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Influence of the Radial Index on the Stability of Laguerre-Gaussian Vortex Beams in Turbulent Media

This paper demonstrates that Laguerre-Gaussian vortex beams with higher radial indices exhibit superior stability against atmospheric turbulence, enabling the derivation of an analytical expression to determine the necessary receiver aperture based on the beam's radial and azimuthal indices.

Original authors: Aleksandr S. Losev, Anastasia Fominova, Natalia I. Matveeva

Published 2026-07-03

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CC BY 4.0

Original authors: Aleksandr S. Losev, Anastasia Fominova, Natalia I. Matveeva

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 secret message using a flashlight beam through a stormy night. In the world of light, this "flashlight" is a special kind of beam called a Laguerre-Gaussian (LG) beam. These beams are unique because they don't just shine straight; they swirl like a tornado or a spiral staircase. Scientists use these swirls to carry information, much like how different lanes on a highway carry different cars.

However, the "storm" in this story is atmospheric turbulence—the hot air rising from the ground, wind, and weather that makes the air wobble. When these swirling beams hit the turbulence, they usually get messed up, causing the message to get garbled or lost.

This paper is like a guidebook for finding the "indestructible" versions of these light beams. Here is what the authors discovered, explained simply:

1. The Two Types of Swirls: "Twist" vs. "Rings"

The authors explain that these light beams have two main settings, like knobs on a radio:

The Twist (Azimuthal Index): This is how many times the light spirals around the center. Think of this like the number of threads on a screw.
The Rings (Radial Index): This is how many concentric circles or "rings" of light the beam has. Think of this like the ripples in a pond when you drop a stone, but the ripples are made of light.

2. The Problem with the "Twist"

The paper points out a major flaw in using the "Twist" setting to send messages. When a beam with a high twist goes through turbulence, the single, perfect spiral in the center breaks apart.

The Analogy: Imagine a single, strong tornado. When it hits a chaotic windstorm, it doesn't just wobble; it shatters into many tiny, weak tornadoes spinning in a circle.
The Result: If you were trying to read the message by looking at the center of the beam, you would see a mess of tiny swirls instead of one big one. You can no longer tell what the original message was.

3. The Solution: The "Rings" are Tougher

The authors found that if you use the Ring setting (the radial index) instead, the beam behaves much better.

The Analogy: Imagine a target with many concentric rings (like a bullseye). When the wind blows, the rings might wiggle or stretch a little, but they stay as rings. They don't shatter into tiny pieces.
The Discovery: The paper shows that if you choose a beam with many rings (a high radial index), it is incredibly stable. It resists the "wobble" of the atmosphere much better than the twisted beams.

4. The "Math Magic" Behind the Stability

To prove this, the authors used a mathematical tool called Zernike polynomials.

The Analogy: Think of a Zernike polynomial as a "distortion detector." It breaks down the shape of the light beam into a list of common errors (like a lens being slightly blurry or tilted).
The Finding: When they looked at beams with many rings, they found that the "distortion detector" barely registered any errors. The math showed that the "unstable" parts of the beam were almost zero. In simple terms, the beam is so well-structured that the atmosphere simply can't mess it up easily.

5. Catching the Beam: How Big of a Net Do You Need?

There is one catch: Beams with many rings are physically wider.

The Analogy: A beam with 1 ring is like a narrow stream of water. A beam with 15 rings is like a wide, spreading river.
The Practical Advice: If you want to catch this wide, stable river of light, you need a bigger bucket (a larger camera sensor or receiver). The authors did the math to tell you exactly how big that bucket needs to be. They found that for these super-stable beams, you need a receiver that is about 10 times wider than the tightest part of the beam.
The Good News: They calculated that modern cameras are already big enough to catch these beams easily, so this isn't a problem.

The Bottom Line

If you want to send information through the air using light beams without it getting ruined by the weather:

Don't rely on the "Twist" (it breaks apart in the wind).
Do rely on the "Rings" (specifically, beams with many rings).
Use a slightly bigger camera to catch the wider beam.

By following these rules, you can send messages that stay clear and strong, even when the air is turbulent.

Technical Summary: Influence of the Radial Index on the Stability of Laguerre-Gaussian Vortex Beams in Turbulent Media

Problem Statement
The widespread adoption of free-space optical (FSO) communication is currently hindered by the degradation of spatially structured light modes caused by atmospheric turbulence. While Laguerre-Gaussian (LG) beams are attractive for high-capacity information encoding due to their orbital angular momentum (OAM) and eigenmode properties, their stability in turbulent media remains a critical challenge. Specifically, modes encoded via the azimuthal index ( $\ell$ ) suffer from the decay of their central phase singularity into multiple unit-charge singularities when propagating through turbulence. This phenomenon complicates mode-division multiplexing (MDM) by making the unambiguous identification of the original OAM state difficult or impossible without adaptive optics. Conversely, the stability of modes encoded via the radial index ( $p$ ) and the specific criteria for selecting robust modes have not been fully characterized in this context.

Methodology
The authors investigate the stability of LG modes by decomposing their wavefronts into an orthogonal Zernike polynomial basis. This approach is chosen because Zernike polynomials naturally characterize wavefront distortions (optical aberrations) and possess circular symmetry, aligning with the geometry of LG beams.

Theoretical Framework: The study utilizes the complex amplitude distribution of LG beams, simplifying the propagation model to the waist region ( $z \approx 0$ ) under the assumption of a weakly converging beam ( $z_R \gg z$ ).
Decomposition Analysis: The LG modes are expanded into a series of Zernike polynomials. The authors derive an analytical expression for the expansion coefficients ( $S_{p,\ell}^{n,m}$ ), which quantify the contribution of specific aberration types to the beam's wavefront.
Stability Criterion: The analysis focuses on the coefficients corresponding to Zernike modes where the radial degree equals the absolute value of the azimuthal frequency ( $n = |\ell|$ ). Previous research indicates these specific modes undergo the maximum transformation (highest root-mean-square error) during propagation. Therefore, minimizing the magnitude of these coefficients is proposed as a metric for enhancing beam stability.
Aperture Calculation: To address practical implementation, the authors derive the required receiver aperture size to capture 99% of the beam's energy for high-order radial modes ( $p > 8$ ), moving beyond standard second-moment definitions which are inaccurate for multi-ring intensity profiles.

Key Contributions and Results

Radial Index as a Stability Factor: The study demonstrates that LG modes with higher radial indices ( $p$ ) exhibit significantly smaller decomposition coefficients for the unstable Zernike modes ( $n = |\ell|$ ). Specifically, for $p > 8$ , the coefficients $S_{p,\ell}^{(n=|\ell|)}$ drop to values $\leq 0.04$ .
Structural Resilience: Unlike azimuthal-indexed modes, which lose their central singularity structure in turbulence, modes with higher radial indices maintain their concentric ring structure. This persistence allows for simpler detection via ring counting in the interference pattern or intensity profile, avoiding the complexity of "fork" dislocation analysis.
Analytical Aperture Formula: The authors derive a practical formula for the minimum receiver aperture diameter ( $D_{aperture}$ ) required to capture 99% of the energy for modes with $p > 8$ :
$D_{aperture} \approx 2w_0\sqrt{2p + |\ell| + 3}$
where $w_0$ is the beam waist radius.
Feasibility Assessment: The analysis shows that for typical scientific camera active areas (15–25 mm diagonal) and standard beam waist sizes (0.5–3.0 mm), detecting modes with $p \leq 15$ is feasible without fundamental hardware limitations, provided the beam scale is matched correctly.

Significance and Claims
The paper claims to provide a clear criterion for filtering out less resilient modes in turbulent media by leveraging the radial index. The primary significance lies in the demonstration that increasing the radial index $p$ (and/or the azimuthal index $|\ell|$ ) minimizes the contribution of unstable Zernike modes, thereby enhancing the propagation stability of LG beams.

The authors conclude that utilizing radial indices for discrete mode encoding offers a distinct advantage over azimuthal encoding in turbulent channels. This approach simplifies the detection process to counting rings rather than analyzing complex fork dislocations, making it a robust strategy for information transmission in atmospheric optical channels without the immediate necessity for complex wavefront correction systems. The derived analytical expressions for receiver aperture requirements are presented as directly applicable guidelines for experimental implementations involving high-order LG modes.