1. Bibliographic Information

1.1. Title

Robust transmission of pin-like vortex beams in plasma sheath turbulence

1.2. Authors

Chengzhao Liu, Xu Zhou, Wenhai Wang, Wentao Hu, Zhengda Hu, JiCheng Wang, and Yun Zhu

1.3. Journal/Conference

Published in a journal associated with Optica Publishing Group. Optica (formerly OSA - The Optical Society) is a prominent professional society for optics and photonics scientists, engineers, educators, and technicians. Its journals are highly regarded in the field, indicating a peer-reviewed publication of good standing.

1.4. Publication Year

Published on August 1, 2025.

1.5. Abstract

This study investigates the propagation characteristics of pin-like vortex beams (PLVBs) when they travel through plasma sheath turbulence. The research employs the random phase-screen method for this analysis. The performance of PLVBs is compared against conventional Laguerre-Gaussian beams (LGBs) across several metrics: intensity dispersion, detection probability of orbital angular momentum (OAM), bit error rate (BER), and channel capacity. The findings indicate that PLVBs exhibit superior performance in plasma sheath turbulence. Specifically, they show detection probabilities 9%–12.5% higher and BER 0.03–0.067 lower than LGBs over propagation distances ranging from 0.1 to 0.4 m. Furthermore, PLVBs demonstrate enhanced channel capacity compared to LGBs, underscoring their greater robustness against plasma sheath turbulence. The study also explores how the beam modulation parameter and wavelengths influence PLVB performance, concluding that higher beam modulation parameters and longer wavelengths lead to reduced BER and increased channel capacity. These results suggest that PLVBs are robust candidates for optical communication systems operating in turbulent plasma environments.

1.6. Original Source Link

/files/papers/694216bbce7364c2304cbca2/paper.pdf (publication status: officially published on 2025-08-01)

2. Executive Summary

2.1. Background & Motivation

The core problem the paper aims to solve is the degradation of communication signals due to plasma sheath turbulence around hypersonic vehicles. When high-speed aircraft enter the atmosphere, they experience severe aerodynamic heating, leading to gas ionization and the formation of a plasma sheath—a thin ionized layer enveloping the vehicle. The turbulence within this plasma sheath severely interferes with communication signals between the aircraft and ground stations, degrading antenna performance and even causing communication interruptions, commonly known as a "blackout." This phenomenon poses a significant challenge for stable communication in such environments.

This problem is crucial in the current field due to the continuous expansion of human activities involving high-speed aircraft with military and economic value. Realizing stable communication within plasma sheath turbulence is therefore a significant research objective.

Prior research has investigated various methods to mitigate the impact of turbulence on light beams, often focusing on decreasing the effective interaction area between the beams and the turbulent medium or developing new vortex beams with intrinsic non-diffracting or self-focusing properties. However, previous studies on pin-like vortex beams (PLVBs) have primarily focused on their transmission in conventional atmospheric or oceanic turbulence. The specific impact of plasma sheath turbulence on the propagation of PLVBs in orbital angular momentum (OAM)-based wireless optical communication (WOC) systems remained unexplored.

The paper's entry point or innovative idea is to investigate the robustness of PLVBs in the previously unaddressed plasma sheath turbulence environment, comparing them against conventional Laguerre-Gaussian beams (LGBs). This fills a critical gap in understanding how advanced beam types perform under the challenging conditions of plasma sheath turbulence, specifically for OAM-based WOC systems.

2.2. Main Contributions / Findings

The paper makes several primary contributions and reaches key conclusions:

• Superior Robustness of PLVBs: The study demonstrates that pin-like vortex beams (PLVBs) significantly outperform conventional Laguerre-Gaussian beams (LGBs) in plasma sheath turbulence. This superiority is observed across multiple key metrics including intensity dispersion, detection probability of orbital angular momentum (OAM), bit error rate (BER), and channel capacity.

• Quantitative Performance Improvement:

  • Detection probabilities for PLVBs are shown to be 9%–12.5% higher than LGBs over propagation distances from 0.1 to 0.4 m.
  • The BER of PLVBs is 0.03–0.067 lower than that of LGBs in the same distance range.
  • PLVBs exhibit enhanced channel capacity compared to LGBs.

• Impact of Beam Modulation Parameter ($\gamma$): The research reveals that a higher beam modulation parameter ($\gamma$) in PLVBs leads to reduced BER and increased channel capacity. This is attributed to the convergent effect induced by larger $\gamma$ values, which results in a narrower beam width and a smaller effective interaction area with the turbulent medium, thereby mitigating mode distortion.

• Impact of Wavelengths: Longer wavelengths are found to reduce BER and increase channel capacity, primarily because larger wavelengths lead to a reduction in the scattering effect, making the beam less prone to severe intensity fluctuations from local turbulence.

• Influence of Turbulence Parameters: The study also details how various turbulence parameters (refractive index fluctuation, outer scale, anisotropy factor) affect PLVB performance, showing that stronger turbulence (larger refractive index fluctuations, lower outer scale, lower anisotropic factor) results in higher BER and lower channel capacity.

• Potential for Optical Communication: These findings strongly suggest the potential of PLVBs as robust candidates for optical communication in challenging turbulent plasma environments, offering a promising solution to the "blackout" problem experienced by hypersonic vehicles.

  These findings solve the specific problem of identifying and characterizing robust optical communication links within plasma sheath turbulence, providing crucial insights for designing and optimizing WOC systems in such extreme environments.

3. Prerequisite Knowledge & Related Work

3.1. Foundational Concepts

To understand this paper, a foundational understanding of several key optical and communication concepts is necessary.

• Vortex Beams: These are special types of light beams that carry orbital angular momentum (OAM). Unlike conventional light beams, their wavefronts twist around their propagation axis, forming a helical phase front. This twist is characterized by a topological charge, $m_0$, which is an integer. Different topological charges correspond to different OAM states, enabling the transmission of multiple data channels on a single beam, thereby increasing communication capacity. The intensity profile of a vortex beam typically has a dark core (a doughnut shape) because the intensity at the center is zero.

• Laguerre-Gaussian Beams (LGBs): These are a common type of vortex beam that serve as an orthogonal basis for light beams carrying OAM. They are characterized by two indices: the azimuthal index ($l$, which corresponds to the topological charge or OAM state) and the radial index ($p$). LGBs have a well-defined dark core and a helical phase front. They are often used as a benchmark for comparison with new types of vortex beams due to their well-understood properties.

• Pin-Like Vortex Beams (PLVBs): These are a specific class of vortex beams engineered to exhibit superior stability and non-diffracting or self-focusing properties compared to conventional Gaussian beams or even LGBs. The "pin-like" characteristic refers to their ability to maintain a narrow main lobe over extended propagation distances, effectively reducing the interaction area with turbulent media. This makes them more robust against turbulence-induced distortions. The paper introduces a beam modulation parameter $\gamma$ that governs these propagation characteristics.

• Orbital Angular Momentum (OAM): Light can carry two forms of angular momentum: spin angular momentum (related to polarization) and orbital angular momentum. OAM is associated with the helical phase front of a light beam. Each OAM state (defined by its topological charge) is orthogonal to others, meaning they can be transmitted independently without interference. This property allows OAM to be used for multiplexing in optical communication, where different data streams are encoded onto different OAM states, significantly boosting channel capacity.

• Plasma Sheath Turbulence: This is a turbulent ionized layer of gas that forms around hypersonic vehicles as they re-enter the atmosphere. The intense aerodynamic heating causes atmospheric gases to ionize, creating a plasma. The interaction of the vehicle with the surrounding air generates turbulence within this plasma sheath. This turbulence causes rapid, random fluctuations in the refractive index of the medium, which severely distorts propagating electromagnetic waves (including light beams), leading to signal degradation and communication blackouts.

• Random Phase-Screen Method: This is a widely used numerical simulation technique for modeling wave propagation through turbulent media. Instead of continuously modeling the turbulence along the propagation path, the method approximates the turbulent medium as a series of thin, discrete phase screens placed at regular intervals. Each phase screen imparts a random phase distortion to the propagating wave, simulating the cumulative effect of refractive index fluctuations. The phase distortion on each screen is generated based on the statistical properties (e.g., power spectrum) of the turbulence. This method is computationally efficient for studying long-distance propagation.

• Bit Error Rate (BER): In digital communication, BER is the number of bit errors divided by the total number of bits transmitted over a studied time interval. It is a key metric for evaluating the reliability and quality of a communication link. A lower BER indicates better communication performance. In OAM-based WOC systems, turbulence can cause crosstalk between OAM modes, leading to detection errors and thus a higher BER.

• Channel Capacity: This refers to the maximum rate at which information can be reliably transmitted over a communication channel. In the context of OAM-based WOC, it quantifies how much data can be sent per unit time through the turbulent plasma sheath without excessive errors. Higher channel capacity implies a more efficient and robust communication system.

• Refractive Index Fluctuations ($\langle \Delta n^2 \rangle$): The refractive index of a medium determines how light propagates through it. In a turbulent medium, the refractive index is not constant but fluctuates randomly. The variance of the refractive index fluctuation ($\langle \Delta n^2 \rangle$) is a measure of the strength of these fluctuations. Larger $\langle \Delta n^2 \rangle$ values indicate stronger turbulence and thus more significant distortion of light beams.

• Outer Scale ($L_0$) and Inner Scale ($l_0$): These are characteristic length scales in turbulence. The outer scale ($L_0$) represents the largest eddies (swirls) in the turbulent flow, where energy is injected into the turbulence. The inner scale ($l_0$) represents the smallest eddies, where the kinetic energy of the turbulence is dissipated into heat due to viscosity. These scales define the range over which turbulence affects wave propagation.

• Anisotropy Factor ($\mu_x, \mu_y$): In many real-world turbulent environments, turbulence is not uniform in all directions; it can be anisotropic. This means that the turbulent eddies are stretched or compressed along certain axes. Anisotropy factors ($\mu_x, \mu_y$) quantify the degree of this directional stretching in the x and y directions, respectively. When $\mu_x = \mu_y = 1$, the turbulence is isotropic (uniform in all directions). Anisotropy influences how a beam's shape and OAM modes are affected by turbulence.

3.2. Previous Works

The paper builds upon a body of research related to hypersonic vehicles, plasma sheaths, turbulence modeling, and vortex beam propagation.

• Plasma Sheath Formation and Communication Interference:

  • Research by Munk et al. [1] highlights the importance of high-speed aircraft.
  • Yuan et al. [2], Guo et al. [3,4], and Gong et al. [5] discuss the formation of plasma sheaths due to aerodynamic heating and gas ionization, and how this plasma sheath turbulence interferes with communication signals, leading to "blackouts." This establishes the critical problem addressed by the paper.

• Modeling Plasma Sheath Turbulence:

  • To quantitatively assess refractive index fluctuation in plasma sheath turbulence, several models have been developed. Zhao et al. [6] used nanometer plane laser scattering to visualize hypersonic turbulent mixing layers, providing insights into flow field structures.
  • Li et al. [7] hypothesized plasma sheath turbulence as locally homogeneous and isotropic at certain scales and developed a fractal model and a three-dimensional (3D) non-Obukhov—Kolmogorov power spectrum based on fractal dimensions from hypersonic turbulence experiments.
  • They further developed a two-dimensional (2D) power spectrum and constructed a phase screen to model refractive index fluctuations using a band-limited Weierstrass fractal function [8].
  • Later work processed experimental images of hypersonic plasma sheath to derive plasma sheath turbulence power spectra based on the von Karman spectrum [9] and a modified von Karman spectrum incorporating an orientation factor, which better matched observed turbulence behaviors. The paper specifically uses the power spectrum described by Deng et al. [28] for its plasma sheath turbulence model.
  • Lin [29] provided theoretical and experimental evidence suggesting that plasma sheath turbulence should be anisotropic, influencing the power spectrum calculation. This led to the introduction of anisotropic factors ($\mu_x, \mu_y$) in the spatial wavenumber [30-32].

• Vortex Beams and Turbulence Mitigation:

  • Vortex beams are known to increase channel capacity in OAM-based wireless optical communication (WOC) systems, as discussed by Paterson [13].
  • However, turbulence severely distorts their phase front, causing crosstalk between OAM modes and signal degradation.
  • Solutions proposed include adaptive optics [14] for phase wavefront correction and spherical concave mirrors or focusing mirrors [15] to reduce OAM crosstalk.
  • A key method to mitigate turbulence impact is reducing the effective interaction area between beams and turbulence [16]. This led to interest in non-diffracting or self-focusing vortex beams.
  • Examples of such beams include Bessel-Gaussian beams [17], Whittaker-Gaussian beams [18], autofocusing Airy beams [19], autofocusing hypergeometric Gaussian beams [20], and twisted Hermite-Gaussian Schell-model beams [21].

• Development of Pin-Like Optical Beams:

  • Zhang et al. [22] designed "optical pin beams" with stable wavefronts to mitigate diffraction and turbulence effects, showing superior performance over conventional Gaussian beams in maintaining peak intensity.
  • Li et al. [23] proposed anti-diffracting optical pin-like beams with adjustable main lobe size via an exponential parameter.
  • The same team later provided theoretical and experimental demonstrations of pin-like vortex beams (PLVBs) in free space [24]. These PLVBs showed superior transmission performance in atmospheric turbulence, including kilometer-scale free-space optical communication [16].
  • Partially coherent PLVBs also demonstrated enhanced stability in oceanic turbulence over distances up to 200 m compared to Gaussian vortex beams [25].

• Propagation in Plasma Sheath Turbulence:

  • Recent investigations have explored propagation characteristics in plasma sheath turbulence for various beam types, such as those by Nobahar et al. [10], Chen et al. [11], and Deng et al. [12], including vortex beams and partially coherent beams.
  • The paper specifically references Lin [29] and Andrews et al. [30] for aspects of anisotropic turbulence and Yu et al. [33] for Markov approximation.
  • Studies by Liu et al. [31] and Hassan et al. [32] systematically explored the impact of anisotropic turbulence on optical beam propagation, providing guidance for parameter selection.

3.3. Technological Evolution

The field has evolved from understanding the detrimental effects of plasma sheaths on communication to developing sophisticated models for plasma sheath turbulence and exploring advanced beam shaping techniques to counteract these effects.

1. Early Recognition of the Problem (1960s-1970s): Initial research focused on identifying the "blackout" phenomenon and characterizing plasma sheaths around re-entry vehicles (e.g., Lin [29]).
2. Modeling Turbulence (1990s-2000s): Development of power spectrum models for various turbulent media (e.g., von Karman spectrum for atmospheric turbulence) and early attempts to apply these to plasma sheath turbulence [6-9]. The introduction of fractal models and anisotropic considerations marked progress towards more accurate representations.
3. Emergence of Vortex Beams (2000s): The realization that light can carry OAM and the potential of vortex beams to increase channel capacity in free-space optical communication systems (e.g., Paterson [13]).
4. Turbulence Mitigation Techniques (2000s-Present): Research into adaptive optics [14] and specialized mirrors [15] to correct turbulence-induced phase distortions. The concept of reducing the effective interaction area between the beam and turbulence gained prominence [16].
5. Advanced Beam Engineering (2010s-Present): Development of non-diffracting and self-focusing beams (e.g., Bessel-Gaussian, Airy, Whittaker-Gaussian [17-21]) to intrinsically resist turbulence. This led to the creation of pin-like optical beams [22,23] and subsequently pin-like vortex beams (PLVBs) [24,25], specifically designed for enhanced robustness.
6. Application to Plasma Sheath Environments (Recent): The current paper represents a crucial step in applying these advanced beam types (specifically PLVBs) to the challenging and less-explored environment of plasma sheath turbulence [10-12, 28, 31, 32], aiming to address the "blackout" problem in a more effective manner.

3.4. Differentiation Analysis

Compared to the main methods in related work, the core differences and innovations of this paper's approach lie in its specific focus and comparative analysis:

• Focus on PLVBs in Plasma Sheath Turbulence: While PLVBs have been shown to be robust in atmospheric and oceanic turbulence [16, 25], this paper is among the first to systematically investigate their propagation characteristics and performance in plasma sheath turbulence. This is a unique and highly challenging turbulent environment due to its specific physical properties (e.g., high electron density, anisotropic nature).

• Comprehensive Performance Metrics: The study goes beyond simple intensity dispersion to include detection probability of OAM, bit error rate (BER), and channel capacity. This provides a holistic evaluation of PLVBs' suitability for practical OAM-based WOC systems in this environment.

• Direct Comparison with LGBs: The paper rigorously compares PLVBs against conventional Laguerre-Gaussian beams (LGBs), which are standard vortex beams. This direct comparison quantitatively highlights the superior performance of PLVBs and substantiates their claim as a more robust candidate.

• Parameter Optimization: The paper systematically analyzes the impact of beam modulation parameter ($\gamma$) and wavelength on PLVB performance within plasma sheath turbulence. This provides practical guidance for designing and optimizing PLVB transmitters for such environments, something often overlooked in general beam propagation studies.

• Low-Frequency Compensation in Phase Screens: The methodology incorporates a subharmonic compensation method in the random phase-screen model for plasma sheath turbulence. This technique addresses insufficient low-frequency sampling in traditional power spectrum inversion, improving the fidelity of large-scale turbulence effects and leading to more accurate simulations compared to simpler phase screen models.

  In essence, the innovation lies in the specific application of advanced PLVB technology to a critical, previously underexplored turbulent environment, supported by a comprehensive comparative analysis and parameter study using an improved simulation methodology.

4. Methodology

The paper investigates the propagation characteristics of pin-like vortex beams (PLVBs) through plasma sheath turbulence using the random phase-screen method. The methodology involves several key steps: defining the PLVB at the source, modeling plasma sheath turbulence with a compensated phase screen, propagating the beam through this turbulent medium, and finally analyzing the beam's orbital angular momentum (OAM) content to determine performance metrics like BER and channel capacity.

4.1. Principles

The core idea is to simulate the complex interaction between a structured light beam (PLVB) and a fluctuating medium (plasma sheath turbulence) to assess communication performance. The theoretical basis relies on the extended Huygens-Fresnel principle for wave propagation and statistical models for turbulence power spectra. The random phase-screen method approximates the continuous turbulent medium as discrete layers that impart random phase distortions. The intuition is that by comparing PLVBs with LGBs under the same turbulent conditions, the benefits of PLVBs' unique properties (like self-focusing or non-diffracting behavior) in mitigating turbulence effects can be quantified.

4.2. Core Methodology In-depth (Layer by Layer)

4.2.1. Pin-Like Vortex Beam (PLVB) Definition at Source Plane

The paper begins by defining the complex amplitude of PLVBs at the source plane ($z=0$). This is the initial state of the beam before it enters the turbulent medium.

The complex amplitude of PLVBs at the source plane $( z = 0 )$ can be expressed as: $$E ( \rho , \phi , 0 ) = A i ^ { | m _ { 0 } | } \rho ^ { - \gamma / 2 } \sqrt { \frac { 2 - \gamma } { 2 \pi k \gamma C } } \exp \big [ { - i ( k C \rho ^ { \gamma } + m _ { 0 } \phi ) } \big ]$$ Here, the variables and parameters are defined as follows:

• $(\rho, \phi, 0)$: Cylindrical coordinates at the source plane, where $\rho$ is the radial distance from the beam center, $\phi$ is the azimuthal angle, and $z=0$ indicates the source plane.

• $A$: A normalization constant associated with the beam power. This ensures that the total power carried by the beam is consistent.

• $m_0$: The topological charge of the beam, which denotes the OAM state at the signal source. It is an integer and characterizes the helical phase front.

• $\gamma$: The beam modulation parameter. This crucial parameter determines the propagation characteristics and spatial confinement of PLVBs. It ranges from $(0, 2)$.

  • For $0 < \gamma < 1$, the beam spreads with propagation (divergent).
  • For $\gamma = 1$, the beam remains diffraction-free, exhibiting a Bessel-like profile.
  • For $1 < \gamma < 2$, the beam narrows during propagation, forming a "pin-like" (convergent) structure.

• $C$: A constant defined as $C = C_{\rho} / \omega_{\rho}^{\gamma}$.

• $C_{\rho}$: A phase scaling parameter capable of assuming any value, adjusted to achieve comparable peak intensities at a given propagation distance.

• $\omega_{\rho}$: A phase normalization factor.

• $k$: The wavenumber, defined as $k = 2\pi / \lambda$.

• $\lambda$: The wavelength of the light beam.

  The paper highlights that for enhanced performance in WOC systems, especially to mitigate turbulence effects, $\gamma = 1.5$ is selected as a representative case, as it leads to a convergent beam profile. The phase scaling parameter $C_{\rho}$ is adjusted based on $\gamma$ to ensure comparable peak intensities, with values of 5.96, 3.77, and $3.12 \mu\text{m}$ for $\gamma = 0.5$, 1, and 1.5, respectively.

4.2.2. Free-Space Propagation using Extended Huygens-Fresnel Principle

After being defined at the source, the PLVB propagates through space. The paper uses the extended Huygens-Fresnel principle to describe this propagation. Although the ultimate goal is to propagate through turbulence, understanding free-space propagation is a foundational step before introducing turbulent effects.

The field of PLVBs after propagation through a distance $\mathcal{Z}$ in free space can be written as: \begin{array} { r l r } & { } & { E ( r , \varphi , z ) = - \displaystyle \frac { i k \exp ( i k z ) } { 2 \pi z } \int _ { 0 } ^ { \infty } \int _ { 0 } ^ { 2 \pi } \rho \mathrm { d } \rho \mathrm { d } \phi E ( \rho , \phi , 0 ) } \\ & { } & { \times \exp \left\{ \displaystyle \frac { i k } { 2 z } \left[ \rho ^ { 2 } + r ^ { 2 } - 2 \rho r \cos ( \varphi - \phi ) \right] \right\} , \end{array} Here, the terms are:

• $E(r, \varphi, z)$: The complex amplitude of the beam at the receiver plane, at a propagation distance $z$.

• $(r, \varphi)$: Radial and azimuthal coordinates at the receiver plane.

• $k$: The wavenumber.

• $z$: The propagation distance.

• $E(\rho, \phi, 0)$: The initial complex amplitude of the PLVB at the source plane, as defined in the previous equation.

• $\rho \mathrm{d}\rho \mathrm{d}\phi$: The differential area element in cylindrical coordinates at the source plane.

• The exponential term $\exp \left\{ \displaystyle \frac { i k } { 2 z } \left[ \rho ^ { 2 } + r ^ { 2 } - 2 \rho r \cos ( \varphi - \phi ) \right] \right\}$: This is the Fresnel kernel, which describes the propagation of a spherical wave from a point source at $(\rho, \phi, 0)$ to a point at $(r, \varphi, z)$ under the paraxial approximation.

  Figure 1 of the paper (shown below) visually demonstrates how the beam modulation parameter $\gamma$ influences the intensity profiles of PLVBs in free space.

  该图像是图表，展示了不同传播距离下，带有不同调制参数 $\gamma$（分别为0.5、1.0和1.5）的针状涡旋束（PLVBs）的归一化强度分布。每一行代表一种调制参数，列依次表示传播距离 $z = 0.1 \mathrm{~ m}$、$z = 0.2 \mathrm{~ m}$、$z = 0.3 \mathrm{~ m}$ 和 $z = 0.4 \mathrm{~ m}$。底部的白色曲线表示对应的输出光束强度分布。

Figure 1. Normalized intensity profiles of PLVBs with $\lambda = 532 \mathrm{ nm}$, $m_0 = 2$, and (a) $\gamma = 0.5$, (b) $\gamma = 1$, and (c) $\gamma = 1.5$, at various propagation distances in free space. The distances are $z = 0.1 \mathrm{ m}$, $z = 0.2 \mathrm{ m}$, $z = 0.3 \mathrm{ m}$, and $z = 0.4 \mathrm{ m}$, from left to right. The white solid line represents the output beam intensity distribution corresponding to the transmission distance.

This figure clearly illustrates the divergent behavior for $\gamma = 0.5$, the Bessel-like non-diffracting behavior for $\gamma = 1$, and the convergent, pin-like behavior for $\gamma = 1.5$. The maximum propagation distance considered is 0.4 m, consistent with the experimentally observed short distances of plasma sheaths [9].

4.2.3. Plasma Sheath Turbulence Modeling

The core of the simulation involves modeling the plasma sheath turbulence. The paper adopts a specific power spectrum for this turbulence, which is crucial for generating realistic phase screens.

The power spectrum of plasma sheath turbulence is expressed as: $$\Phi ( \kappa ) = a \frac { 6 4 \pi \langle \Delta n ^ { 2 } \rangle L _ { 0 } ^ { 2 } ( s - 1 ) } { ( 1 + 1 0 0 \kappa L _ { 0 } ^ { 2 } ) ^ { s } } \exp \left( - \frac { \kappa } { \kappa _ { 0 } } \right) ,$$ Here, the parameters are:

• $\Phi(\kappa)$: The power spectral density of the refractive index fluctuations in plasma sheath turbulence. This function describes how the energy of the turbulence is distributed across different spatial frequencies (wavenumbers).

• $\kappa$: The spatial wavenumber. In anisotropic turbulence, this is typically denoted as $\kappa = \sqrt { k_{z}^2 + ( \mu_x k_x )^2 + ( \mu_y k_y )^2 }$, where $k_x$ and $k_y$ are the spatial frequency components in the x and y directions, respectively. $k_z$ is neglected due to the Markov approximation [33].

• $a$: A fitting parameter, approximately equal to $475 (\kappa_0)^{2s}$.

• $\langle \Delta n^2 \rangle$: The variance of the refractive index fluctuation of plasma sheath turbulence. This quantifies the strength of the turbulence.

• $L_0$: The outer scale of plasma sheath turbulence, representing the largest turbulent eddies. It is related to the inner scale $l_0$ by the expression $L_0 / l_0 = R_e^{3/4}$, where $R_e = 5 \times 10^5$.

• $s$: A parameter defined as $s = 4 - d$, where $d$ is the fractal dimension. The value $d=2.6$ is determined from experiments [6].

• $\kappa_0$: A parameter related to the inner scale, given by $\kappa_0 = (2\pi / l_0)^{(s - 0.7)}$.

• $l_0$: The inner scale of plasma sheath turbulence, representing the smallest turbulent eddies.

• $\mu_x, \mu_y$: The anisotropic factors that account for the directional stretching of turbulent cell scales in the corresponding x and y directions [30]. If $\mu_x = \mu_y = 1$, the turbulence is isotropic.

  Figure 2 (shown below) provides a visual comparison of plasma sheath turbulence and atmospheric turbulence, highlighting the significantly stronger and more localized phase fluctuations produced by plasma sheath turbulence.

  该图像是图表，展示了（a1）等离子体鞘层湍流和（b1）大气湍流的二维和三维相位分布。图（a2）和图（b2）分别为相应的三维相位表现，使用改进的冯·卡门谱建模。

Figure 2. 2D and 3D phase distributions of (a1)—(a2) plasma sheath turbulence and (b1)(b2) atmospheric turbulence modeled by the modified von Karman spectrum.

4.2.4. Random Phase-Screen Method with Subharmonic Compensation

To simulate the effect of turbulence, the paper employs the random phase-screen method using power spectrum inversion, with an important enhancement: subharmonic compensation.

The random phase-screen method approximates the continuous turbulent medium as a series of discrete phase screens. Each screen introduces a phase shift to the beam. To address the insufficient low-frequency sampling (which affects the representation of large-scale turbulence effects) in traditional power spectrum inversion, subharmonic compensation is used.

The total phase introduced by a screen is given by: $$\psi _ { \mathrm { t o t } } = \psi _ { \mathrm { h i g h } } + \psi _ { \mathrm { l o w } } ,$$ Where:

• $\psi_{\mathrm{tot}}$: The total phase screen applied to the beam.
• $\psi_{\mathrm{high}}$: The high-frequency phase screen, which captures the fine-scale fluctuations of the turbulence. It is calculated as: $$\psi _ { \mathrm { h i g h } } = \mathrm { R e } \{ F ^ { - 1 } [ b ( k _ { x } , k _ { y } ) \sqrt { \Phi ( k _ { x } , k _ { y } ) } ] \}$$ Here:
  • $\mathrm{Re}\{\cdot\}$: Denotes the real part of the complex number.
  • $F^{-1}[\cdot]$: Represents the inverse Fourier transform.
  • $b(k_x, k_y)$: A Gaussian random matrix whose elements are normally distributed with values scaled between 0 and 1. This introduces the randomness characteristic of turbulence.
  • $\Phi(k_x, k_y)$: The power spectral density of the plasma sheath turbulence (as defined previously), which shapes the statistical properties of the phase screen.
• $\psi_{\mathrm{low}}$: The subharmonic compensation phase screen, which accounts for the large-scale, low-frequency fluctuations that are often undersampled in standard power spectrum inversion. It is calculated as: $$\psi _ { \mathrm { l o w } } = \mathrm { R e } \{ F ^ { - 1 } [ \dot { b } ( k _ { x , \mathrm { l o w } } , k _ { y , \mathrm { l o w } } ) \times \sqrt { \Phi ( k _ { x , \mathrm { l o w } } , k _ { y , \mathrm { l o w } } ) } ] \}$$ Here:

  • $\dot{b}(k_{x, \mathrm{low}}, k_{y, \mathrm{low}})$: Another Gaussian random matrix, similar to $b(k_x, k_y)$ but for the low-frequency components.

  • $k_{x, \mathrm{low}} = k_x / N^\xi$ and $k_{y, \mathrm{low}} = k_y / N^\xi$: The spectral grid sizes for the low-frequency screen are adjusted by a factor related to the subharmonic order $N$ and a parameter $\xi$. This downscaling of wavenumbers allows for better representation of low-frequency effects.

  • $N$: The subharmonic order.

  • $\xi$: A parameter representing the subharmonic order variant with values ranging from 1 to $N$.

    Figure 3 (shown below) schematically illustrates this process:

    该图像是示意图，展示了PIN-like涡旋光束（PLVBs）在等离子体鞘层湍流中传播的过程。左侧为激光发射器，产生涡旋光束源，中央部分展示了传播距离和等离子体湍流对光束的影响，右侧为接收器，显示了不同模式下的光束检测结果。整体展示了PLVBs在复杂环境下的传播特性。

Figure 3. Schematic diagram illustrating the propagation of PLVBs through plasma sheath turbulence using the random phase-screen method.

The diagram shows the PLVB source, followed by multiple plasma screens that introduce turbulence, and finally the receiver.

4.2.5. Beam Propagation Through Multiple Phase Screens

As the PLVB propagates, it sequentially passes through these phase screens. Each time it passes a screen, its phase is distorted. The complex amplitude of the PLVB at the received plane after propagating through multiple screens can be described using a split-step Fourier method, which combines free-space propagation with phase distortion at each screen.

The complex amplitude of PLVBs at the received plane can be described as: $$\begin{array} { r l r } { { \tilde { \boldsymbol { E } } ( \boldsymbol { r } , \varphi , z ) = \boldsymbol { F } ^ { - 1 } \{ \boldsymbol { F } [ \boldsymbol { E } ( \boldsymbol { r } , \varphi , z = 0 ) \exp ( i \psi _ { \mathrm { t o t } } ) ] } } \\ & { } & { \qquad \times \exp ( \frac { i z } { 2 } \sqrt { k ^ { 2 } - 4 \pi ^ { 2 } ( k _ { x } ^ { 2 } + k _ { y } ^ { 2 } ) } ) \} . } \end{array}$$ This equation represents a simplified form, likely representing a single step of propagation (or cumulative effect across multiple steps in a split-step approach). Let's break it down in the context of split-step propagation:

• $\tilde{\boldsymbol{E}}(\boldsymbol{r}, \varphi, z)$: The complex amplitude of the beam at the receiver plane, after propagating a total distance $z$ through turbulence.

• $\boldsymbol{E}(\boldsymbol{r}, \varphi, z=0)$: The complex amplitude of the beam at the beginning of a propagation step (or initial source). This is usually the field before it hits the current phase screen.

• $\exp(i \psi_{\mathrm{tot}})$: The phase modulation introduced by the total phase screen ($\psi_{\mathrm{tot}}$) at a specific propagation step. This term distorts the phase of the beam.

• $\boldsymbol{F}[\cdot]$: The Fourier transform operator. This transforms the beam's spatial domain representation into the spatial frequency domain.

• $\exp ( \frac { i z } { 2 } \sqrt { k ^ { 2 } - 4 \pi ^ { 2 } ( k _ { x } ^ { 2 } + k _ { y } ^ { 2 } ) } )$: This is the transfer function for free-space propagation in the spatial frequency domain (angular spectrum method). It propagates the beam over a distance $z$ (or the distance between two phase screens).

  • $k$: Wavenumber.
  • $k_x, k_y$: Spatial frequency components.

• $\boldsymbol{F}^{-1}[\cdot]$: The inverse Fourier transform operator, which transforms the beam back to the spatial domain.

  In a typical split-step algorithm, this process (phase screen application, Fourier transform, free-space propagation, inverse Fourier transform) is repeated for each phase screen along the total propagation distance.

4.2.6. OAM Mode Decomposition and Detection Probability

After the beam has propagated through the turbulent medium, its OAM content needs to be characterized to assess how much crosstalk has occurred and how well the original OAM state can be detected.

Any light field $\tilde{E}(r, \varphi, z)$ can be expanded on an orthogonal basis of spiral harmonics $\exp(-im\varphi)$. For a given OAM number $m$, the expansion coefficient corresponding to the $m$-th spiral harmonic is defined as: $$a _ { m } ( r , z ) = \frac { 1 } { \sqrt { 2 \pi } } \int _ { 0 } ^ { 2 \pi } \tilde { E } ( r , \varphi , z ) \exp { ( - i m \varphi ) } \mathrm { d } \varphi .$$ Here:

• $a_m(r, z)$: The expansion coefficient for OAM mode $m$ at radial position $r$ and propagation distance $z$. This coefficient quantifies the contribution of the $m$-th OAM mode to the total field.

• $\tilde{E}(r, \varphi, z)$: The complex amplitude of the beam at the receiver plane.

• $\exp(-im\varphi)$: The spiral harmonic (or OAM basis function) for topological charge $m$.

  The energy of the $m$-th spiral harmonic (or OAM mode) is then represented as: $$C ( m | m _ { 0 } ) = \int _ { 0 } ^ { \infty } | a _ { m } ( r , z ) | ^ { 2 } r \mathrm { d } r .$$ Here:

• $C(m | m_0)$: The energy contained in OAM mode $m$ when the initial OAM mode was $m_0$.

• $|a_m(r, z)|^2$: The intensity contribution of OAM mode $m$ at a given radial position.

• $\int_0^\infty \cdot r \mathrm{d}r$: Integration over the entire radial extent of the beam to sum up the energy.

  Finally, the energy weight of OAM mode $m$ received at the receiving plane (which corresponds to the detection probability) is calculated by normalizing this energy: $$P ( m | m _ { 0 } ) = \frac { C ( m | m _ { 0 } ) } { \displaystyle \sum _ { t = - \infty } ^ { \infty } C ( t | m _ { 0 } ) } .$$ Here:

• $P(m | m_0)$: The detection probability of OAM mode $m$, given that the original transmitted mode was $m_0$.

• \displaystyle \sum _ { t = - \infty } ^ { \infty } C ( t | m _ { 0 } ): The total energy across all OAM modes at the receiver plane.

• When $m = m_0$, P(m | m_0)_{m=m_0} represents the detection probability of the intended OAM mode.

• For $m \neq m_0$, $P(m | m_0)_{m \neq m_0}$ is the crosstalk probability of the OAM mode, describing the probability of energy migrating from the original OAM signal mode $m_0$ to an adjacent OAM mode $m$.

4.2.7. Signal-to-Noise-and-Crosstalk Ratio (SNCR), Bit Error Rate (BER), and Channel Capacity

These metrics are crucial for evaluating the communication performance of the OAM-based WOC system.

The signal-to-noise-and-crosstalk ratio (SNCR) of the OAM mode $m_0$ is defined as [39]: $$\mathrm { S N C R } = \frac { P ( m | m _ { 0 } ) _ { m = m _ { 0 } } } { \displaystyle \sum _ { n = - \infty } ^ { \infty } P ( m | m _ { 0 } ) _ { m \neq m _ { 0 } } + 1 0 ^ { - \frac { \mathrm { S N R } _ { 0 } ( \mathrm { d B ) } } { 1 0 } } } ,$$ Here:

• $\mathrm{SNCR}$: The signal-to-noise-and-crosstalk ratio. It quantifies the power of the desired signal relative to the sum of noise and crosstalk from other OAM modes.

• P(m | m_0)_{m=m_0}: The detection probability of the signal OAM mode $m_0$.

• \displaystyle \sum _ { n = - \infty } ^ { \infty } P ( m | m _ { 0 } ) _ { m \neq m _ { 0 } }: The sum of crosstalk probabilities from all other OAM modes ($n \neq m_0$).

• $10^{- \frac{\mathrm{SNR}_0 (\mathrm{dB})}{10}}$: This term represents the power of the background electrical noise, derived from the background signal-to-noise ratio ($\mathrm{SNR}_0$) in decibels (dB). A higher $\mathrm{SNR}_0$ (meaning less background noise) results in a smaller value for this term.

  The BER of OAM channels is derived as [40]: $${ \mathrm { B E R } } = { \frac { 1 } { 2 } } \mathrm { e r f c } \left( { \sqrt { \frac { \mathrm { S N C R } } { 2 } } } \right) ,$$ Here:

• $\mathrm{BER}$: The bit error rate.

• $\operatorname{erfc}(\bullet)$: The complementary error function. This is a special mathematical function related to the Gaussian probability distribution, commonly used in communication theory to calculate error probabilities based on signal-to-noise ratios. A higher SNCR leads to a lower BER.

  By using the concept of information capacity of multilevel symmetric channels, the average capacity of an optical communication link with $N$ symmetric OAM channels can be defined as [41]: $$\begin{array} { l } { C = \log _ { 2 } N + ( 1 - \mathrm { B E R } ) } \\ { \qquad \times \log _ { 2 } ( 1 - \mathrm { B E R } ) + \mathrm { B E R } \log _ { 2 } \frac { \mathrm { B E R } } { N - 1 } . } \end{array}$$ Here:

• $C$: The channel capacity in bits per symbol (or per OAM mode transmission).

• $N$: The total number of symmetric OAM channels used for communication. If OAM modes span from $m = -L, \ldots, L$, then $N = 2L+1$.

• $\mathrm{BER}$: The bit error rate.

  This formula, derived from Shannon's channel capacity theorem principles for discrete memoryless channels with specific error characteristics, quantifies the maximum achievable data rate. A lower BER and a higher number of available channels ($N$) generally lead to a higher channel capacity.

5. Experimental Setup

The simulation parameters for investigating PLVB performance in plasma sheath turbulence are detailed in Table 1.

5.1. Datasets

This study is entirely simulation-based and does not use traditional "datasets" in the sense of a collection of real-world measurements or labeled examples. Instead, the "data" for the simulation are the input pin-like vortex beams (PLVBs) and Laguerre-Gaussian beams (LGBs), and the "environment" is the plasma sheath turbulence generated numerically.

The properties of the plasma sheath turbulence are defined by parameters derived from experimental observations and theoretical models, such as the fractal dimension (2.6 from Ref. [6]) and the anisotropic nature [29]. The propagation distance is limited to 0.4 m, consistent with experimental observations of plasma sheath existence [9].

5.2. Evaluation Metrics

For every evaluation metric mentioned in the paper, here is a complete explanation:

1. Intensity Dispersion:

   • Conceptual Definition: Intensity dispersion refers to how the spatial distribution of the beam's intensity changes or spreads out as it propagates through a medium. In the context of turbulence, it quantifies the degree to which the beam's energy is scattered away from its original path or central lobe, leading to a wider and less concentrated beam profile. Less dispersion indicates a more stable and focused beam.
   • Mathematical Formula: The paper does not provide a specific mathematical formula for "intensity dispersion" as a single metric. Instead, it is implicitly evaluated by observing and comparing the normalized intensity profiles (as shown in Figure 4a1 and 4b1) and the intensity profiles at a specific distance (Figure 4a2 and 4b2). A common way to quantify beam spread (related to dispersion) is through metrics like beam width (e.g., RMS beam width or $1/e^2$ beam width), which would typically be calculated as: $ w_x^2 = \frac{\int \int x^2 I(x,y,z) dx dy}{\int \int I(x,y,z) dx dy} $ Where I(x,y,z) is the intensity distribution, and $w_x$ is the beam width in the x-direction. Similar for $w_y$. The overall beam size can then be given by $\sqrt{w_x^2 + w_y^2}$. However, the paper evaluates this visually rather than with a single numerical metric formula.
   • Symbol Explanation:
     • I(x,y,z): Intensity distribution of the beam in the transverse plane at a propagation distance $z$.
     • x, y: Transverse spatial coordinates.
     • dx, dy: Differential elements for integration.

2. Detection Probability of Orbital Angular Momentum (OAM):

   • Conceptual Definition: In OAM-based communication, the detection probability of an OAM mode refers to the likelihood that a transmitted OAM state ($m_0$) is correctly identified as $m_0$ at the receiver, despite the distortions caused by the turbulent channel. It quantifies the fidelity of OAM mode transmission. A higher detection probability indicates less crosstalk and better signal integrity.
   • Mathematical Formula: $ P ( m | m _ { 0 } ) = \frac { C ( m | m _ { 0 } ) } { \displaystyle \sum _ { t = - \infty } ^ { \infty } C ( t | m _ { 0 } ) } $ Where for detection probability, $m = m_0$. And $C ( m | m _ { 0 } ) = \int _ { 0 } ^ { \infty } | a _ { m } ( r , z ) | ^ { 2 } r \mathrm { d } r$.
   • Symbol Explanation:
     • $P(m | m_0)$: The detection probability of OAM mode $m$, given that the original transmitted mode was $m_0$.
     • $C(m | m_0)$: The energy contained in OAM mode $m$ when the initial OAM mode was $m_0$.
     • \displaystyle \sum _ { t = - \infty } ^ { \infty } C ( t | m _ { 0 } ): The total energy across all OAM modes at the receiver plane.
     • $a_m(r, z)$: The expansion coefficient for OAM mode $m$ at radial position $r$ and propagation distance $z$.
     • $|a_m(r, z)|^2$: The intensity contribution of OAM mode $m$ at a given radial position.
     • $r$: Radial coordinate.
     • $z$: Propagation distance.
     • $\mathrm{d}r$: Differential element for integration.

3. Bit Error Rate (BER):

   • Conceptual Definition: BER is a critical performance metric in digital communication, representing the ratio of erroneously received bits to the total number of transmitted bits. It directly measures the reliability of the communication link. In OAM-based systems, BER is influenced by crosstalk between OAM modes and background noise. A lower BER signifies a more robust and error-free communication.
   • Mathematical Formula: $ { \mathrm { B E R } } = { \frac { 1 } { 2 } } \mathrm { e r f c } \left( { \sqrt { \frac { \mathrm { S N C R } } { 2 } } } \right) $ Where SNCR is calculated as: $ \mathrm { S N C R } = \frac { P ( m | m _ { 0 } ) _ { m = m _ { 0 } } } { \displaystyle \sum _ { n = - \infty } ^ { \infty } P ( m | m _ { 0 } ) _ { m \neq m _ { 0 } } + 1 0 ^ { - \frac { \mathrm { S N R } _ { 0 } ( \mathrm { d B ) } } { 1 0 } } } $
   • Symbol Explanation:
     • $\mathrm{BER}$: Bit error rate.
     • $\operatorname{erfc}(\bullet)$: The complementary error function, defined as $\operatorname{erfc}(x) = \frac{2}{\sqrt{\pi}} \int_x^\infty e^{-t^2} dt$.
     • $\mathrm{SNCR}$: Signal-to-noise-and-crosstalk ratio.
     • P(m | m_0)_{m=m_0}: Detection probability of the signal OAM mode $m_0$.
     • \displaystyle \sum _ { n = - \infty } ^ { \infty } P ( m | m _ { 0 } ) _ { m \neq m _ { 0 } }: Sum of crosstalk probabilities from all other OAM modes ($n \neq m_0$).
     • $\mathrm{SNR}_0 (\mathrm{dB})$: Background signal-to-noise ratio in decibels.

4. Channel Capacity:

   • Conceptual Definition: Channel capacity represents the theoretical maximum rate at which information can be transmitted reliably over a communication channel without error. In OAM-based WOC, it quantifies the data throughput potential, considering the number of available OAM channels and the BER. A higher channel capacity indicates a more efficient and capable communication system.
   • Mathematical Formula: $ \begin{array} { l } { C = \log _ { 2 } N + ( 1 - \mathrm { B E R } ) } \ { \qquad \times \log _ { 2 } ( 1 - \mathrm { B E R } ) + \mathrm { B E R } \log _ { 2 } \frac { \mathrm { B E R } } { N - 1 } . } \end{array} $
   • Symbol Explanation:
     • $C$: Channel capacity in bits per symbol.
     • $N$: Total number of symmetric OAM channels (e.g., $N = 2L+1$ for modes $m = -L, \ldots, L$).
     • $\mathrm{BER}$: Bit error rate.

5.3. Baselines

The paper primarily compares pin-like vortex beams (PLVBs) against Laguerre-Gaussian beams (LGBs).

• Laguerre-Gaussian Beams (LGBs): LGBs are a conventional and well-established type of vortex beam that carry orbital angular momentum (OAM). They are widely used in theoretical and experimental studies of OAM-based communication and propagation through turbulence. They serve as a representative baseline because:

  • They are commonly understood OAM carriers.

  • Their propagation characteristics in turbulence are well-studied, providing a standard against which new beam types can be evaluated.

  • By comparing PLVBs directly with LGBs, the paper can quantify the performance advantages of the pin-like structure in mitigating turbulence effects under identical turbulent conditions.

    The comparison is made across all key metrics: intensity dispersion, detection probability of OAM, BER, and channel capacity, to provide a comprehensive evaluation of PLVBs' superior robustness.

The following are the results from Table 1 of the original paper:

6. Results & Analysis

6.1. Core Results Analysis

The simulation results consistently demonstrate the superior performance and robustness of pin-like vortex beams (PLVBs) compared to Laguerre-Gaussian beams (LGBs) when propagating through plasma sheath turbulence.

The initial comparison in Figure 4 illustrates the fundamental difference in how PLVBs and LGBs respond to plasma sheath turbulence.

该图像是图表，展示了在等离子体包层湍流中，(a1) PLVBs 和 (b1) LGBs 的传播侧视图，以及在距离 $z = 0.4 \, ext{m}$ 时的强度分布 (a2) 和 (b2)。图(c)显示了不同传播距离下，PLVBs 和 LGBs 的检测概率，实线表示两者的检测概率差异。

Figure 4. Side views of (a1) PLVBs and (b1) LGBs during propagation through plasma sheath turbulence; intensity profiles of (a2) PLVBs and (b2) LGBs at the distance $z = 0.4 \mathrm{ m}$ and (c) detection probabilities of PLVBs and LGBs at various propagation distances $\mathcal{Z}$. The solid line in (c) represents the difference in detection probabilities between PLVBs and LGBs.

• Intensity Dispersion (Figure 4a1, 4b1, 4a2, 4b2): As shown in the side views and intensity profiles, both beams experience beam spread and intensity fluctuation due to turbulence. However, LGBs (Figure 4b1, 4b2) exhibit more noticeable energy dispersion and dramatic distortion of the intensity distribution compared to PLVBs (Figure 4a1, 4a2). This visual evidence suggests that PLVBs maintain a more confined and stable intensity profile, even under strong turbulence. The reason for this, as discussed in the methodology, is the convergent property of PLVBs with $\gamma=1.5$, which results in a narrower beam width and thus a smaller effective interaction area with the turbulent medium.

• Detection Probability of OAM (Figure 4c): This metric directly quantifies the ability to correctly detect the transmitted OAM mode. As propagation distance $\mathcal{Z}$ increases, the cumulative effect of turbulence leads to crosstalk (energy spreading into neighboring OAM modes), reducing detection probabilities for both beams. However, PLVBs consistently show a higher detection probability than LGBs. The difference becomes more pronounced with increasing distance, with PLVBs being 9%–12.5% higher than LGBs over distances from 0.1 m to 0.4 m. This is a direct consequence of PLVBs' reduced intensity dispersion, which limits OAM crosstalk.

  The implications for communication performance are further highlighted by the Bit Error Rate (BER) and channel capacity metrics, as shown in Figure 5.

  该图像是图表，展示了在等离子鞘层湍流中，针状涡旋束（PLVBs）与常规拉盖尔-伽乌斯束（LGBs）在传输距离 $\mathcal{z}$ 下的比特错误率（BER）和信道容量的变化情况。图（a）显示了BER的变化，PLVBs的BER显著低于LGBs，并标注了关键数值0.03和0.067。图（b）则展示了信道容量，PLVBs的容量优于LGBs。

Figure 5. (a) BER and (b) channel capacity of PLVBs and LGBs in plasma sheath turbulence as a function of propagation distance $\mathcal{z}$.

• Bit Error Rate (BER) (Figure 5a): Consistent with the higher detection probability, PLVBs exhibit a significantly lower BER than LGBs. A higher detection probability directly translates to a higher SNCR (signal-to-noise-and-crosstalk ratio), which in turn reduces BER. The BER of PLVBs is 0.03–0.067 lower than that of LGBs over the 0.1–0.4 m propagation range. Furthermore, the growth rate of BER for PLVBs is noticeably slower than for LGBs as distance increases, indicating better scalability with propagation distance.

• Channel Capacity (Figure 5b): Channel capacity shows an inverse trend to BER. Because PLVBs have lower BERs, they naturally achieve higher channel capacity compared to LGBs. This means PLVBs can transmit more information reliably through plasma sheath turbulence.

  These core results strongly validate the effectiveness of PLVBs in plasma sheath turbulence. Their inherent convergent properties minimize beam spread and OAM crosstalk, leading to improved detection probability, reduced BER, and enhanced channel capacity compared to LGBs.

6.2. Ablation Studies / Parameter Analysis

The paper also investigates the impact of various beam parameters and turbulence parameters on the performance of PLVBs.

6.2.1. Impact of Beam Modulation Parameter ($\gamma$)

Figure 6 illustrates the influence of the beam modulation parameter $\gamma$ on BER and channel capacity for PLVBs compared to LGBs, for different OAM numbers.

该图像是图表，展示了不同光束调制参数 eta 下，PLVBs 与 LGBs 在比特错误率 (BER) 和信道容量方面的比较，涵盖了多种轨道角动量 (OAM) 数值。数据体现了不同参数对性能的影响。

Figure 6. BER and capacity of PLVBs with different beam modulation parameters $\gamma$, compared with those of LGBs for various OAM numbers.

• Performance vs. $\gamma$: Across all OAM numbers, PLVBs demonstrate consistently lower BER and higher channel capacity than LGBs. For PLVBs, increasing $\gamma$ (e.g., from 0.5 to 1.5) leads to a lower BER and higher channel capacity. This is attributed to the convergent effect induced by larger $\gamma$ values (as shown in Figure 1). A narrower main lobe width reduces the effective interaction area with the turbulent medium, mitigating mode distortion and energy spreading.
• OAM Order Dependence: The performance advantage of increasing $\gamma$ becomes less pronounced at higher OAM orders. This suggests that while $\gamma$ enhances robustness, the intrinsic turbulence sensitivity of higher-order OAM modes still imposes a performance limitation.
• Optimal $\gamma$: The results indicate that $\gamma = 1.5$ yields comparatively superior performance, making it suitable for achieving more stable transmission in this environment.

6.2.2. Impact of Channel Number ($N$) and Wavelength ($\lambda$)

Figure 7 explores how the channel number $N$ and wavelength $\lambda$ affect the channel capacity of PLVBs.

该图像是三维柱状图，展示了不同信道数 $N$ 和波长 $\lambda$ 对 PLVBs 在通过等离子体鞘层湍流时的容量的影响。图中展示的容量值在 $N$ 变化为 9 到 21 及波长范围为 460 nm 到 1550 nm 的情况下，显示出随波长和信道数的变化而变化的容量。

Figure 7. Capacity of PLVBs passing through plasma sheath turbulence with different channel numbers $N$ and wavelengths $\lambda$.

• Wavelength Effect: A larger wavelength leads to a higher channel capacity for a fixed channel number $N$. This is because longer wavelengths reduce the scattering effect of turbulence, making the beam less prone to severe intensity fluctuations from local turbulence disturbances [42]. This implies a more stable signal and lower BER, hence higher capacity.

• Channel Number Effect: For a fixed wavelength, a larger channel number ($N$) also leads to an increase in channel capacity. This is intuitive: more channels mean the communication system has more independent pathways for transmitting signals, inherently increasing the channel capacity.

  The paper notes that while longer wavelengths offer performance benefits in simulation, practical deployment must consider constraints like laser source availability and atmospheric transmission characteristics. Wavelengths around 1550 nm are often preferred in practice due to a good balance of low atmospheric loss, eye-safety, and compatibility with InGaAs detectors [43].

6.2.3. Impact of Refractive Index Fluctuation ($\langle \Delta n^2 \rangle$)

Figure 8 demonstrates the relationship between the variance of the refractive index fluctuation $\langle \Delta n^2 \rangle$ and BER and channel capacity.

该图像是图表，展示了PLVBs在等离子体鞘波动中不同 $\langle \Delta n ^{2} \rangle$ 下的比特错误率（BER）和信道容量随传播距离 $z$ 变化的关系。左侧（图 (a)）显示了BER的曲线，右侧（图 (b)）则展示了信道容量的变化。可以看到，随着传播距离的增加，BER逐渐上升，而信道容量则呈下降趋势，反映了PLVBs在不稳定环境中的传输性能。

Figure 8. (a) BER and (b) channel capacity of PLVBs in plasma sheath turbulence against different $\mathcal{Z}$ for different $\langle \Delta n^2 \rangle$.

• Effect of $\langle \Delta n^2 \rangle$: A larger refractive index fluctuation $\langle \Delta n^2 \rangle$ implies stronger turbulence. As expected, a decrease in $\langle \Delta n^2 \rangle$ (weaker turbulence) leads to a pronounced decrease in BER (Figure 8a) and a corresponding increase in channel capacity (Figure 8b). Stronger turbulence causes greater wavefront distortions, reducing signal stability and thus degrading communication performance.

6.2.4. Impact of Outer Scale ($L_0$) and Anisotropy Factor ($\mu_x$)

Figures 9a and 9b illustrate how the outer scale $L_0$ and anisotropy factor $\mu_x$ influence BER and channel capacity at a specific propagation distance.

该图像是图表，展示了在等离子体鞘层湍流下，pin-like vortex beams（PLVBs）在$z = 0.4 \ \mathrm{m}$时的比特错误率（BER）和信道容量。图（a）显示了不同外尺度$L_0$和各个各向异性因子$u_x$下的BER变化，图（b）则展示了相应的信道容量。数据表明，随着$L_0$的增加，BER逐渐下降，而信道容量趋于稳定。

Figure 9. (a) BER and (b) channel capacity of PLVBs in plasma sheath turbulence at $z = 0.4 \mathrm{ m}$ with different outer scales $L_0$ and anisotropy factors $u_x$.

• Outer Scale Effect: As the outer scale $L_0$ increases, the BER decreases (Figure 9a) and channel capacity increases (Figure 9b). A larger $L_0$ means the largest turbulent eddies are larger, implying a weaker impact of turbulence on the beam relative to the beam size. Notably, when $L_0 \ge 0.2 \mathrm{ m}$, the turbulence has a negligible interference effect on PLVBs, as the thickness of the anisotropic hypersonic flow field becomes comparable to or smaller than this scale [9].
• Anisotropy Factor Effect: A larger anisotropy factor (i.e., less isotropic turbulence, or specific directional stretching) results in a weaker impact of turbulence on the beam, leading to lower BER and higher channel capacity. The paper suggests that larger anisotropy factors mean turbulence eddies exhibit higher curvature, altering their focusing characteristics [44]. This effectively reduces amplitude fluctuations and the scintillation index, thereby mitigating turbulence-induced degradation.

6.2.5. Convergence Analysis of Statistical Realizations

To ensure the statistical reliability of the simulation outcomes, a sensitivity analysis was performed on the number of independent turbulence realizations. Figure 10 presents the results.

该图像是条形图，展示了统计次数对检测概率、误比特率（BER）和信道容量的影响。检测概率在统计次数为10时达到最高，BER则在100左右波动，信道容量呈现小幅上升趋势。这些结果反映了在不同统计次数下光束性能的变化。

Figure 10. Influence of statistics times on the convergence of detection probability, BER, and capacity.

• Convergence: The figure shows that when the number of realizations exceeds 100, all key metrics (detection probability, BER, channel capacity) exhibit stable behavior, with fluctuations constrained to within 3%. This validates the choice of 100 statistical realizations as a reasonable balance between computational cost and simulation accuracy.

  In summary, these parameter studies provide valuable insights for optimizing PLVBs for WOC in plasma sheath turbulence. Key takeaways include favoring higher $\gamma$ (within practical limits), longer wavelengths, and designing systems to be resilient to stronger turbulence conditions (smaller $L_0$, lower $\mu_x$, larger $\langle \Delta n^2 \rangle$).

7. Conclusion & Reflections

7.1. Conclusion Summary

This study rigorously investigated the propagation characteristics of pin-like vortex beams (PLVBs) through plasma sheath turbulence using the random phase-screen method, which included subharmonic compensation for enhanced accuracy. The findings conclusively demonstrate that PLVBs offer superior robustness compared to conventional Laguerre-Gaussian beams (LGBs) in this challenging environment. Specifically, PLVBs exhibited lower intensity dispersion, 9%–12.5% higher detection probability of OAM modes, and a bit error rate (BER) that was 0.03–0.067 lower than LGBs over propagation distances of 0.1 to 0.4 m. Consequently, PLVBs also showed enhanced channel capacity.

Furthermore, the research revealed that specific beam parameters can significantly optimize PLVB performance. A higher beam modulation parameter ($\gamma > 1.0$) notably reduced BER and increased channel capacity, especially for lower OAM modes, by promoting a convergent beam profile that minimizes interaction with turbulence. Longer wavelengths were also found to reduce BER and boost channel capacity by mitigating scattering effects. Conversely, stronger plasma sheath turbulence—characterized by larger refractive index fluctuations ($\langle \Delta n^2 \rangle$), a lower outer scale ($L_0$), and a lower anisotropic factor ($\mu_x$)—resulted in higher BER and lower channel capacity for PLVBs. The findings highlight the importance of selecting appropriate source parameters, such as a large $\gamma$ and longer wavelengths, to ensure strong resistance to turbulence. This positions PLVBs as promising candidates for stable optical communication in turbulent plasma environments, potentially mitigating communication blackouts for hypersonic vehicles.

7.2. Limitations & Future Work

The authors did not explicitly list limitations in a dedicated section. However, some implicit limitations and potential future work can be inferred:

• Simulation vs. Experiment: The study is entirely simulation-based. While the random phase-screen method with subharmonic compensation is robust, real-world plasma sheath turbulence can exhibit complexities not fully captured by current models. Experimental validation of these theoretical findings would be a crucial next step.
• Model Simplifications: The Markov approximation was used to neglect the spatial wavenumber component $k_z$. While common, this is an approximation that might not hold perfectly in all plasma sheath scenarios. Further refinement of the turbulence model could be considered.
• Specific Turbulence Model: The paper uses a specific power spectrum for plasma sheath turbulence. Different plasma sheath conditions (e.g., varying altitudes, vehicle speeds, atmospheric compositions) might lead to different power spectra, which could affect PLVB performance. Investigating a broader range of plasma sheath models could be beneficial.
• Interaction with Platform: The study focuses solely on beam propagation through the plasma sheath. The interaction of the PLVB generation and detection systems with the hypersonic vehicle platform itself (e.g., vibrations, heat, integration challenges) is not considered, but would be critical for practical deployment.
• Beam Power and Nonlinear Effects: The paper does not discuss the impact of high beam powers, which might introduce nonlinear propagation effects in dense plasma. For long-distance or high-power communication, these effects might become relevant.
• Dynamic Turbulence: The phase screens represent a static snapshot of turbulence for each realization. Real plasma sheath turbulence is highly dynamic. Investigating the impact of rapidly changing turbulence on PLVB tracking and communication systems would be important.
• Adaptive Optics Integration: While PLVBs show inherent robustness, combining them with adaptive optics systems (as mentioned in related work) could offer even greater resilience. Future work could explore this synergy.
• Impact of Multiple OAM Modes: The paper primarily focuses on the detection probability and crosstalk of a single OAM mode. Exploring the performance of PLVBs when simultaneously transmitting multiple OAM modes (i.e., OAM multiplexing) in plasma sheath turbulence would be a valuable extension.

7.3. Personal Insights & Critique

This paper presents a compelling argument for pin-like vortex beams as a viable solution for optical communication in plasma sheath turbulence. The rigorous simulation methodology, including subharmonic compensation for phase screens, adds credibility to the results. The comprehensive comparison with Laguerre-Gaussian beams across multiple metrics is particularly valuable, clearly quantifying the advantages of PLVBs.

One key insight is the profound impact of the beam modulation parameter $\gamma$. The ability to actively tune a beam's convergence or divergence to specifically mitigate turbulence effects is a powerful design principle. This suggests that future optical communication systems in turbulent environments might not only rely on passive robustness but also on actively shaped beams that adapt their spatial profile for optimal transmission. The finding that the advantage of higher $\gamma$ diminishes for higher OAM orders is also crucial, indicating a fundamental trade-off that needs to be considered when designing OAM multiplexing systems for such extreme conditions.

The analysis of wavelength dependence is also highly relevant. While longer wavelengths are beneficial for turbulence resistance, practical considerations like eye safety and detector availability (e.g., 1550 nm window) are wisely acknowledged. This highlights the gap between theoretical optimal performance and real-world engineering constraints, which future research needs to bridge.

A potential area for improvement or further investigation could be a more detailed theoretical explanation or modeling of the physical mechanism by which anisotropy reduces amplitude fluctuations and scintillation index for PLVBs. While the paper cites a reference, a more integrated discussion within the methodology could deepen understanding. Additionally, a direct comparison of the computational cost of simulating PLVBs versus LGBs could be useful, especially if PLVBs require more complex propagation algorithms or parameter tuning.

The paper's conclusions are significant for hypersonic vehicle technology and space exploration, where plasma sheaths are a major communication hurdle. The concept of PLVBs could potentially be transferred to other extreme turbulent environments, such as underwater optical communication through highly scattering and fluctuating water, or even inter-satellite links affected by atmospheric re-entry conditions. The systematic approach to parameter optimization is also broadly applicable to designing robust free-space optical links.