1. Bibliographic Information

1.1. Title

Color image information transmission in plasma sheath turbulence based on orbital angular momentum mode

1.2. Authors

Haimeng Liu, Dong ZhiI, Long Huang, HUFENG Liu, Yunfei Li, And Yong Tan

Their affiliations indicate a focus on aerospace and applied physics:

• Haimeng Liu, Yong Tan: Anhui University of Science and Technology, Anhui, China.
• Dong ZhiI, Long Huang, HUFENG Liu, Yunfei Li: China Aerodynamics Research and Development Center, Hypervelocity Aerodynamics Institute, Mianyang, China; and National Key Laboratory of Aerospace Physics in Fluids, Mianyang, China. This suggests a strong background in fluid dynamics, high-speed aerodynamics, and the physical challenges of communication in extreme environments like plasma sheaths.

1.3. Journal/Conference

Published by Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement. Optica (formerly OSA) is a highly reputable professional society for optics and photonics, publishing leading journals in the field. This indicates the research has undergone peer review and is recognized within the optics community.

1.4. Publication Year

2025 (Published at UTC: 2025-04-29T00:00:00.000Z)

1.5. Abstract

This study conducts a numerical investigation into the transmission of color images through a 0.4 m plasma sheath turbulence (PST) channel. The core methodology involves utilizing the orbital angular momentum (OAM) mode of Gaussian vortex beams for encoding and decoding image pixels, building upon an anisotropic power spectrum model for refractive-index fluctuations within the PST. The research systematically analyzes how refractive index fluctuation variance ($n_1^2$), outer scale ($L_0$), and anisotropy parameters ($\xi_x$) impact the peak signal-to-noise ratio (PSNR) of the received image and the bit error rate (BER) in free-space optical (FSO) communication. Key findings reveal that an increase in $n_1^2$, a decrease in $L_0$, and a decrease in $\xi_x$ all lead to a reduction in PSNR and an increase in BER. The study demonstrates promising BER values, achieving $1.25 \times 10^{-4}$ under weak turbulence and 0.447 even under strong turbulence conditions. The successful reconstruction of a $256 \times 256$-pixel color image validates the feasibility and robustness of the proposed encoding and decoding scheme in challenging PST environments.

1.6. Original Source Link

/files/papers/694216bca8811a6da9575214/paper.pdf (This link indicates the paper is likely an official publication and may be a PDF available from the journal's website or a preprint repository).

2. Executive Summary

2.1. Background & Motivation

The paper addresses the critical challenge of reliable communication through plasma sheath turbulence (PST), particularly relevant for high-speed aerospace vehicles during reentry. This phenomenon, caused by intense friction between the spacecraft and air, creates a time-varying PST layer that severely distorts optical signals due to random fluctuations in wavefront amplitude and phase. This distortion can scatter wave energy, leading to speckle patterns that contain or even lose encoded information, thereby significantly limiting the performance of free-space optical (FSO) communication links.

The current field of FSO communication faces increasing demand for higher transmission capacity, driven by imminent capacity shortages. Various techniques, such as spatial division multiplexing, frequency division multiplexing, and mode division multiplexing, are being explored. Image transmission, a common type of communication, has also seen advancements using different optical modes like high-order vector beams, spatial mode superposition, and twisted beams. However, a significant gap remains in ensuring robust image transmission through highly disturbed channels like PST.

The paper's innovative idea centers on leveraging orbital angular momentum (OAM) modes of Gaussian vortex beams. OAM beams possess distinct helical wavefront characteristics and their different topological charges are orthogonal to each other, providing an additional degree of freedom for encoding. This property makes OAM a promising candidate for increasing communication capacity and enhancing resilience against channel disturbances. The paper aims to bridge the gap by developing a robust encoding and decoding scheme for color image transmission specifically tailored for the challenging PST environment, an area where detailed studies on image transmission performance are limited.

2.2. Main Contributions / Findings

The primary contributions and key findings of this paper are:

• Novel Image Encoding/Decoding Scheme for PST: The study proposes and validates a comprehensive simulation model for color image information transmission through a plasma sheath turbulence (PST) channel. This scheme utilizes orbital angular momentum (OAM) modes of Gaussian vortex beams for encoding image pixels and a mode-matching method for decoding. This is a significant contribution as it addresses the specific challenges of communication in a highly disturbed aerospace environment.
• Detailed Analysis of PST Parameters on Image Quality: The paper systematically investigates the impact of critical PST parameters—namely, refractive index fluctuation variance ($n_1^2$), outer scale ($L_0$), and anisotropy parameters ($\xi_x$)—on the peak signal-to-noise ratio (PSNR) of the received image and the bit error rate (BER) in FSO communication. This provides crucial insights into how different turbulence characteristics affect communication performance.
• Quantification of Turbulence Impact: The results demonstrate that increasing $n_1^2$, decreasing $L_0$, and decreasing $\xi_x$ progressively lead to a decrease in PSNR and an increase in BER. This quantification helps in understanding the severity of turbulence effects.
• Feasibility Demonstration under Varying Turbulence: The study shows that the proposed scheme can achieve a BER as low as $1.25 \times 10^{-4}$ under weak turbulence conditions. Even under strong turbulence, a BER of 0.447 is achieved, indicating a notable level of robustness.
• Successful Color Image Reconstruction: The successful reconstruction of a $256 \times 256$-pixel color image after transmission through the simulated PST channel validates the practical feasibility and effectiveness of the proposed encoding and decoding scheme.
• Comparison with Conventional Schemes: The proposed method demonstrates superior performance compared to conventional QAM, PSK, and prior OAM-based encoding schemes in terms of channel capacity, BER, and turbulence resilience. It achieves 24 bits per pixel transmission, outperforming 16-ary systems, maintains a respectable BER under strong turbulence, and sustains a high $PSNR (> 20 dB)$ under significant refractive index fluctuations.
• Guidance for FSO Link Improvement: The findings offer valuable insights for improving FSO link performance in PST environments, suggesting that minimizing drastic variations in PST parameters and optimizing outer scale are effective strategies.

3. Prerequisite Knowledge & Related Work

3.1. Foundational Concepts

Free-Space Optical (FSO) Communication

Free-space optical (FSO) communication is a wireless communication technology that uses light to transmit data through the atmosphere or space. Instead of fiber optic cables, FSO links use lasers or LEDs to transmit information through the air, forming an optical link between two points.

• How it works: A transmitter converts electrical signals into optical signals (e.g., laser pulses), which are then sent through the air. A receiver at the other end detects these optical signals and converts them back into electrical signals.
• Advantages: High bandwidth, high data rates, license-free operation, rapid deployment, and high security.
• Challenges: Susceptibility to atmospheric conditions like fog, rain, snow, and turbulence, which can attenuate or scatter the optical beam, leading to signal degradation or loss.

Plasma Sheath Turbulence (PST)

Plasma sheath turbulence (PST) refers to the turbulent, ionized gas layer that forms around a high-speed vehicle (like a spacecraft) as it re-enters the atmosphere. The intense friction between the vehicle and the air causes the air molecules to heat up and ionize, creating a plasma. This plasma layer is not uniform but exhibits random, time-varying fluctuations in properties like electron density, temperature, and collision frequency, leading to turbulence.

• Impact on FSO: The turbulent variations in the plasma's refractive index cause scattering, absorption, and dispersion of electromagnetic waves (including optical beams), leading to amplitude scintillations (random intensity fluctuations) and phase distortions in the transmitted light. This severely degrades communication quality.

Orbital Angular Momentum (OAM)

Orbital angular momentum (OAM) is a property of light beams that describes the twisting or helical phase front of the light wave. Unlike spin angular momentum, which relates to the polarization of light, OAM is associated with the spatial distribution of the electromagnetic field.

• How it works: A light beam carrying OAM has a phase that varies azimuthally (around the beam's center) as $\exp(il\phi)$, where $l$ is the topological charge (an integer) and $\phi$ is the azimuthal angle. This results in a helical wavefront. Beams with different integer values of $l$ are orthogonal to each other, meaning they can be distinguished at the receiver even if they propagate through the same space.
• Advantages for Communication: The orthogonality of different OAM modes allows for mode division multiplexing (MDM), where multiple data streams can be transmitted simultaneously on different OAM modes, significantly increasing the channel capacity. The topological charge $l$ also provides an additional degree of freedom for encoding information.
• Gaussian Vortex Beams: A Gaussian vortex beam is a type of light beam that combines the properties of a standard Gaussian beam (a common laser beam with a smooth intensity profile) with the helical phase front of an OAM beam. It typically has a dark center (a phase singularity) where the intensity is zero.

Peak Signal-to-Noise Ratio (PSNR)

Peak signal-to-noise ratio (PSNR) is a widely used metric to quantify the quality of reconstruction of lossy compression codecs, and in this context, the quality of a received image after transmission through a noisy or turbulent channel. It compares the maximum possible power of a signal to the power of corrupting noise that affects the fidelity of its representation.

• Interpretation: A higher PSNR generally indicates a higher quality image, meaning less distortion or noise introduced during transmission.
• Formula (General Concept): PSNR is typically expressed in decibels (dB) and is derived from the mean squared error (MSE) between the original and reconstructed images.

Bit Error Rate (BER)

Bit error rate (BER) is a key performance metric in digital communication systems, representing the number of bit errors divided by the total number of bits transmitted over a given time interval.

• Interpretation: A lower BER indicates better communication quality and reliability. For example, a BER of $10^{-6}$ means that, on average, one bit out of every million transmitted bits is received incorrectly.

Wavelength Division Multiplexing (WDM) and Mode Division Multiplexing (MDM)

These are two techniques used to increase the capacity of optical communication systems.

• WDM: Wavelength Division Multiplexing combines multiple optical signals, each transmitted at a different wavelength (color of light), onto a single optical channel. At the receiver, a Wavelength Division Demultiplexer (WDD) separates these wavelengths.
• MDM: Mode Division Multiplexing transmits multiple data streams simultaneously on different spatial modes of light (e.g., different OAM modes or other higher-order modes) within the same optical channel. At the receiver, a Mode Division Demultiplexer (MDD) separates these modes.
• Combined Use: By combining WDM and MDM, even more data can be transmitted through a single physical channel, leveraging both wavelength and spatial dimensions.

3.2. Previous Works

The paper builds upon a foundation of research in FSO communication, OAM applications, and beam propagation through turbulent media.

• FSO Capacity Enhancement: References [1-11] highlight ongoing efforts to improve FSO capacity using various techniques like spatial division multiplexing [3,4], frequency division multiplexing [5-7], and mode division multiplexing [8-11]. These works establish the context for the need for higher capacity solutions and the potential of MDM.
• Image Transmission in FSO: Previous studies [12-15] have explored image transmission by converting images into numerical sequences and encoding them onto different optical modes. Examples include high-order vector beams [16], spatial mode superposition [17], and twisted beams [18]. This paper specifically extends this to OAM beams [19] for color image encoding.
• OAM for Communication: The use of optical vortex beams with orbital angular momentum (OAM) for communication is well-established [27-30]. The orthogonality of different topological charges offers an additional degree of freedom [31-33]. OAM multiplexing [8,34-36] and OAM encoding/decoding (OAM shift-keying) [40,41] are key methods.
  • Reference [42] showed high-fidelity transmission of grayscale and color images in scattering media using OAM multiplexing mechanisms with lower BER.
  • Reference [43] presented an experimental analysis of spatial modes for image transmission over a 3 km turbulence channel.
  • Reference [44] proposed a turbo-coded 16-ary OAM shift keying FSO communication system to reduce BER. These works confirm the viability of OAM for image transmission and BER reduction in turbulence.
• Turbulence Modeling and Effects: The understanding of beam propagation through turbulent media, particularly PST, is crucial. References [20,21] describe how turbulence scatters wave energy, causing random fluctuations in wavefront amplitude and phase, leading to signal distortion. The paper specifically uses an anisotropic power spectrum of refractive-index fluctuations for PST, based on previous models [25,52] that account for the asymmetric and nonuniform nature of turbulent shear flow. The split-step beam propagation method combined with the multiple random phase screen approach [49,50] is a standard technique for simulating beam propagation through turbulence.
• Scintillation Index: The scintillation index is a common metric to characterize intensity fluctuations due to turbulence. The paper validates its simulation against theoretical models for scintillation index of Gaussian beams in anisotropic hypersonic plasma turbulence [53,54].

3.3. Technological Evolution

The evolution of FSO communication has moved from basic line-of-sight links to increasingly sophisticated systems that employ various multiplexing techniques to overcome bandwidth limitations. Early FSO systems often struggled with atmospheric turbulence, leading to the development of adaptive optics and robust modulation schemes. The introduction of OAM as a degree of freedom for encoding and multiplexing represents a significant step, moving beyond conventional wavelength and polarization multiplexing. This paper pushes the boundary further by applying OAM to image transmission, specifically addressing the highly challenging PST environment relevant to aerospace applications. The use of an anisotropic turbulence model reflects a more accurate representation of real-world scenarios compared to simpler isotropic models.

3.4. Differentiation Analysis

Compared to previous works, the core differences and innovations of this paper's approach are:

• PST-Specific Focus: While many studies address atmospheric turbulence, this paper specifically focuses on plasma sheath turbulence (PST), which has distinct characteristics (anisotropy, high ionization) and is highly relevant to aerospace reentry scenarios. This makes the channel model and validation particularly novel.
• Integrated WDM-MDM-OAM Scheme for Color Images: The paper proposes a comprehensive encoding and decoding scheme that integrates Wavelength Division Multiplexing (WDM) and Mode Division Multiplexing (MDM) using OAM modes of Gaussian vortex beams specifically for color image transmission. Previous works often focused on grayscale images or simpler encoding without this hybrid multiplexing approach in PST.
• Detailed Parameter Impact Analysis in PST: The study provides a detailed numerical analysis of how specific PST parameters ($n_1^2$, $L_0$, $\xi_x$) influence image quality (PSNR) and communication reliability (BER). This quantitative analysis is crucial for understanding and mitigating turbulence effects in this unique environment.
• Feasibility under Strong Turbulence: The paper rigorously tests and demonstrates the feasibility of image transmission even under strong turbulence conditions in PST, showing a BER of 0.447. While this BER is high, demonstrating transmission under such extreme conditions is significant.
• Robustness and Performance Claims: The paper claims superior performance in channel capacity, BER, and turbulence resilience compared to conventional QAM, PSK, and even other OAM-based schemes, achieving 24 bits per pixel transmission and a $PSNR > 20 dB$ under challenging conditions. The mode-matching decoding method is highlighted as a key enabler for this resilience.

4. Methodology

The paper proposes a simulation model for color image information transmission through a plasma sheath turbulence (PST) channel, utilizing orbital angular momentum (OAM) modes of Gaussian vortex beams for encoding and a mode-matching method for decoding. The overall process can be divided into encoding, transmission, and decoding stages.

4.1. Principles

The core idea is to leverage the unique properties of OAM modes, specifically their orthogonality, to encode information. By mapping binary representations of image pixel values to different topological charges of Gaussian vortex beams, and using Wavelength Division Multiplexing (WDM) for different color channels, a high-capacity encoding scheme is achieved. The mode-matching method at the receiver then correlates the received distorted beams with ideal OAM modes to recover the encoded information. The PST channel is modeled using a multiple random phase screen approach based on an anisotropic power spectrum to accurately simulate turbulence effects.

4.2. Core Methodology In-depth

4.2.1. Image Pixel Encoding

The encoding process converts a 256 × 256 color image into OAM modes of Gaussian vortex beams.

1. Color Channel Decomposition: The input color image is first decomposed into its three independent primary color channels: Red (R), Green (G), and Blue (B).
2. Pixel Value Extraction: For each channel, the pixel values (ranging from 0 to 255) are extracted in a left-to-right, top-to-bottom order.
3. Decimal to Binary Conversion: Each pixel value (a decimal number) is converted into an 8-bit binary sequence.
4. Binary to OAM Mapping: The 8-bit binary sequence corresponding to each pixel value is mapped onto Gaussian vortex beams with different topological charges (M). This mapping is performed in a right-to-left order for the binary sequence.

   • Each bit position in the 8-bit sequence corresponds to a specific topological charge. If a bit is 1, a vortex beam with the corresponding topological charge is used for encoding.

   • Wavelength Assignment: Each color channel (R, G, B) uses vortex beams with a different wavelength ($\lambda_1$ for Red, $\lambda_2$ for Green, $\lambda_3$ for Blue). This enables WDM.

     The following figure (Figure 1 from the original paper) shows the image pixel encoding scheme:

     

     Example:

• Suppose a pixel has R: 65, G: 130, B: 160.

• Red Channel (R=65): Binary is 0100 0001. The first and seventh bits (from right to left) are 1. So, two vortex beams with wavelength $\lambda_1$ and topological charges $M=1$ and $M=7$ are used.

• Green Channel (G=130): Binary is 1000 0010. The second and eighth bits are 1. So, two vortex beams with wavelength $\lambda_2$ and topological charges $M=2$ and $M=8$ are used.

• Blue Channel (B=160): Binary is 1010 0000. The fifth and seventh bits are 1. So, two vortex beams with wavelength $\lambda_3$ and topological charges $M=5$ and $M=7$ are used.

  The Gaussian vortex beam is expressed by the following equation: $$E ( x , y , z = 0 ) = A _ { 0 } \exp \left( - \frac { x ^ { 2 } + y ^ { 2 } } { \omega _ { 0 } ^ { 2 } } \right) ( x + i y ) ^ { | M | }$$ Where:

• E ( x , y , z = 0 ) represents the electric field amplitude of the Gaussian vortex beam in the xy-plane at $z=0$ (the source plane).

• $A_0$ is the input beam intensity, a constant representing the amplitude.

• $\exp \left( - \frac { x ^ { 2 } + y ^ { 2 } } { \omega _ { 0 } ^ { 2 } } \right)$ is the Gaussian envelope, which defines the intensity distribution of the beam. It ensures the beam intensity peaks at the center and decays radially.

• $\omega_0$ is the beam waist (or spot size) of the vortex beam, representing the radius where the electric field amplitude falls to $1/e$ of its peak value.

• $( x + i y ) ^ { | M | }$ is the vortex phase term. In polar coordinates, $x = r \cos \phi$ and $y = r \sin \phi$, so $x+iy = r e^{i\phi}$. Thus, $(x+iy)^{|M|} = r^{|M|} e^{iM\phi}$. This term introduces the helical wavefront characteristic of OAM beams.

• $|M|$ is the absolute value of the topological charge. $M$ determines the number of $2\pi$ phase twists around the beam's center. A non-zero $M$ indicates an OAM beam. The sign of $M$ indicates the direction of the twist (clockwise or counter-clockwise).

4.2.2. Image Pixel Transmission

During the transmission stage, the encoded signals are combined and sent through the PST channel.

1. Signal Combination: The encoded signals (vortex beams with different wavelengths and OAM modes) are first combined using Wavelength Division Multiplexing (WDM) and Mode Division Multiplexing (MDM). This creates a single synthesized beam carrying all the encoded information.

2. PST Channel Propagation: The synthesized beam then propagates through the 0.4 m Plasma Sheath Turbulence (PST) channel.

   The following figure (Figure 2 from the original paper) shows the image pixel transmission scheme:

   该图像是示意图，展示了基于轨道角动量模式的颜色图像像素传输方案。图中展示了通过光纤多路复用(WDM)和模态多路复用(MDM)发送图像信号，再通过等离子体鞘层湍流(PST)进行传输，最后通过光纤多路解复用(WDD)和模态多路解复用(MDD)接收图像。

4.2.3. PST Channel Model and Beam Propagation

The PST channel is modeled as a series of random phase screens. The beam propagation process is simulated using the split-step beam propagation method combined with the multiple random phase screen approach.

The general Schrödinger equation for beam propagation through a medium can be solved iteratively using this method: $$E ( x , y , z ) = \prod _ { n } ^ { N } \exp ( - \frac { i } { 2 k } \Delta z \nabla _ { \perp } ^ { 2 } ) \exp [ i S ( x , y , z _ { n } ) ] E ( x , y , z _ { n - 1 } )$$ Where:

• E ( x , y , z ) is the electric field of the beam at position $z$.

• $\prod _ { n } ^ { N }$ indicates a product (or sequential application) over $N$ propagation steps.

• $\exp ( - \frac { i } { 2 k } \Delta z \nabla _ { \perp } ^ { 2 } )$ is the diffraction operator (in the spatial frequency domain, this corresponds to multiplying by a quadratic phase term). It models free-space propagation over a small distance $\Delta z$.

• $\nabla _ { \perp } ^ { 2 } = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}$ is the transverse Laplacian operator.

• $k = 2 \pi / \lambda$ is the wavenumber of the incident light, where $\lambda$ is the wavelength.

• $\Delta z = z_n - z_{n-1}$ is the thickness of each phase screen, representing the propagation step size.

• $\exp [ i S ( x , y , z _ { n } ) ]$ is the phase modulation operator. It models the phase distortion introduced by the $n$-th plasma sheath turbulence screen.

• $S ( x , y , z _ { n } )$ represents the random phase modulation caused by the PST at step $n$.

  The phase distribution $S ( x , y , z _ { n } )$ for a turbulence screen of thickness $\Delta z$ is obtained as: $$S ( x , y , z _ { n } ) = F ^ { - 1 } \left[ \alpha \sqrt { 2 \pi k ^ { 2 } \Delta z \phi _ { n } ( \kappa ) } \right]$$ Where:

• $F^{-1}$ represents the inverse Fourier transform. This means the phase screen is constructed in the spatial frequency domain and then transformed back to the spatial domain.

• $\alpha$ follows a standard Gaussian random distribution, introducing the stochastic nature of turbulence.

• $\phi_n(\kappa)$ is the anisotropic power spectrum of refractive-index fluctuations for the PST, which determines the statistical properties of the turbulence.

• $\kappa$ is the spatial frequency vector.

  The anisotropic power spectrum of refractive-index fluctuations is based on the generalized anisotropic von Kármán spectrum, accounting for asymmetry and nonuniformity of turbulence vortices: $$\phi _ { n } ( \kappa ) = a \frac { 6 4 \pi \left. n _ { 1 } ^ { 2 } \right. L _ { 0 } ^ { 2 } ( m - 1 ) } { \left( 1 + 1 0 0 \kappa L _ { 0 } ^ { 2 } \right) ^ { m } } \exp \left( - \frac { \kappa } { \kappa _ { 0 } } \right)$$ Where:

• $\kappa = \sqrt { \xi _ { x } ^ { 2 } \kappa _ { x } ^ { 2 } + \xi _ { y } ^ { 2 } \kappa _ { y } ^ { 2 } + \kappa _ { z } ^ { 2 } }$ is the modified spatial frequency. This anisotropic form means that the turbulence statistics are different in different directions.

• $\xi_x$ and $\xi_y$ are two anisotropy parameters representing scale-dependent stretching along the $x$ and $y$ directions. They quantify the degree of anisotropy.

• $\left. n _ { 1 } ^ { 2 } \right.$ is the variance of the refractive index fluctuations, a measure of the strength of the turbulence.

• $l_0$ is the inner scale of the turbulence, representing the smallest scale at which turbulence kinetic energy is dissipated into heat.

• $L_0$ is the outer scale of the turbulence, representing the largest scale at which energy is injected into the turbulent flow.

• $R_e$ is the Reynolds number, and the relationship between inner and outer scales is given by $L_0 / l_0 = R_e^{3/4}$.

• $a = 475 (2\pi / l_0)^{2.683}$ is a constant.

• $\kappa_0 = (2\pi / l_0)^{m-0.7}$ is related to the inner scale.

• $m = 4 - d$, where $d$ is the fractal dimension. For fully developed turbulence in a high-speed turbulent mixing layer, $d = 2.6$, thus $m = 1.4$.

  The following figure (Figure 4 from the original paper) illustrates the propagation path of vortex beams through PST:

  该图像是示意图，展示了激光通过等离子体鞘层湍流（PST）传播的过程。图中标示出激光源、输出平面和接收器的位置，以及PST的影响区域。

4.2.4. Image Pixel Decoding

At the receiving end, the distorted beam is processed to recover the original image information.

1. Signal Separation: The received beam is first separated and extracted using Wavelength Division Demultiplexing (WDD) to distinguish between the R, G, and B channels, and then Mode Division Demultiplexing (MDD) to separate the different OAM modes.

2. Mode-Matching Method: The core of the decoding process is the mode-matching method, which identifies the original OAM modes from the received signals. This is done by calculating the correlation between the received signal and a set of ideal original OAM modes.

   The correlation function used to evaluate the degree of correlation is defined as: $$C o r r ( A ^ { \prime } , A ) = \frac { C o \nu ( A ^ { \prime } , A ) } { \sqrt { V a r [ A ^ { \prime } ] V a r [ A ] } }$$ Where:

• Corr(A', A) is the correlation coefficient between the received signal $A'$ and the original (ideal) signal $A$. A higher value indicates a better match.
• $Co\nu(A', A)$ represents the covariance between the received signal $A'$ and the original signal $A$. Covariance measures how two variables change together.
• Var[A'] and Var[A] denote the variances of the received signal $A'$ and the original signal $A$, respectively. Variance measures the spread of a set of data points around their mean.

Decoding Steps using Correlation:

• For each received optical signal (corresponding to a specific wavelength and OAM mode), its correlation with nine original signals (representing topological charges from $M=0$ to $M=8$, assuming an 8-bit encoding which means $2^8=256$ possible values, hence $M$ up to 7 in this case) is calculated, resulting in nine correlation coefficient values ($Corr_0 - Corr_8$).
• The maximum correlation coefficient indicates the most likely original OAM mode transmitted.
• Example (continued from encoding):

  • For the red channel (wavelength $\lambda_1$), if the maximum correlation coefficients are found for modes $M=1$ and $M=7$, then the binary-decoded value for this channel is 0100 0001, which corresponds to the pixel value 65.

  • The same process is applied for the green ($\lambda_2$) and blue ($\lambda_3$) channels to determine their respective pixel values.

    The following figure (Figure 3 from the original paper) shows the image pixel decoding scheme:

    该图像是示意图，展示了图像像素解码方案。左侧部分呈现了不同模式的匹配过程，通过最大相关性选取对应的模式，右侧则展示了颜色通道（R、G、B）的解码过程，包括对应的十进制值转换。此图用于说明如何从光束模式中提取颜色信息，以实现图像的解码。

Through this pixel-by-pixel transmission approach, the original image information is fully decoded and reconstructed.

4.2.5. Received Light Intensity Model

In a real system, power loss due to the environment and detector noise are considered. The received light intensity at the detector is expressed as: $$I ( x , y , z ) = \lvert E ( x , y , z ) \rvert ^ { 2 } \exp ( - \rho z ) + N ( 0 , \delta )$$ Where:

• I ( x , y , z ) is the received light intensity at a given position (x, y) and propagation distance $z$.
• $\lvert E ( x , y , z ) \rvert ^ { 2 }$ is the intensity of the optical beam after propagating through the PST channel (i.e., the square of the electric field amplitude).
• $\exp ( - \rho z )$ accounts for power loss due to environmental factors (e.g., absorption, scattering not captured by phase screens).
• $\rho$ is the power loss coefficient, fixed at $6.5 \times 10^{-5} \mathrm{m^{-1}}$ in this study.
• $N ( 0 , \delta )$ represents random Gaussian noise introduced by the detector.
• 0 is the mean of the Gaussian noise.
• $\delta$ is the variance of the detector noise.

4.2.6. Rytov Variance for Turbulence Classification

To classify the turbulence conditions (weak or strong), the Rytov variance ($\sigma_B^2$) is used. For a Gaussian beam wave in plasma sheath turbulence, the expression for the Rytov variance is approximately: $$\sigma _ { B } ^ { 2 } \cong 3 . 8 6 \sigma _ { R } ^ { 2 } \left\{ 0 . 4 0 \left[ ( 1 + 2 \Theta ) ^ { 2 } + 4 \Lambda ^ { 2 } \right] ^ { 5 / 1 2 } \times \cos \left[ \frac { 5 } { 6 } \tan ^ { - 1 } \frac { 1 + 2 \Theta } { 2 \Lambda } \right] - \frac { 11 } { 16 } \Lambda ^ { 5 / 6 } \right\}$$ Where:

• $\sigma _ { B } ^ { 2 }$ is the Rytov variance for a Gaussian beam.
• $\sigma _ { R } ^ { 2 }$ is the Rytov variance for a plane wave, defined as: $$\sigma _ { R } ^ { 2 } = 8 \pi ^ { 2 } k ^ { 2 } z \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { + \infty } \kappa \Phi _ { n } ( \kappa ) \left[ 1 - \cos \left( \frac { z \kappa ^ { 2 } \xi } { k } \right) \right] d \kappa d \xi$$
  • $k = 2\pi/\lambda$ is the wavenumber.
  • $z$ is the propagation distance.
  • $\kappa$ is the spatial frequency.
  • $\Phi_n(\kappa)$ is the power spectrum of refractive-index fluctuations (as defined previously for PST).
  • $\xi$ is an integration variable from 0 to 1.
• $\Theta$ and $\Lambda$ are beam parameters related to the beam's divergence and focusing properties, expressed as: $$\Theta = \frac { \Theta _ { 0 } } { \Theta _ { 0 } ^ { 2 } + \Lambda _ { 0 } ^ { 2 } } , \Lambda = \frac { \Lambda _ { 0 } } { \Theta _ { 0 } ^ { 2 } + \Lambda _ { 0 } ^ { 2 } } , \Lambda _ { 0 } = \frac { 2 z } { k \omega _ { 0 } ^ { 2 } }$$

  • $\Theta_0 = 0$ for Gaussian beams (initially collimated, flat phase front).

  • $\Lambda_0$ is the diffraction parameter of the beam, where $\omega_0$ is the initial beam waist.

    The Rytov variance acts as a criterion:

• $\sigma _ { B } ^ { 2 } \ge 1$: Corresponds to strong turbulence.
• $\sigma _ { B } ^ { 2 } < 1$: Indicates weak turbulence.

5. Experimental Setup

5.1. Datasets

The paper uses a 256 × 256-pixel color image for transmission experiments. While the specific image used is not explicitly named as a standard dataset, the visual examples provided in Figure 1 and Figure 6 clearly show an image of a monkey.

The following figure (Figure 1 from the original paper) shows an example of the input color image:

该图像是一个示意图，展示了图像像素编码方案。左侧是猴子的彩色图像，右侧则是对应的十进制和二进制编码，反映红色（R）、绿色（G）和蓝色（B）通道的数值及其二进制形式，最后是量子编码的结果。

The choice of a 256 × 256-pixel color image is typical for demonstrating image transmission schemes, providing a balance between visual complexity and computational load for simulation. It is effective for validating the method's ability to encode, transmit, and decode color information and assess image quality metrics like PSNR under various turbulence conditions.

5.2. Evaluation Metrics

The quality of the transmitted image and the performance of the communication system are evaluated using two primary metrics: Peak Signal-to-Noise Ratio (PSNR) and Bit Error Rate (BER).

Peak Signal-to-Noise Ratio (PSNR)

PSNR is used to measure the quality of the reconstructed image compared to the original image. A higher PSNR value indicates a better-quality image, meaning less distortion from the original.

Conceptual Definition: PSNR quantifies the ratio between the maximum possible power of a signal and the power of corrupting noise that affects the fidelity of its representation. Because many signals have a very wide dynamic range, PSNR is usually expressed in terms of the logarithmic decibel scale.

Mathematical Formula: The overall PSNR for a color image is the average of the PSNR for each color channel (Red, Green, Blue). The PSNR for a single channel is defined as: $$P S N R = \frac { P S N R _ { R } + P S N R _ { G } + P S N R _ { B } } { 3 }$$ For each channel, $PSNR_{channel}$ is calculated as: $$P S N R _ { c h a n n e l } = 1 0 \log _ { 1 0 } \left[ \frac { ( 2 ^ { p } - 1 ) ^ { 2 } } { M S E _ { c h a n n e l } } \right]$$ Where $MSE_{channel}$ is the Mean Squared Error for that channel: $$M S E _ { c h a n n e l } = \frac { 1 } { Q } \sum _ { x , y } \left[ T ( x , y ) - O ( x , y ) \right] ^ { 2 }$$ Symbol Explanation:

• PSNR: The average Peak Signal-to-Noise Ratio for the color image (in dB).
• $PSNR_R, PSNR_G, PSNR_B$: The PSNR for the Red, Green, and Blue color channels, respectively.
• $p$: The bit depth of each channel. In this study, $p = 8$, meaning each pixel value ranges from 0 to $2^8-1 = 255$.
• $(2^p - 1)^2$: Represents the maximum possible squared pixel value (i.e., the square of the peak signal value). For an 8-bit image, this is $(2^8 - 1)^2 = 255^2$.
• $MSE_{channel}$: The Mean Squared Error for a specific color channel.
• $Q$: The total number of pixels in the image. For a $256 \times 256$ image, $Q = 256 \times 256 = 65536$.
• $\sum_{x,y}$: Summation over all pixel coordinates (x, y) in the image.
• T(x,y): The pixel value at coordinate (x,y) in the transmitted (received and decoded) image.
• $O(x,y)$: The pixel value at coordinate (x,y) in the original image.
• $\left[ T ( x , y ) - O ( x , y ) \right] ^ { 2 }$: The squared difference between the transmitted and original pixel values at (x,y).

Bit Error Rate (BER)

BER is used to assess the reliability of the digital communication link, representing the fraction of bits that are incorrectly received.

Conceptual Definition: BER is the number of erroneous bits received divided by the total number of bits transmitted over a communication channel during a specific time interval. It is a direct measure of the accuracy of data transmission.

Mathematical Formula: $$BER = \frac{\text{Number of bit errors}}{\text{Total number of bits transmitted}}$$ Symbol Explanation:

• $\text{Number of bit errors}$: The count of individual bits that were transmitted as one value (e.g., '0') but received as another (e.g., '1'), or vice versa.
• $\text{Total number of bits transmitted}$: The total count of all bits sent through the communication channel.

5.3. Baselines

The paper implicitly compares its OAM-based encoding scheme with conventional FSO communication approaches and other OAM schemes by discussing their limitations and the superior performance of the proposed method. While explicit numerical baselines are not presented in tables within the results section, the discussion in Section 4 (and Conclusion) makes comparisons against:

• Conventional FSO Schemes:

  • 64-QAM (Quadrature Amplitude Modulation): A common digital modulation scheme used in FSO. The paper mentions its limited symbol space, spectral efficiency issues, and requirement of 38.5 dB OSNR with turbulence-sensitive BER thresholds.
  • PSK (Phase-Shift Keying): Another digital modulation technique. The paper notes its modulation-order constraints, phase-fluctuation-induced error floors, and a 14 dB SNR penalty.

• Prior OAM-based Encoding Schemes: The paper mentions that prior OAM implementations are restricted by turbulence-induced mode degradation despite their theoretical infinite-mode advantages, often degrading to $BER ≈ 10⁻¹$ under strong turbulence.

• Turbo-coded 16-ary OAM Shift Keying: Specifically mentioned in Ref. [44], the paper claims its method can achieve a BER two orders of magnitude better than the residuals after Turbo coding ($4.01 \times 10^{-4}$).

  The paper highlights its method's superiority in channel capacity, BER, and turbulence resilience compared to these categories of baselines.

6. Results & Analysis

The results section analyzes the impact of PST parameters ($n_1^2$, $L_0$, and $\xi_x$) on the quality of the received image (measured by PSNR) and the reliability of the communication link (measured by BER). The numerical simulations are validated against theoretical models, and the feasibility of transmitting a 256 × 256-pixel color image is demonstrated.

6.1. Core Results Analysis

6.1.1. Validation of Beam Propagation Model

The simulation program for beam propagation through PST is validated by comparing on-axis scintillation results of a Gaussian beam with theoretical predictions.

• Simulation Parameters:

  • Initial beam width $\omega_0 = 1 \mathrm{cm}$

  • Wavelength $\lambda = 532 \mathrm{nm}$

  • Sampling grid: $512 \times 512$

  • Sampling interval: $0.97 \mathrm{mm}$

  • Turbulence parameters for validation: $\left. n _ { 1 } ^ { 2 } \right. = 0.73 \times 10^{-20}$, $L_0 = 0.25 \mathrm{m}$, $\xi_x = 3$.

  • Thickness of each phase screen $\Delta z = 0.04 \mathrm{m}$.

  • Propagation step $N = 4$.

    The following figure (Figure 5 from the original paper) shows the on-axis scintillation of Gaussian beams for theory and simulation results:

    该图像是图表，展示了理论和仿真结果的高斯光束典轴闪烁随 $L_0$ 变化的关系。图中的蓝线代表理论结果，红色圆点表示仿真数据。

The plot in Figure 5 shows the on-axis scintillation index $\sigma_{Simulation}^2 = \langle I^2 \rangle / \langle I \rangle^2 - 1$ (the theoretical calculation using the Rytov approximation theory from Ref. [54] and the simulation results) as a function of propagation distance. The close match between the simulation results (red circles) and the theoretical curve (blue line) demonstrates the validity of the computer program for simulating beam propagation in PST.

6.1.2. Impact of Refractive Index Fluctuation Variance ($n_1^2$) on Image Quality (PSNR)

The impact of varying refractive index fluctuation variance ($n_1^2$) on the transmitted image quality is analyzed, keeping other parameters constant ($L_0 = 0.25 \mathrm{m}$, $\xi_x = 3$).

The following figure (Figure 6 from the original paper) illustrates the evolution of transmitted images under varying $n_1^2$:

该图像是传输图像在不同 PSNR 条件下的示意图，其中 (a) PSNR = Inf，(b) PSNR = 16.42，(c) PSNR = 13.19，(d) PSNR = 12.13，(e) PSNR = 11.57，(f) PSNR = 11.29。图中表现出随着 PSNR 值的下降，图像质量逐渐变差。

As shown in Figure 6, under very weak turbulence conditions (a) where $\left. n _ { 1 } ^ { 2 } \right. = 1 \times 10^{-22}$, the image is decoded without errors (PSNR = Inf, indicating a perfect match). As $\left. n _ { 1 } ^ { 2 } \right.$ increases, the quality of the received image gradually deteriorates. This is visually evident from the increasing blurriness and distortion from (b) to (f).

• Analysis: A larger value of $\left. n _ { 1 } ^ { 2 } \right.$ signifies greater differences among irregular structures in the PST, leading to stronger turbulence effects. This stronger turbulence causes more severe distortion to the optical beam, resulting in poorer image transmission performance and a decrease in PSNR. The PSNR values presented below each subfigure confirm this trend, decreasing from 16.42 dB to 11.29 dB as $n_1^2$ increases. This directly demonstrates that the variance of refractive index fluctuations is a critical factor determining image fidelity in PST.

6.1.3. Impact of Outer Scale ($L_0$) on Image Quality (PSNR)

The relationship between PSNR and the outer scale ($L_0$) for different values of refractive index fluctuation variance ($n_1^2$) is investigated, with $\xi_x = 3$.

The following figure (Figure 7 from the original paper) shows the variation of PSNR with $L_0$:

该图像是一个三维柱状图，展示了在不同的 $L_0$ 值下，各种 $_{1}^{2}$ 值对 PSNR 的影响。图中显示的 PSNR（dB）随 $L_0$（m）变化，并以不同的颜色表示 $_{1}^{2}$ 的不同取值，揭示了参数变化对信号质量的影响。

• Analysis: Figure 7 shows that as $L_0$ increases, the PSNR generally increases, indicating improved image quality. Conversely, as $L_0$ decreases, the quality of the beam degrades, leading to a reduction in PSNR.
• Influence of $L_0$ on PSNR: The influence ofL_0on PSNR is more pronounced whenn_1^2is relatively small (i.e., under weaker turbulence). When $L_0$ is small, the modulation of the beam spot by turbulent vortices becomes stronger, resulting in a more uneven wavefront. This leads to increased decoding errors and a reduction in PSNR.
• Physical Explanation: When $L_0$ approaches the thickness of the flow field, the impact of turbulent vortices on the propagation of the beam becomes weaker because larger-scale eddies (which $L_0$ represents) have less impact on beam spreading compared to smaller scales when they are constrained. However, if $L_0$ is too small, it means the dominant energy-containing eddies are small, leading to stronger local phase distortions.

6.1.4. Impact of Anisotropy Parameter ($\xi_x$) on Image Quality (PSNR)

The variation of PSNR with the anisotropy parameter ($\xi_x$) is analyzed for different values of refractive index fluctuation variance ($n_1^2$), with $L_0 = 0.25 \mathrm{m}$.

The following figure (Figure 8 from the original paper) illustrates the variation of PSNR with $\xi_x$:

该图像是三维柱状图，展示了不同 $<n_1^2>$ 值下的 PSNR 和 $\xi_{x}$ 之间的关系。随着 $<n_1^2>$ 值的增加，PSNR 有所变化，具体数据和趋势通过色彩深浅体现。

• Analysis: Figure 8 indicates that as $\xi_x$ increases, the PSNR generally increases. This means that enhancing the anisotropic characteristics of PST by increasing $\xi_x$ can mitigate turbulence effects and improve beam transmission performance.

• Influence of $\xi_x$ on PSNR: Similar to $L_0$, the impact of\xi_xon PSNR is more pronounced whenn_1^2is relatively small.

• Physical Explanation: An increase in $\xi_x$ corresponds to irregular stretching and compression of vortices perpendicular to the propagation direction. This stretching effectively reduces the scattering effect of turbulent vortices on the beam. When $\xi_x$ increases, the anisotropic turbulent vortices act as lenses with longer curvature radii, causing less beam deviation from the propagation direction. However, when $\xi_x$ becomes significantly larger than\xi_y, its impact on PSNR becomes `negligible`, suggesting a saturation point for this effect. ### 6.1.5. Impact of PST Parameters on Bit Error Rate (BER) The `BER performance` of the `FSO link` is analyzed as a function of $n_1^2$, $L_0$, and $\xi_x$. The `Rytov variance` ($\sigma_B^2$) is used to classify turbulence conditions: $\sigma_B^2 \ge 1$ for `strong turbulence` and $\sigma_B^2 < 1$ for `weak turbulence`. The following figure (Figure 9 from the original paper) shows the relationship between BER and $L_0$ under different values of $n_1^2$:  *该图像是图表，展示了在不同 $ _1^2$ 值下，$L_0$ 和 `ext{log}(BER)` 之间的关系。随着 $L_0$ 的变化，BER 的对数值显示出明显的下降趋势，体现了湍流对信号传输的影响。* The following figure (Figure 10 from the original paper) shows the relationship between BER and $\xi_x$ under different values of $n_1^2$:  *该图像是图表，展示了不同 $<n_{1}^{2}>$ 值下，误比特率（BER）与 $\xi_{x}$ 的关系。可以看到，随着 $\xi_{x}$ 的增加，BER 的对数值逐渐降低，反映出边界条件对传输性能的影响。* * **Analysis:** * **Impact of $n_1^2$**: Both Figure 9 and Figure 10 show that `BER increases with`n_1^2. This is consistent with the PSNR analysis: stronger refractive index fluctuations (higher $n_1^2$) lead to more severe turbulence, causing more errors in bit transmission. The BER can reach 0.447 under strong turbulence conditions.

  • Impact of $L_0$: Figure 9 clearly shows that BER decreases with increasingL_0. As $L_0$ increases, the influence of turbulent vortices on beam propagation `weakens`, leading to a `sharp decline in BER`. This is because larger outer scales imply larger, less disruptive eddies relative to the beam size, reducing scattering. * **Impact of $\xi_x$**: Figure 10 demonstrates that `BER decreases as`\xi_x`increases`. Similar to PSNR, increasing $\xi_x$ enhances the beam's resistance to turbulence, reducing scattering and thus errors. However, the effect of $\xi_x$ on BER `becomes negligible when`\xi_x`is much larger than`\xi_y, indicating a saturation of the benefit.
  • Combined Effects: The impact of bothL_0and\xi_xon BER is more significant with smaller values ofn_1^2 (i.e., under weaker turbulence). For instance, when $L_0 = 0.4 \mathrm{m}$ (likely for a lower $n_1^2$ value, though not explicitly stated for this specific point), the BER can be as low as $1.25 \times 10^{-4}$. * **Conclusion for BER:** To improve `FSO link performance` in `PST environments`, it is crucial to `minimize drastic variations in PST parameters` (i.e., reduce $n_1^2$) and `optimize parameters` like $L_0$ and $\xi_x$ to reduce turbulence effects. ## 6.2. High-Resolution and High Bit-Depth Requirements The paper also addresses the practical demands of `high-resolution` and `high bit-depth` transmission. * **Resolution Scaling:** Increasing resolution from `256 × 256` to `1024 × 1024` pixels results in a `16-fold increase in pixel count`. With a `per-pixel processing time of 0.5 µs`, the image encoding/decoding duration scales linearly from `32.768 ms` to `524.288 ms`. * **High Bit-Depth:** To support higher bit-depth, the study implements a `wavelength-extension approach`. For example, the red channel uses `dual wavelengths` ($\lambda_{11}$ and $\lambda_{12}$) to encode the `upper and lower 8-bit data` respectively, achieving `16-bit depth per channel`. This allows for `48-bit color transmission` across the three RGB channels. * **Transmission Rate:** Coupled with a `360 Hz refresh-rate spatial light modulator`, a `256 × 256` image transmission rate reaches `8640 bits/s`, representing a `threefold enhancement` over conventional schemes (2880 bits/s) mentioned in Ref. [15]. This significantly improves `real-time performance`. ## 6.3. Comparison with Conventional and Prior OAM Schemes The proposed method demonstrates superior performance compared to conventional `QAM`, `PSK`, and prior `OAM-based encoding schemes` in `channel capacity`, `BER`, and `turbulence resilience`. * **Channel Capacity:** * `64-QAM` suffers from `limited symbol space` and `spectral efficiency`. * `PSK` faces `modulation-order constraints`. * `Prior OAM implementations` are `restricted by turbulence-induced mode degradation`. * The proposed hybrid `WDM and MDM architecture` achieves `24 bits per pixel transmission`, enabling `256-ary modulation`. This significantly `outperforms 16-ary systems` and ensures `distortion-free reconstruction` of `256 × 256-pixel color images`, effectively overcoming `channel capacity limitations`. * **BER Performance:** * `64-QAM` requires `38.5 dB OSNR` with `turbulence-sensitive BER thresholds`. * `PSK` suffers from `phase-fluctuation-induced error floors`. * `Prior OAM schemes` degrade to $BER ≈ 10⁻¹$ under `strong turbulence`. * The proposed system, by integrating `mode-matching decoding`, maintains a `BER of 0.447 in intense turbulence` and achieves a `BER of`1.25 \times 10^{-4}`under mild turbulence`. Through `outer-scale optimization`, the BER can be further reduced to the 10^{-5}level, representing an improvement of two orders of magnitude compared to the residuals after Turbo coding ($4.01 \times 10^{-4}$).

• Turbulence Resilience:

  • The proposed method addresses modal mismatch and wavefront distortion. Its dynamically tuned outer-scale strategy sustains $PSNR > 20 dB$ at a refractive index fluctuation variance of1 \times 10^{-20}. * This `outperforms QAM` (which requires complex `DSP compensation`) and `PSK` (which incurs a `14 dB SNR penalty`). * Leveraging `OAM's inherent resistance to wavefront perturbations`, the method establishes a new paradigm for `reliable optical communication in complex environments`. # 7. Conclusion & Reflections ## 7.1. Conclusion Summary This study successfully conducted a numerical investigation into `color image transmission` through a `plasma sheath turbulence (PST)` channel. A detailed simulation model encompassing `pixel encoding`, `transmission`, and `decoding` processes was presented. The model leverages `orbital angular momentum (OAM)` modes of `Gaussian vortex beams` for encoding and a `mode-matching method` for decoding. The key findings demonstrate the significant influence of `PST parameters`—specifically, `refractive index fluctuation variance` ($n_1^2$), `outer scale` ($L_0$), and `anisotropy parameters` ($\xi_x$)—on `image transmission performance`. The results consistently show that `PSNR` increases and `BER` decreases with `decreasing`n_1^2, increasingL_0, andincreasing $\xi_x$. The most pronounced effects of $L_0$ and $\xi_x$ were observed under weaker turbulence conditions (smaller $n_1^2$). While a high BER (0.447) was observed under strong turbulence, the system achieved a respectable $1.25 \times 10^{-4}$ BER under weak turbulence, showcasing its potential. The successful reconstruction of a 256 × 256-pixel color image validates the feasibility and effectiveness of the proposed encoding and decoding scheme. The method also shows superior performance in channel capacity, BER, and turbulence resilience compared to conventional and prior OAM-based schemes.

7.2. Limitations & Future Work

The authors highlight several implications for improving FSO link performance in PST environments:

• Minimizing $n_1^2$: A reduction in $n_1^2$ decreases the irregular structural differences within the PST, weakening turbulence activity.

• Increasing $L_0$: An increase in $L_0$ causes the turbulent vortices to contract (or rather, the dominant scale of turbulent eddies to become larger relative to the beam, thereby reducing beam energy dissipation), making the beam less susceptible to small-scale distortions.

• Increasing $\xi_x$: Increasing $\xi_x$ induces lateral stretching of the beam, enhancing its resistance to turbulence and reducing the scattering effect of turbulence on the beam.

• Saturation of $\xi_x$ effect: The influence of $\xi_x$ on PSNR and BER becomes negligible when $\xi_x$ is much larger than $\xi_y$, suggesting a limit to the benefits of increasing anisotropy in one dimension.

  While the paper demonstrates numerical feasibility, it is a simulation study. Practical implementation would face additional engineering challenges not fully explored, such as:

• Real-time adaptation: The dynamically tuned outer-scale strategy mentioned is key, but the mechanisms for real-time sensing of PST parameters and adaptive optimization are not detailed.

• Hardware limitations: The practical realization of high-order OAM mode generation, detection, and precise mode-matching in a dynamic, turbulent environment presents significant hardware and computational challenges.

• Other atmospheric effects: While PST is the focus, other atmospheric factors (e.g., absorption, scattering by particles) are simplified into a fixed power loss coefficient $\rho$, which may not fully capture their dynamic effects.

  Future work could focus on:

• Experimental validation: Translating the simulation results into a laboratory or field experiment to confirm the theoretical and numerical predictions.

• Adaptive optics integration: Exploring advanced adaptive optics techniques to actively compensate for PST-induced wavefront distortions in conjunction with the OAM encoding.

• Higher-order OAM and larger image sizes: Scaling the system to even higher topological charges and larger image resolutions, considering the associated computational complexities and potential mode crosstalk.

• Optimization of encoding/decoding algorithms: Further refining the mode-matching algorithm or exploring machine learning-based decoding for enhanced robustness.

7.3. Personal Insights & Critique

This paper provides a timely and relevant contribution to FSO communication, especially for aerospace applications where PST is a critical concern. The utilization of OAM for color image transmission is an elegant approach that leverages the fundamental physics of light to address a complex engineering problem.

One key insight is the detailed quantification of how specific PST parameters (variance, outer scale, anisotropy) individually affect image quality and BER. This level of analysis is invaluable for system designers who need to understand the operating limits and potential mitigation strategies in such extreme environments. The finding that PSNR and BER are significantly influenced by these parameters, and that optimizing $L_0$ and $\xi_x$ can substantially improve performance, provides clear guidance for engineering robust FSO links.

A potential area for deeper exploration or an unverified assumption lies in the mode-matching method itself. While powerful, the effectiveness of correlation-based mode identification can be highly sensitive to the signal-to-noise ratio and the degree of mode crosstalk introduced by severe turbulence. The paper mentions OAM's inherent resistance to wavefront perturbations, but in strong turbulence, mode scattering and power redistribution among modes can be substantial, making perfect correlation challenging. Further investigation into the robustness of mode-matching under various crosstalk scenarios would be beneficial.

Another aspect to consider is the computational complexity. While the paper touches upon the scaling for higher resolution and bit-depth, the real-time processing demands for generating and detecting multiple OAM modes, especially with adaptive strategies, can be substantial. The 0.5 µs per-pixel processing time for encoding/decoding is promising, but the overall latency and throughput in a full system incorporating turbulence compensation would need careful consideration.

Overall, the paper successfully demonstrates a feasible pathway for high-fidelity image transmission in challenging PST environments, making its methods and conclusions highly transferable to other FSO scenarios experiencing strong turbulence, such as underwater optical communication or long-distance atmospheric links. It inspires further research into hybrid multiplexing schemes and adaptive techniques to harness the full potential of OAM in real-world communication systems.