Method for Suppressing Scintillation in Up-Link Optical Communication Using Optical Pin-like Beams Propagating Through Atmospheric Turbulence

Abstract

Free space optical communication (FSOC) systems operating in the space–atmosphere channel are susceptible to severe turbulence-induced scintillation, particularly in up-link configurations where the adaptive optics (AO) pre-correction becomes ineffective due to anisoplanatic constraints. This study presents a novel scintillation suppression strategy utilizing self-focusing optical pin-like beams (OPBs) with tailored phase modulation, combining theoretical derivation and numerical simulation. It is found that increasing the shape factor γ and modulation depth C elevates the average received power and reduces the scintillation index at the focal point. Meanwhile, quantitative evaluation of the five OPB configurations shows that the parameter set γ = 1.4 and C = 7 × 10⁻⁵ gives a peak scintillation suppression efficiency. It shows that turbulence induced scintillation is suppressed by 44% with the turbulence intensity D/r₀ = 10, demonstrating exceptional effectiveness in up-link transmission. The findings demonstrate that OPB with optimized γ and C establish an approach for uplink FSOC, which is achieved through suppressed scintillation and stabilized power reception.

1. Introduction

For decades, the growing demand for faster and more efficient data transmission has driven significant advancements in free-space optical communication (FSOC) [1,2,3]. FSOC has expanded into multiple application domains, such as satellite laser communication [4], satellite ground laser communication [5], and unmanned aerial vehicle communication [6], etc. However, atmospheric turbulence can severely degrade beam transmission by distorting both the power distribution and wavefront shape, affecting the communication efficiency of laser-based systems [7,8,9]. In particular, when propagating through turbulent channels, the Gaussian beam (GB) is highly susceptible to intensity scintillation [10]. In FSOC, down-link systems leverage adaptive optics (AO) to mitigate atmospheric turbulence effectively, due to their optical path enabling precise ground-based wavefront correction [11]. The up-link systems face inherent limitations. Because the point-ahead angle often exceeds the isoplanatic angle, which results in anisoplanatic effects. These effects render AO precorrection ineffective [12]. While laser guide stars enhance AO pre-correction capabilities, their performance is constrained by the limited correction field imposed by isoplanatic angle restrictions, inherent luminance limitations, and reduced sensitivity to oblique wavefront aberrations [13]. The multi-guide-star configurations have been proposed to address these issues, but such approaches introduce higher system complexity and significant cost challenges. Spatial diversity technology has also been used to suppress optical intensity fluctuations [14], but it requires complex hardware and signal processing algorithms. Therefore, it is necessary to develop efficient methods to mitigate intensity scintillation, which remains a critical and challenge for up-link FSOC systems.

The optical pin-like beam (OPB), which is first proposed by Zhang et al. in 2019 [15], represents a novel beam profile with unique propagation characteristics. It is proven that the OPB exhibits remarkable turbulence suppression capabilities while maintaining stable propagation over kilometer-scale distances [16]. Beyond optical communications [17,18,19,20], OPBs have found applications in diverse fields, including optical trapping and particle manipulation [21,22], as well as nonlinear optics [23]. Previous research has primarily focused on analyzing far-field intensity distributions and energy fluctuations of OPB, while the application on scintillation suppression has not enough. Although a few studies have explored the impact of OPB on intensity scintillation [24], their scope has been limited to a single OPB configuration. Therefore, the correlation between beam parameters and scintillation characteristics has not been explored. In this study, we systematically investigate the scintillation suppression characteristics of OPB in atmospheric turbulence through theoretical derivation and numerical simulations. Based on the wave equation, an analytical expression for the focal length of OPB has been derived. This formula contributes to the design of a set of OPBs with identical focal lengths and different beam parameters. Through spatial transmission simulations, we analyze the propagation characteristics, scintillation index (SI), and average received power (ARP) stability in different turbulent channels. The results indicate that optimized OPB significantly suppress turbulence-induced intensity scintillation, a capability particularly advantageous in up-link communication scenarios. This suppression capability effectively contributes to the stable transmission of the beam in FSOC.

2. Theory and Method

The wave equation is the fundamental equation, which describes electromagnetic waves. In cylindrical coordinates, the solution to the wave equation is

$$u\left(r,\varnothing ,z\right)={\displaystyle \frac{k{e}^{ikz}}{i2\pi z}}{\int }_0^r{\int }_0^{2\pi }{u}_0(\rho ,\varnothing ,0)e^{ik{\displaystyle \frac{r^2+{\rho }^2-2\rho rcos(\varnothing -\theta )}{2z}}}\rho d\rho d\theta .$$

(1)

The wavenumber is defined as $k=2\pi /\lambda$, where λ represents the wavelength. $\rho$ represents the radial distance in the polar coordinate system defined on the plane perpendicular to the optical axis. The i represents the imaginary unit. And the radial r and azimuthal $\mathrm{\varnothing }$ coordinates, along with the longitudinal distance z, form the spatial coordinate system. The initial wavefront at $z=0$ is expressed as $u_0(\rho ,\varnothing ,0)=A(\rho )exp[ikC{({\displaystyle \frac{\rho }{{\omega }_{\rho }}})}^{\gamma }]$, where $A\left(\rho \right)$ is amplitude function, ρ is the polar diameter of the polar coordinate system perpendicular to the optical axis direction, ${\omega }_{\rho }$ is the phase normalization coefficient, C (dimensionless) is modulation depth, and γ is shape factor. Subsequent studies show that the modulation depth C determines the magnitude of the wavefront phase perturbations, while the shape factor γ controls the rate at which the beam forms. Together, these parameters influence the propagation characteristics of the optical beam through turbulent media. Using the Gaussian amplitude $e^{-{\displaystyle \frac{{\rho }^2}{{\omega }_0}}}$ as the initial $A\left(\rho \right)$, substituting the polar coordinate amplitude function $A\left(\rho \right)$ into u₀ and evaluating the Fresnel integral (1) along the optical axis (r = 0) gives the on-axis light field distribution in Equation (2):

$$u(0,z)={\displaystyle \frac{k}{iz}}{e}^{ikz}{\int }_0^{\infty }{e}^{-{\displaystyle \frac{{\rho }^2}{{\omega }_0}}}{e}^{{ik[(C{\displaystyle \frac{\rho }{{\omega }_{\rho }}})}^{\gamma }+{\displaystyle \frac{{\rho }^2}{2z}}]}\rho d\rho .$$

(2)

Due to the axial symmetry of the input optical field, the integral is evaluated in one-dimensional polar coordinates (${\omega }_{\rho }=1$) using the phase stabilization method. The phase term of the integrand is

$$\varphi \left(z\right)=C{\left(\rho \right)}^{\gamma }+{\displaystyle \frac{{\rho }^2}{2z}}.$$

(3)

An equilibrium point satisfies

$${\displaystyle \frac{d\varphi }{d{\rho }_0}}=0.$$

(4)

This yields the equilibrium point:

$${\rho }_0={\left(\right|C\left|\gamma z\right)}^{1/(2-\gamma )}.$$

(5)

The second derivative at ${\rho }_0$ is

$${\varphi }^{″}\left({\rho }_0\right)={\displaystyle \frac{2-\gamma }{z}}.$$

(6)

Since ${\varphi }^{″}\left({\rho }_0\right)>0$, we introduce the phase factor $e^{i{\displaystyle \frac{\pi }{4}}}$. Applying the phase stabilization formula gives

$${\int }_0^{\infty }{e}^{-{\displaystyle \frac{{\rho }^2}{{\omega }_0}}}{e}^{ik\varphi \left(\rho \right)}\rho d\rho \approx e^{-{\displaystyle \frac{{\rho }^2}{{\omega }_0}}}{\rho }_0\sqrt{{\displaystyle \frac{2\pi z}{k\left(2-\gamma \right)}}}{e}^{ik\varphi \left({\rho }_0\right)+{\displaystyle \frac{i\pi }{4}}}.$$

(7)

Here, $\varphi \left({\rho }_0\right)$ is defined as

$$\varphi \left({\rho }_0\right)={\displaystyle \frac{\left|C\right|\gamma }{2-\gamma }}{\left(\left|C\right|\gamma z\right)}^{{\displaystyle \frac{\gamma }{2-\gamma }}}+{\displaystyle \frac{{\rho }_0^2}{2z}}.$$

(8)

It should be noted that this formula serves as a simplified representation to facilitate focal position calculations and does not accurately reflect the actual optical field distribution.

The complex amplitude distribution on the optical axis is

$$u\left(0,z\right)={\displaystyle \frac{k}{iz}}{e}^{ikz}{e}^{-{\displaystyle \frac{{\rho }^2}{{\omega }_0}}}{\rho }_0\sqrt{{\displaystyle \frac{2\pi z}{k\left(2-\gamma \right)}}}{e}^{ik\varphi \left({\rho }_0\right)+{\displaystyle \frac{i\pi }{4}}}.$$

(9)

This holds for $0<\gamma <2,C>0,z>0$. The expression of light intensity distribution on the final optical axis is

$$I\left(0,z\right)={\left|u\left(0,z\right)\right|}^2={\displaystyle \frac{2\pi k}{\left(z-\gamma \right)z}}{\left(\left|C\right|\gamma z\right)}^{{\displaystyle \frac{2}{2-\gamma }}}{e}^{-{\displaystyle \frac{2}{{\omega }_0^2}}{\left(\left|C\right|\gamma z\right)}^{{\displaystyle \frac{2}{2-\gamma }}}}.$$

(10)

To find the intensity maximum, we compute the first derivative:

$${\displaystyle \frac{dI}{dz}}={\displaystyle \frac{2\pi k}{{(2-\gamma )}^2}}{\left(\left|C\right|\gamma \right)}^{{\displaystyle \frac{2}{2-\gamma }}}{e}^{-{\displaystyle \frac{2}{{\omega }_0^2}}{\left(\left|C\right|\gamma z\right)}^{{\displaystyle \frac{2}{2-\gamma }}}}{z}^{\left({\displaystyle \frac{2\gamma -2}{2-\gamma }}\right)}\left[\gamma -{{\displaystyle \frac{4}{{\omega }_0^2}}\left(\left|C\right|\gamma z\right)}^{{\displaystyle \frac{2}{2-\gamma }}}\right].$$

(11)

Because z ≠ 0, $z_{peakintensity}$ denotes the position of the peak intensity on the optical axis, which is

$$z_{peakintensity}={\left({\displaystyle \frac{\gamma {\omega }_0^2}{4}}\right)}^{{\displaystyle \frac{2-\gamma }{2}}}\left(\left|C\right|\gamma \right),\mathrm{with}0<\gamma <2,C>0,z>0.$$

(12)

Equation (12) shows that the peak intensity position of the OPB depends on the shape factor γ, waist radius ω₀, and modulation depth C. This enables generating different OPB with identical peak intensity positions by adjusting γ, ω₀, and C.

The analytical model (Equations (1)–(12)) characterizes OPB propagation in vacuum. In the context of weak fluctuation conditions that satisfy the Rytov approximation, atmospheric turbulence is incorporated via the stochastic phase perturbation ψ(r,z) in $u(r,z)=u_0\left(r,z\right)e^{\psi \left(r,z\right)}$. Here, $u_0\left(r,z\right)$ denotes the field in the absence of turbulence, and $\psi \left(r,z\right)$ represents the perturbation to the turbulent phase. While turbulence distorts the wavefront and shifts the focal position, it does not redefine the fundamental focusing mechanism governed by γ and C. The scintillation index ${\sigma }_I^2$ is defined as the normalized variance of irradiance fluctuations [25], which is given by

$${\sigma }_I^2={\displaystyle \frac{〈I_S^2〉}{{〈I_S〉}^2}}-1.$$

(13)

Here, the symbol $I_S$ denotes the received power in the receiving aperture. OPB has the capability for stable energy over transmission distances, so it inherently resists atmospheric turbulence. These characteristics hold significant research value for FSOC. This study investigates the scintillation behavior of OPB under varying turbulence conditions and optimizes the system design for practical applications. The experiment utilized a rectangular sampling screen with a length of 559.8 mm, and the number of sampling points was set to N = 1024. The 1550 Nm Gaussian beam with a waist radius ${\omega }_0=15\mathrm{m}\mathrm{m}$ is generated and collimated. The beam has been modulated in OPB phase using either a spatial light modulator or a phase mask. Following modulation, the OPB is expanded by a factor of 9.33 through an expansion system before transmitting into space. At the receiver, an aperture-restricted screen captures the far-field spot to analyze scintillation effects (Figure 1). The experiments compare the scintillation performance of OPB and conventional Gaussian beams (GB) under the same turbulence conditions. By simulating atmospheric turbulence and calculating the ARP and SI, this work proposed a method to design the OPB to suppress light intensity scintillation. The results demonstrate that OPB exhibits superior stability in scenarios with turbulence, offering a viable solution for enhancing the reliability of FSOC systems.

The turbulence model is the Kolmogorov turbulence theory, which ${\varphi }_n\left(k\right)$ shows

$${\varphi }_n\left(k\right)=0.033C_n^2{k}^{-{\displaystyle \frac{11}{3}}},$$

(14)

where the $C_n^2$ denotes the refractive index structure parameter and the $r_0$ is the atmospheric coherence length, in accordance with the formula: $r_0={\left[0.423k^2{C}_n^{2}L\right]}^{-{\displaystyle \frac{5}{3}}}$. The L, D denote transmission distance and telescope aperture, respectively. $k$ represents wave vector and $k={\displaystyle \frac{2\pi }{\lambda }}$. In addition, D/r₀ can be calculated using the above formula and also serves as an indicator of turbulence intensity. To more accurately characterize the effect of atmospheric turbulence on the beam, 300 sets of random turbulent wavefronts were used to represent the same turbulence intensity when calculating each set of SI and ARP. The detailed analysis of SI and ARP will be presented in the following section to quantify these effects.

3. Numerical Simulation and Analysis

As demonstrated in Section 2, the theoretical analysis shows numerous OPB types at a given focal distance. This section aims to explore how adjusting beam parameters enables OPBs to achieve optimal scintillation suppression effectiveness. The numerical simulations will be employed to demonstrate and compare their scintillation suppression capabilities. Firstly, we systematically investigate the focal distance of the OPB under two distinct parametric configurations: (1) with a fixed beam modulation depth C = 0.00005, varying the shape factor γ (0.3–1.2, step 0.1) and beam waist radius ${\omega }_0$∈{40 mm, 80 mm, 120 mm}; and (2) with a fixed ${\omega }_0$ = 70 mm, scanning γ over the same range while adjusting the modulation depth C ∈ {0.00006, 0.00008, 0.0001}. Locate the position of peak intensity using the spatial transmission simulation method without turbulence and compare it with the value calculated from Equation (12). The simulation results are presented in Figure 2. Comparative analysis of the focal positions calculated by both methods shows a maximum deviation of less than 200 m for the OPB. This consistency validates the formula-based approach as a reliable method for designing OPB in practical applications.

In order to examine the distinct propagation patterns, the comparative analysis was conducted of OPB and GB under both turbulence and turbulence-free conditions. The OPB phase used is shown in Figure 3. Far-field spot distributions for all four configurations were measured at 400 m intervals under turbulence conditions (D/r₀ = 5), with results illustrated in Figure 4 and Figure 5.

The oscillating white curve traces the on-axis normalized intensity I/I₀ versus propagation distance, capturing turbulence-induced signal fluctuations along the beam path. It is exhibited that atmospheric turbulence disrupts OPB self-focusing, inducing premature peak intensity extrema along the propagation axis (900 m earlier than transmission without turbulence conditions). In contrast, GB display distinct propagation dynamics. They sustain continuous divergence profiles and avoid forming focal points during transmission (Figure 4c). However, turbulent environments induce intensity fluctuations. This phenomenon directly accounts for the apparent quasi-focal behavior observed in the statistical ARP of GB, originating from turbulence-induced phase distortion effects rather than intrinsic self-focusing mechanisms.

In order to investigate the selection of OPB with the same focal length in practical applications, a study was conducted in which five types of OPB with a focal length of 1500 m were designed and their far-field scintillation characteristics were studied. Firstly, the five designed OPB with a focal length of 1500 m are propagated together with a GB under four turbulence conditions: D/r₀ = 5, D/r₀ = 10, D/r₀ = 15, and D/r₀ = 20. Data are collected at 50 m intervals along the propagation path. The receiving aperture is 50 mm. For a more intuitive analysis of the SI values of the OPB and the GB at the focal point (specifically, at 1500 m), Figure 6 is presented.

The outcomes reveal that the OPB, characterized by maximum modulation depth and shape factor, exhibits the least intense scintillation across various turbulence scenarios. Notably, as turbulence intensity increases, the scintillation index at the focal point does not continually escalate. This phenomenon is primarily due to turbulence impairing the OPB′s self-focusing property. Consequently, the focal point emerges earlier, and the entire self-focusing region shifts anteriorly. Within the self-focusing regime, the scintillation index of the OPB initially decreases and then increases. Therefore, the greater the turbulence, the greater the reduction in the scintillation index. In reality, as turbulence intensifies, the scintillation index increases overall along the entire propagation trajectory.

In the entire propagation path, SI is calculated shown in Figure 7. The OPB with a focal length of 1500 m shows superior performance compared to the Gaussian beams within a reception range of 2000 m, as their scintillation factor is consistently lower in most cases. The SI along the propagation path shows a characteristic trend of increasing and then decreasing. In particular, the far-field SI of the OPB is minimized under conditions where the shape factor γ and the modulation depth C are maximized in the transmission path. Furthermore, the scintillation factor increases more rapidly with increasing turbulence intensity. Particularly, the minimum scintillation factor is observed after the beam passes through the focal point, as shown by the lowest SI at 2000 m. In addition to the SI, the ARP is a critical parameter for investigating the far-field characteristics of beams. Therefore, the ARP of the six beam types (five OPBs and one Gaussian beam) was calculated across the entire propagation path under the four turbulence conditions (D/r₀ = 5, D/r₀ = 10, D/r₀ = 15, and D/r₀ = 20). The results are presented in Figure 8. Compared to the Gaussian beam, the OPB with a focal length of 1500 m consistently shows a higher ARP within the 2000 m reception range. The ARP of the OPB shows a characteristic trend of increasing and then decreasing along the propagation path. In particular, the maximum ARP is observed between 200 and 600 m. Furthermore, as the turbulence intensity increases, the position of the maximum ARP shifts towards shorter distances. It is also observed that larger values of shape factor γ and modulation depth C result in the highest far-field ARP for the OPB. Therefore, it is essential to consider the shape factor γ and modulation depth C as critical parameters when designing OPB with the same focal length. In particular, larger values of shape factor γ and modulation depth C result in higher ARP over the entire propagation path and lower SI at the focal point.

To further investigate the OPB′s performance under long-range propagation conditions (2–10 km), we conducted additional simulations by adjusting the beam’s focal parameters (γ = 0.8, C = 0.000049) while maintaining consistent system configurations. The result is in Figure 9.

The results demonstrate that the OPB can maintain relatively low scintillation levels over an extended range, with the scintillation index exhibiting a characteristic pattern of initial increase followed by reduction and subsequent rise at longer distances. So, OPB has great potential for applications in suppressing intensity scintillation over longer distances. To further evaluate OPB′s performance in long-haul transmission, the link distance was extended to 15 km. With the transmitter aperture held constant, the receiver aperture was increased to 70 mm and the sampling screen expanded to 2099.25 mm, implementing a sampling grid of N = 1024 × 1024 points. Scintillation indices and average received power were computed at 20 equidistant points spanning 0–15 km (750 m intervals), as shown in Figure 10.

Within the 0–15 km range, OPB exhibits notable scintillation suppression efficacy and enhanced average received power. For uplink satellite-to-ground optical communications, where atmospheric turbulence predominantly occurs at altitudes of 15–20 km, the transmission medium transitions to a vacuum beyond this turbulent layer. Consequently, OPB has considerable application potential in space–ground links thanks to its simplified modulation architecture and superior turbulence suppression capability.

4. Conclusions

This study demonstrates a self-focusing OPB for FSOC systems, demonstrating its efficacy in suppressing turbulence-induced intensity scintillation by numerical simulations. Comparative analysis shows that the OPB has superior ARP over conventional GB and lower SI throughout the propagation path. Further investigation indicates a unique distance-dependent characteristic: While the ARP of OPB temporarily drops below GB levels at intermediate distances (prior to achieving stabilization), its SI remains consistently suppressed compared to GB. These results underscore the critical importance of optimizing modulation depth C and shape factor γ for effective scintillation suppression. Through comprehensive statistical evaluation of five OPB configurations, the parameter γ = 1.4 and C = 0.00007 was identified as optimal. At a focal distance of 1500 m and under turbulence D/r₀ = 10, this configuration reduced the SI degradation by 44% compared to the Gaussian beam case. The demonstrated capability of OPB to mitigate intensity fluctuations offers a viable approach for improving FSOC reliability. To realize this potential, it is essential to systematically optimize the parameters of OPB. Future studies should focus on experimental validation and multi-parameter co-design frameworks for harnessing its full potential.