🌀 nonlinear sciences

Multi-ring necklace vortex solitons in Kerr nonlinear media with azimuthally modulated Bessel potentials

This paper demonstrates that Kerr nonlinear media with azimuthally modulated Bessel potentials can support stable single- and multi-ring necklace vortex solitons with high topological charges, overcoming the inherent instability of conventional high-winding-number vortex beams.

Original authors: Ruolan Zhao, Jing Chen, Boris A. Malomed, Rongcao Yang

Published 2026-02-24

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CC BY 4.0

Original authors: Ruolan Zhao, Jing Chen, Boris A. Malomed, Rongcao Yang

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 balance a spinning top on a wobbly table. If the top spins too fast or the table is too uneven, it wobbles apart and falls. This is exactly what happens with optical vortex beams—special beams of light that twist like a corkscrew. In normal conditions, these "twisting" beams are notoriously unstable; they tend to break apart into fragments as they travel, making them hard to use for things like high-speed internet or quantum computing.

This paper presents a clever solution: building a custom "track" for the light to run on.

Here is the breakdown of the research using simple analogies:

1. The Problem: The Spinning Top That Falls Apart

Think of a standard optical vortex beam as a group of dancers holding hands in a circle, spinning around a center point.

The Issue: In a normal environment (like empty space or a standard glass block), the dancers get too close to each other and start pushing apart. The circle breaks, and the formation collapses.
The Consequence: Scientists have struggled to keep these "twisting" light beams stable, especially when they have a high "spin" (called topological charge).

2. The Solution: The "Beaded Necklace" on a Custom Track

The researchers created a special environment using a Kerr nonlinear medium (a special type of glass) and a Bessel lattice potential (a pattern of light that acts like a fence).

The Track (The Bessel Lattice): Imagine a series of concentric circular racetracks (like a target). Usually, these tracks are smooth rings.
The Twist (Azimuthal Modulation): The researchers chopped these smooth rings into segments, like a pizza cut into slices. Now, instead of a smooth circle, you have a ring of "islands" or "wells" separated by "hills."
The Light (The Solitons): The light doesn't flow in a continuous circle anymore. Instead, it gets trapped in these "islands," forming a necklace. Each "bead" on the necklace is a tiny packet of light.

3. The Magic: How the "Spin" Saves the Day

Here is the most surprising part of the discovery. In the real world, usually, the faster something spins, the more likely it is to fly apart. But in this specific setup, spinning faster actually makes the necklace more stable.

The Analogy: Imagine the beads on the necklace are magnets.
- If the light has a low spin, the magnets attract each other too strongly, and the necklace crumples.
- If the light has a high spin, the "magnetic" force between the beads turns into repulsion (they push each other away). This repulsion keeps the beads spaced out perfectly, preventing the necklace from collapsing.
The Result: They successfully created stable necklaces with 1, 2, 3, up to 12 "beads" (monopoles to 12-poles). Some of these even "breathe" (expand and contract rhythmically) without breaking apart.

4. The Different Scenarios

The researchers tested different configurations:

Single Ring: They put the necklace on the innermost track. They found that for certain numbers of beads (like 2, 3, or 4), the necklace is very stable.
Multi-Ring: They tried putting the necklace across two tracks at once.
- The Octupole (8 beads): These showed a fascinating "breathing" motion, where the light flows back and forth between the inner and outer tracks, like a heart pumping, yet staying stable.
- The 12-Pole: They managed to create a stable necklace with 12 beads on the outer tracks, provided the "spin" was high enough.

5. Why Does This Matter?

Think of light as a delivery truck carrying data.

Before: The truck could only carry a few packages (low data capacity) because if it tried to carry too many (high spin), it would crash.
Now: With this "necklace" track, we can build trucks that carry massive amounts of packages (high topological charge) without crashing.

In summary: This paper shows that by building a specific, segmented "racetrack" for light, we can force twisting light beams to stay together. Surprisingly, making them spin faster helps them stay stable. This opens the door to creating more complex, stable light patterns for better optical communications, faster data processing, and advanced quantum technologies.

1. Problem Statement

Optical vortex beams (OVBs) carry orbital angular momentum (OAM) and are crucial for applications in optical tweezers, communications, and quantum information. However, their propagation in nonlinear media faces significant stability challenges:

Collapse and Diffraction: In self-focusing Kerr media, high-power beams collapse, while low-power beams diffract.
Azimuthal Instability: Vortex beams with high topological charges (winding numbers) are inherently unstable against azimuthal perturbations, causing them to split into separate fragments.
Limitations of Existing Solutions: While competing nonlinearities (e.g., cubic-quintic) or external potentials (e.g., harmonic oscillators) can stabilize some vortices, the stability of complex, high-order vortex structures (necklace solitons) with large winding numbers in azimuthally modulated Bessel lattices remains unexplored.

The paper aims to investigate the existence, stability, and dynamics of single-ring and multi-ring necklace vortex solitons in Kerr nonlinear media subjected to a Bessel-lattice potential modulated in the azimuthal direction.

2. Methodology

The study employs a theoretical and numerical approach based on the following framework:

Governing Equation: The propagation is modeled using the Nonlinear Schrödinger Equation (NLSE) in cylindrical coordinates $(r, \theta, z)$ :
$i \frac{\partial \Psi}{\partial z} + \frac{1}{2} \nabla^2 \Psi + |\Psi|^2 \Psi - p V(r, \theta) \Psi = 0$
where $\Psi$ is the normalized field, $p$ is the potential depth, and $V(r, \theta)$ is the external potential.
Potential Design: The external potential is a first-order Bessel lattice modulated azimuthally:
$V(r, \theta) = V_0(r) - V_1(r) \cos(k\theta)$
where $V_0$ and $V_1$ involve Bessel functions ( $J_1$ ), and $k$ is the azimuthal modulation parameter. This creates concentric rings of fragmented potential wells (beads) separated by barriers.
Stationary Solutions: The authors seek stationary solutions of the form $\Psi(r, \theta, z) = \phi(r, \theta) \exp(i\mu z)$ , where $\mu$ is the propagation constant. These are solved numerically using the Newton-conjugate gradient method.
Stability Analysis:
- Linear Stability: Small perturbations are added to the stationary solution to derive a linear eigenvalue problem. Stability is determined if all eigenvalues ( $\lambda$ ) are purely imaginary ( $\text{Re}(\lambda) = 0$ ).
- Nonlinear Dynamics: The stability is further verified via direct numerical simulations of the perturbed NLSE using the split-step Fourier method.
Physical Realization: The model applies to spatial domain bulk waveguides, spatiotemporal optical cavities, and effectively 2D Bose-Einstein Condensates (BECs).

3. Key Contributions

Discovery of Stable High-Charge Vortices: Contrary to the conventional wisdom that high topological charges lead to instability, this work demonstrates that vortex necklace solitons with large winding numbers can be stable in azimuthally modulated Bessel lattices.
Mechanism of Stabilization: The paper identifies that increasing the topological charge ( $m$ ) increases the phase difference between adjacent "beads" (soliton fragments) in the necklace. When this phase difference exceeds a critical value (specifically $\pi/2$ ), the interaction between beads becomes repulsive, preventing the structure from collapsing or splitting.
Multi-Ring Dynamics: The study characterizes solitons trapped in single rings as well as those spanning multiple adjacent rings (e.g., octupoles in rings 1 & 2, 12-poles in rings 2 & 3), revealing complex breathing dynamics and power transfer mechanisms.

4. Key Results

A. Zero-Vorticity ( $m=0$ ) Solitons

Structure: For modulation indices $k=1$ to $5$, stable monopole, dipole, tripole, and quadrupole solitons were found in the first ring.
Instability: Pentapole ( $k=5$ ) and sextupole ( $k=6$ ) solitons in the first ring were found to be unstable due to strong attractive interactions between closely spaced beads.
Ring Dependence: Solitons in outer rings (2nd, 3rd) with $k=6$ showed instability across nearly their entire existence region due to weaker confinement and bead interactions.

B. Non-Zero Vorticity ( $m \neq 0$ ) Solitons

Stabilization by Topological Charge:
- For $k=6$ (first ring), solitons with $m=1$ were unstable, but $m=2$ solitons were stable in a specific parameter range ( $\mu < -10.34$ ).
- Reasoning: For $m=2$ , the phase difference between adjacent beads is $2\pi/3$ ( $> \pi/2$ ), inducing repulsive forces that stabilize the necklace.
Octupoles ( $k=8$ ):
- Solitons with $m=0$ and $m=2$ exhibited stable breathing dynamics (oscillations between the 1st and 2nd rings).
- However, for $k=8$ , increasing $m$ generally induced instability because the oscillatory motion between rings disrupted the intra-ring phase coherence required for stability.
12-Pole Solitons ( $k=12$ ):
- High-order solitons ( $m=3$ ) trapped in the 2nd and 3rd rings were found to be stable for specific propagation constants ( $\mu \le -13.66$ ).
- These solitons demonstrated long-distance stable propagation, whereas slightly higher power ( $\mu > -13.66$ ) led to unstable breathing and collapse.

C. General Trends

Power vs. Structure: Higher-order multipole solitons require higher power to localize.
Ring Shift: As the azimuthal modulation parameter $k$ increases, the stable location of multi-ring solitons shifts toward rings with larger radii.
Stability Criterion: Stability is a delicate balance between the topological charge (phase difference), the modulation period ( $k$ ), and the ring radius.

5. Significance

Overcoming Fundamental Limits: The findings challenge the traditional view that high-topological-charge vortices are inherently unstable in Kerr media, offering a new route to stabilize them.
Complex Light Field Control: The ability to generate stable, multi-ring necklace solitons with high winding numbers (up to 12-poles) provides a vast potential for manipulating complex spatiotemporal light fields.
Applications:
- Optical Communications: Stable high-order vortex states can increase the capacity of transmission channels via OAM multiplexing.
- Quantum Information: Provides robust carriers for quantum information processing.
- BECs: Offers a theoretical basis for creating stable vortex structures in ultracold atomic gases using optical lattices.
Experimental Feasibility: The proposed potentials can be realized using spatial light modulators (SLMs) and laser processing in bulk waveguides, making the theoretical predictions experimentally accessible.

In conclusion, this paper establishes that azimuthally modulated Bessel potentials act as a powerful tool to engineer the interaction between soliton "beads," enabling the creation of stable, high-order vortex necklace solitons that were previously thought impossible in standard Kerr media.