "sing" in the wind. The team's simulations show that these light pulses create a 3D version of that same street, but instead of air or water, it's made of swirling magnetic and electric fields.

The paper doesn't just show pretty pictures; it proves through detailed computer simulations that these structures are incredibly robust. While ordinary light pulses might have these cool swirling patterns for just a split second before they blur away, the ND-STPs keep these "vortex streets" and other complex shapes—like fractal patterns and tiny magnetic knots called skyrmions—intact over arbitrarily long distances. The researchers argue that because these pulses are so stable and don't spread out, they could be perfect for sending information over long distances without losing data, potentially revolutionizing telecommunications. They also suggest these pulses could be used for super-precise imaging or even in remote sensing technologies like lidar. While the paper relies on simulations to prove these pulses can exist, the math is solid, and the team believes that with the right tools, like special "metasurfaces" (ultra-thin, patterned materials), we could actually build these light beams in a real lab. It's a step toward a future where we can ride the "vortex streets" of light to send messages and see the world in ways we never thought possible.

Technical Summary: Nondiﬀracting Supertoroidal Pulses: Optical "K´arm´an Vortex Streets"

Problem Statement
Recent research has established supertoroidal pulses (STPs) as a family of space-time nonseparable free-space electromagnetic excitations characterized by unique topological features, including skyrmionic field configurations, fractal-like singularity patterns, and extended areas of energy backflow. However, a significant limitation of previously studied STPs is their transverse divergence; the complex topological structures they exhibit are typically transient, observable only instantaneously at the focal point before the pulse diffracts and the topology collapses. The challenge addressed in this work is to generate a propagation-robust version of these pulses that can maintain their intricate singular configurations and topological properties over arbitrary propagation distances.

Methodology
The authors derive solutions based on the "electromagnetic directed-energy pulse trains" (EDEPT) theory introduced by Ziolkowski. The electromagnetic fields are constructed from a scalar generating function, $f(r, t)$ , which satisfies the scalar wave equation. The function is defined as:
$f(r, t) = f_0 e^{-s/q_3} (q_1 + i\tau)^{-1} (s + q_2)^{-\alpha}$
where $r = (r, \theta, z)$ represents spatial cylindrical coordinates, $t$ is time, and $c$ is the speed of light. The parameters $q_1, q_2,$ and $q_3$ are real positive constants with units of length, while $\alpha$ is a real dimensionless parameter ( $\alpha \ge 1$ ) ensuring finite pulse energy.

To retrieve the electromagnetic fields, the authors construct Hertz potentials $\Pi(r, t) = \hat{n}f(r, t)$ . By setting the Hertz potential as a transverse vortex vector field ( $\hat{n} = \nabla \times \hat{z}$ ), they generate an azimuthally polarized toroidal pulse. The study focuses on varying the parameter $q_3$ , which defines the degree of transverse divergence. While previous works assumed $q_3 \to \infty$ to generate fundamental toroidal pulses or focused structured pulses, this work explores finite values of $q_3$ . Specifically, the authors investigate the condition where $q_3 = q_1$ and $\alpha = 1$ to achieve a nondiffracting state.

Key Contributions and Results

Generation of Nondiﬀracting Supertoroidal Pulses (ND-STPs): The authors demonstrate that by tuning the transverse divergence parameter to $q_3 = q_1$ , the pulse envelope transforms from a standard diffracting profile into a nondiffracting "X-shaped" structure. These pulses satisfy the definition of nondiffraction, $|E(r, \theta, z, t)| = |E(r, \theta, z + \Delta z, t + \Delta z/v)|$ , allowing them to propagate without spreading.
Propagation-Robust Topology: Unlike standard STPs where complex topological structures (such as self-similar fractal-like patterns and matryoshka-like singularity arrays) exist only momentarily at the focus, ND-STPs maintain these structures over arbitrarily long distances. The electric field singularities evolve in a stable, propagation-robust manner.
Optical "K´arm´an Vortex Streets" (KVS): A central finding is the structural similarity between the magnetic field of ND-STPs and the von K´arm´an vortex street observed in fluid dynamics. The magnetic field distribution exhibits vector singularities of vortex and saddle types. The vortices trace off-axis trajectories, forming staggered arrays of vortex dipoles (clusters of two vortices with opposite circulations).
- This pattern is analogous to the KVS formed by vortex shedding in fluid flow around blunt bodies, but here it is realized in a linear electromagnetic system.
- Unlike the 2D KVS in fluids, the optical KVS in ND-STPs corresponds to a 3D set of vortex rings due to cylindrical symmetry.
Skyrmionic Persistence: The study reveals that electromagnetic skyrmions, which typically collapse rapidly upon propagation in prior works, persist in ND-STPs. These skyrmions exhibit a periodic behavior, alternating between four different types (combinations of opposite polarities and helical angles) as the pulse propagates.
Energy Backflow and Transport: The Poynting vector field of ND-STPs displays layered structures of forward and backward energy flow. Energy transport is mediated by the magnetic vortex arrays, where vortices in the front half of the pulse act as energy sources and those in the rear half act as sinks, creating extended areas of energy backflow between them.

Significance and Claims
The paper claims that ND-STPs represent a significant advancement in structured light, offering a "propagation-robust" medium for complex topological excitations. The authors argue that the strong similarity to the von K´arm´an vortex street provides a novel optical analogy to fluid dynamics, demonstrating that such vortex-street patterns can be constructed from electric or magnetic fields in a linear regime, whereas they are typically nonlinear phenomena in fluids.

The authors suggest that the robust topological structure of ND-STPs, which remains invariant upon propagation, makes them suitable candidates for:

Directed energy channels for telecommunications.
Long-distance information transfer encoded in the topological features of the pulses.
Potential applications in light-matter interactions, superresolution microscopy, remote sensing, and lidar.

Furthermore, the authors note that the field of ND-STPs is exponentially confined, making them potentially less sensitive to the size of the generating metasurface aperture compared to Bessel beams, which could facilitate practical implementation. The work concludes that these pulses unveil intriguing analogies between fluid transport and the flow of energy in structured light.

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