A. Free-Space Optical Communication Channel Model

In typical line-of-sight FSOC channels, atmospheric turbulence vortices of varying scales induce diffraction and scattering effects on propagating beams, resulting in random intensity fluctuations at the receiver plane. Severe intensity scintillation causes signal attenuation, reduced signal-to-noise ratio (SNR), increased bit error rate (BER), and even communication interruptions. The scintillation index (SI) is defined as the normalized variance of received intensity:

View Source \begin{equation*} \delta _I^2 = \frac{{\left\langle {{I^2}} \right\rangle - {{\left\langle I \right\rangle }^2}}}{{{{\left\langle I \right\rangle }^2}}} \tag{1} \end{equation*}

Where I denote the received optical intensity, and <·> represents statistical averaging. Extensive studies have established statistical models for intensity fluctuations under different turbulence conditions [21], [22], [23]. While the log-normal distribution describes weak turbulence, the Gamma-Gamma model proposed by Andrews et al. [23] effectively characterizes intensity scintillation across weak-to-strong turbulence regimes:

View Source \begin{equation*} p(I) = \frac{{2{{(\alpha \beta )}^{\frac{{\alpha + \beta }}{2}}}}}{{\Gamma (\alpha )\Gamma (\beta )}}{(I)^{\frac{{\alpha + \beta }}{2} - 1}}{K_{\alpha - \beta }}(2\sqrt {\alpha \beta I} ),I \geq 0 \tag{2} \end{equation*}

In (2), K_n() represents the modified Bessel function of the second kind of order n. For plane wave approximation (valid in long-distance propagation), the large-scale α and small-scale β turbulence parameters satisfy:

View Source \begin{equation*} \begin{array}{l} \alpha = {\left\{ {\exp \left[\frac{{0.49\sigma _R^2}}{{{{(1 + 1.11\sigma _R^{12/5})}^{7/6}}}}\right] - 1} \right\}^{ - 1}}\\ \beta = {\left\{ {\exp \left[\frac{{0.51\sigma _R^2}}{{{{(1 + 0.69\sigma _R^{12/5})}^{5/6}}}}\right] - 1} \right\}^{ - 1}} \end{array} \tag{3} \end{equation*}

B. MISO System Scintillation Model

According to the additivity of the Gamma function, the sum of multiple random variables following the Gamma distribution also follows the Gamma distribution. The Gamma-Gamma model is extended to multi-aperture transmission systems. For M partially correlated sub-apertures, the combined intensity distribution becomes:

View Source \begin{equation*} p({I_s}) = \frac{{2{{\left( {\overline{\alpha} \overline{\beta} } \right)}^{\frac{{\overline{\alpha} + \overline{\beta} }}{2}}}}} {{\Gamma \left( {\overline{\alpha }} \right)\Gamma \left( {\overline{\beta} } \right)}} \left( {\frac{{{I_s}}}{{\overline{\Omega} }}} \right)^{\frac{\overline{\alpha} + {{\overline{\beta} }}} {2} - {1}} {K_{\overline{\alpha} - \overline{\beta} }}\left( {2\sqrt {\overline{\alpha} \overline{\beta} \frac{{{I_s}}}{{\overline{\Omega} }}} } \right) \tag{4} \end{equation*}

$$\begin{equation*}\begin{array}{l} \overline{\alpha} = h\alpha,\overline{\beta} = h\beta,\overline{\Omega} = M\Omega \\ h = M\left( {1 + \frac{2}{M}\sum\limits_{i = 1}^M {\sum\limits_{j = 1}^M {{\rho _{ij}}} } } \right) \end{array} \tag{5}\end{equation*}$$ View Source \begin{equation*}\begin{array}{l} \overline{\alpha} = h\alpha,\overline{\beta} = h\beta,\overline{\Omega} = M\Omega \\ h = M\left( {1 + \frac{2}{M}\sum\limits_{i = 1}^M {\sum\limits_{j = 1}^M {{\rho _{ij}}} } } \right) \end{array} \tag{5}\end{equation*}

In (5), Ω represents the expectation value of the received intensity, typically normalized to 1. The parameter h degenerates to 1 under ideal independent aperture conditions where turbulence-induced distortions and signal fading across sub-channels are statistically uncorrelated. However, in practical laser communication systems, the physical spacing between apertures cannot be arbitrarily large due to finite transmitter/receiver terminal dimensions (typically constrained by payload size in urban deployments). This spatial limitation necessitates explicit consideration of inter-aperture correlations. The correlation coefficient ρ_ij between the i-th and j-th apertures is defined as:

View Source \begin{equation*} {\rho _{ij}}({s_{ij}}) = \frac{{{B_{ij}}({s_{ij}})}}{{\sqrt {\delta _i^2\delta _j^2} }} \tag{6} \end{equation*}

$$\begin{align*} B(s) \simeq& \exp \left[ \begin{array}{l} \frac{{0.49\delta _R^2}}{{{{\left( {1 + 1.11\delta _R^{12/5}} \right)}^{7/6}}}}{}_1{F_1}\left( {\frac{7}{6};1; - \frac{{k{r^2}{\eta _\chi }}}{{4L}}} \right)\\ + \frac{{0.5}}{{{{\left( {1 + 0.69\delta _R^{12/5}} \right)}^{5/6}}}}{\left( {\frac{{k{r^2}{\eta _\gamma }}}{{4L}}} \right)^{\frac{{12}}{5}}}{K_{{5 \mathord{\left/ {\vphantom {5 6}} \right. } 6}}}\sqrt {\left( {\frac{{k{r^2}{\eta _\gamma }}}{L}} \right)} \end{array} \right]\\ {\eta _\chi } =& \frac{{2.61}}{{(1 + 1.11\delta _R^{12/5})}},{\eta _\gamma } = \frac{3}{{(1 + 0.69\delta _R^{12/5})}} \tag{7} \end{align*}$$ View Source \begin{align*} B(s) \simeq& \exp \left[ \begin{array}{l} \frac{{0.49\delta _R^2}}{{{{\left( {1 + 1.11\delta _R^{12/5}} \right)}^{7/6}}}}{}_1{F_1}\left( {\frac{7}{6};1; - \frac{{k{r^2}{\eta _\chi }}}{{4L}}} \right)\\ + \frac{{0.5}}{{{{\left( {1 + 0.69\delta _R^{12/5}} \right)}^{5/6}}}}{\left( {\frac{{k{r^2}{\eta _\gamma }}}{{4L}}} \right)^{\frac{{12}}{5}}}{K_{{5 \mathord{\left/ {\vphantom {5 6}} \right. } 6}}}\sqrt {\left( {\frac{{k{r^2}{\eta _\gamma }}}{L}} \right)} \end{array} \right]\\ {\eta _\chi } =& \frac{{2.61}}{{(1 + 1.11\delta _R^{12/5})}},{\eta _\gamma } = \frac{3}{{(1 + 0.69\delta _R^{12/5})}} \tag{7} \end{align*}

In (7), ${}_1{F_1}( \cdot )$ denotes the confluent hypergeometric function.

The analysis of the aforementioned model reveals that the spatial arrangement of the optical antenna array fundamentally governs the inter-aperture separation distances s_ij. This study adopts a uniform circular array (UCA) configuration at the transmitter, where multiple beams are symmetrically distributed along a concentric circle with radius R = 10 cm. Each sub-aperture maintains an individual radius of r = 2 cm, with the number of transmitters varied as M = 1, 2, 3, 4, 8 (see Fig. 1). D_M is the center-to-center distance matrix. By integrating (4)–​(7) with these geometric constraints, we can derive the statistical distribution characteristics of the total received intensity in the FSO-MISO system.

Fig. 1.

The distribution of diversity transmission array.

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C. Outage Probability and Bit Error Rate

The performance of FSOC systems under turbulence is quantified through two metrics: outage probability (OP) and BER. For a predetermined intensity threshold I_T, OP characterizes the probability that the received signal intensity I falls below I_T defined by the cumulative distribution function of the Gamma-Gamma model:

View Source \begin{equation*} P(I \leq {I_T}) = \int_{0}^{{{I_T}}}{{p(I)}}dI \tag{8} \end{equation*}

Higher OP values indicate increased susceptibility to communication interruptions, particularly in strong turbulence regimes where deep fades occur frequently [25].

BER measures the reliability of data transmission, expressed as the ratio of erroneously received bits to the total transmitted bits. For on-off keying (OOK) modulation — widely adopted in intensity-modulation/direct-detection (IM/DD) FSO systems — the theoretical BER under Gamma-Gamma turbulence is derived as [26]:

View Source \begin{equation*} \text{BER} = \frac{1}{2}{\mathop{\rm erfc}\nolimits} \left[ {\frac{Q}{{\sqrt 2 }}} \right] \approx \frac{{\exp ({{ - {Q^2}} \mathord{\left/ {\vphantom {{ - {Q^2}} 2}} \right. } 2})}}{{Q\sqrt {2\pi } }} \tag{9} \end{equation*}

D. Simulation Analysis

Based on the theoretical models and experimental constraints, we analyzed the relationship between the $\delta _I^2$, the number of transmitting apertures M, and the refractive index structure constant $C_n^2$ for a transmission distance L = 2 km. The $C_n^2$ was varied from 1 × 10⁻¹⁷ m^−2/3 to 7.5 × 10⁻¹⁴ m^−2/3, with the corresponding Rytov variance $\delta _R^2$ plotted as the horizontal axis. By calculating the correlation coefficients between apertures based on their separation distances and generating Gamma-Gamma-distributed intensity values through Monte Carlo simulations, we obtained the statistical characteristics of $\delta _I^2$

Fig. 2(a) illustrates the relationship between $\delta _R^2$ and $\delta _I^2$ for different M. Under fixed total transmit power, $\delta _I^2$ decreases as M increases. For M = 2 $\delta _I^2$ is reduced by approximately 50% compared to M = 1. However, due to inter-aperture correlations, the mitigation efficiency diminishes for M > 4 with saturation observed at M = 8.

Fig. 2.

(a) The curve of turbulence intensity and SI under M transmitting signals; (b) The curve of turbulence intensity and OP under M transmitting signals; (c) The curve of SI and BER corresponding to M transmitting signals.

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With an outage threshold I_T = 0.5, Fig. 2(b) demonstrates the turbulence-dependent OP. Increasing M significantly lowers OP, but the improvement rate slows for M > 4. The OP performance for M = 8 approaches that of M = 4, indicating limited gains from excessive apertures.

Fig. 2(c) illustrates quantifies the interplay between $\delta _I^2$ and BER under OOK modulation. BER escalates with higher $\delta _I^2$while multi-aperture transmission effectively mitigates this degradation. In weak turbulence, BER decreases by two orders of magnitude when transitioning from M = 1 to M = 4. However, for M > 3, further BER reduction becomes marginal due to persistent correlations between sub-channels.

A. Experimental Configuration

To validate the turbulence mitigation performance of the proposed multi-aperture transmission scheme, a 2.1 km free-space optical communication link was established in an urban environment. The experimental setup (Fig. 3) comprises three key parts:

Fig. 3.

Experimental configuration.

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Transmitter terminal: Three collimators with four-dimensional adjustable mounts (28 mm aperture) were deployed, each maintaining a center-to-center spacing exceeding 12 cm to ensure low inter-channel correlation. The total transmit power was fixed at 72 mW, with equal power allocation among sub-apertures.

Receiver terminal: as shown in Fig. 4(a), The receiver incorporated a custom-designed AFC featuring a 20 mm diameter coupling lens and a 60 μm core multi-mode fiber (MMF). A 98:2 beam splitter diverted 2% of the coupled light to a photodetector (PD) for closed-loop control, while the remaining 98% was routed to communication receivers. The fiber tip was mounted on a cross beam driven by four piezoelectric actuators, as shown in Fig. 4(b), which can provide angular correction of ±550 μrad.

Fig. 4.

The working principle of the AFC: (a) Internal structure and working principle; (b) Results of the offset range test.

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Two-phase experimental protocol: in the first phase, a 1550 nm laser source was used to optimize the closed-loop control algorithm, with PD data sampled at 11 kHz. In the second phase, a 10 Gbit/s OOK signal generated by a small form-factor pluggable (SFP+) module was amplified through an erbium-doped fiber amplifier (EDFA) and transmitted via the multi-aperture array for communication.

B. Algorithm Principle

In this experiment, disturbances introduced by turbulence and platform vibrations can reduce the coupling efficiency. To minimize the impact of these disturbances, we employed a MMF with a core diameter of 60 μm for coupling, which enhances tolerance to perturbation errors [27]. Additionally, we integrated adaptive coupling techniques to further mitigate the interference from dynamic errors. In practice, accurately modeling the fiber coupling process is challenging, and optimization algorithms are commonly used for error correction. Among these algorithms, the Stochastic Parallel Gradient Descent (SPGD) algorithm is the most widely used [28]. The implementation process of the SPGD algorithm is as follows:

1. Random voltage perturbations $\Delta {\mathbf{u}^{(n)}} = \lbrace \delta \mathbf{c}_x^{(n)}, \delta \mathbf{c}_y^{(n)}\rbrace$ obeying the Bernoulli distribution are generated. $\delta$ represents the amplitude of the perturbation. ${\mathbf{c}^{(n)}}$represents the direction. n is the iteration number during the optimization process.

2. The performance metric J, which is the coupled power detected by the PD, is measured under positive and negative perturbations:

   $$\begin{equation*}J_ + ^{(n)} = J({\mathbf{u}^{n - 1}} + \Delta {\mathbf{u}^{(n)}}),J_ - ^{(n)} = J({\mathbf{u}^{n - 1}} - \Delta {\mathbf{u}^{(n)}}).\end{equation*}$$ View Source \begin{equation*}J_ + ^{(n)} = J({\mathbf{u}^{n - 1}} + \Delta {\mathbf{u}^{(n)}}),J_ - ^{(n)} = J({\mathbf{u}^{n - 1}} - \Delta {\mathbf{u}^{(n)}}).\end{equation*}

The gradient is estimated based on the obtained performance metrics: $\Delta {J^{(n)}} = J_ + ^{(n)} - J_ - ^{(n)}$.

3. The control voltage is iteratively refined through: ${\mathbf{u}^{(n)}} = {\mathbf{u}^{(n - 1)}} + \alpha \Delta {\mathbf{u}^{(n)}}\Delta {J^{(n)}}$, $\alpha$is the gain coefficient.

However, traditional SPGD algorithms exhibit limited correction ranges and slow convergence under dynamic turbulence. To mitigate these effects, we develope an ARW algorithm by integrating machine learning optimization principles with adaptive control. The improvements of the ARW algorithm mainly include the following two aspects [29]:

1. First, we introduce a perturbation warm-up module to adjust the amplitude of the perturbations applied. In the initial iterations, the amplitude is amplified, and it gradually decreases as the number of iterations increases. This ensures that the algorithm enhances the initial search range while maintaining stability after convergence. The specific formula is shown below:

   $$\begin{equation*}\delta = \frac{{(\tau + n)}}{n}{\delta _0} \tag{10} \end{equation*}$$ View Source \begin{equation*}\delta = \frac{{(\tau + n)}}{n}{\delta _0} \tag{10} \end{equation*} where ${\delta _0}$is the initial perturbation amplitude, $\tau$is the perturbation warm-up coefficient.

2. Secondly, a gain warm-up module is added. The accumulated gradient squared S undergoes exponential moving average smoothing:

   $$\begin{equation*}{S^{(n)}} = \beta {S^{(n - 1)}} + (1 - \beta )\Delta {J^2} \tag{11} \end{equation*}$$ View Source \begin{equation*}{S^{(n)}} = \beta {S^{(n - 1)}} + (1 - \beta )\Delta {J^2} \tag{11} \end{equation*} where $\beta$controls the historical gradient weight. To enable larger step sizes during initial iterations, the gain warm-up module further scales S by: $$\begin{equation*} {S^{(n)}} = \frac{{{S^{(n)}}}}{{1 - \exp ({{ - n} \mathord{\left/ {\vphantom {{ - n} \gamma }} \right. } \gamma })}} \tag{12} \end{equation*}$$ View Source \begin{equation*} {S^{(n)}} = \frac{{{S^{(n)}}}}{{1 - \exp ({{ - n} \mathord{\left/ {\vphantom {{ - n} \gamma }} \right. } \gamma })}} \tag{12} \end{equation*} where $\gamma$is the gain warm-up rate. And the final voltage update rule becomes: $$\begin{equation*} {\mathbf{u}^{(n)}} = {\mathbf{u}^{(n - 1)}} + \frac{{{\alpha _0}}}{{\sqrt{S} + {{10}^{ - 8}}}}\Delta {\mathbf{u}^{(n)}}\Delta {J^{(n)}} \tag{13} \end{equation*}$$ View Source \begin{equation*} {\mathbf{u}^{(n)}} = {\mathbf{u}^{(n - 1)}} + \frac{{{\alpha _0}}}{{\sqrt{S} + {{10}^{ - 8}}}}\Delta {\mathbf{u}^{(n)}}\Delta {J^{(n)}} \tag{13} \end{equation*} where ${\alpha _0}$is the initial gain rate. The algorithm workflow is shown in Fig. 5.
   

   Fig. 5.

   The control process of the ARW algorithm.

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A. Closed-Loop Performance Under Angular Offsets

To evaluate the effectiveness of the ARW and the SPGD in compensating beam misalignment induced by turbulence and platform vibrations, we conducted closed-loop tests under different angular offsets. The AFC was intentionally initialized with predefined deviation angles (200 μrad, 400 μrad, and 550 μrad). Both the ARW and conventional SPGD algorithms operated at a 1 kHz iteration rate, synchronized with the photodetector's 11 kHz sampling frequency. In Fig. 6(a), the initial angle deviation is approximately 200 μrad. The ARW algorithm achieved convergence within 0.14 s, stabilizing the coupling power at an average performance metric J = 1.79 V. In contrast, SPGD required 0.24 s to reach an average performance metric J = 1.77 V. In Fig. 6(b), the initial angle deviation is increased to 400 μrad. The ARW maintained rapid convergence time (0.16 s) with an average J of about 1.71 V. SPGD's convergence time increased significantly to 0.82 s, with an average J of about 1.52 V. In Fig. 6(c), under extreme misalignment near the AFC's mechanical limit (±550 μrad), ARW successfully achieved convergence within 0.15 s, and the average J is about 1.12 V. However, SPGD fails to converge in closed-loop conditions, with an average J being almost zero, showing no improvement compared to the open-loop. These results demonstrate that the ARW algorithm has an advantage in rapidly correcting angle deviations. Compared to the SPGD algorithm, the effective convergence range of the ARW algorithm is increased by 20%, and the convergence speed is improved by 40%.

Fig. 6.

The closed-loop effect of the ARW algorithm and the SPGD algorithm under different deviation angles: (a) Angle offset-200 μrad; (b) Angle offset-400 μrad; (c) Angle offset-550 μrad.

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B. Algorithm Principle

After closed-loop control, we further investigated the scintillation mitigation capability of the multi-aperture transmission scheme under real atmospheric conditions. In this experiment, the three transmitters working separately were labeled as T.1, T.2, and T.3, respectively. When all three transmitters worked simultaneously, they were labeled as T.123. The total transmission power was kept constant, while the sampling rate of the PD at the receiver was set to 11 kHz, with a collection time of 20 s. Fig. 7(a)–​(c) show the changes in the performance metric J of the fiber coupling power detected from the PD when each of the three transmitters worked individually. Fig. 7(d) shows the changes in the performance metric J when all three transmitters worked simultaneously. The comparative results indicate that when a single transmitter worked independently, the performance metric J exhibited significant differences and more severe fluctuations, with mean values of 1.73 V, 2.2 V, and 0.64 V, and minimum values of 0.08 V, 1.24 V, and 0.42 V, respectively. However, when all three transmitters worked simultaneously, the performance metric J not only remained at a higher power but also significantly reduced the fluctuation amplitude, with a mean value of 1.91 V and a minimum value of 1.45 V.

Fig. 7.

Detection values at the receiving end under different transmitting conditions: (a) T.1 launch; (b) T.2 launch; (c) T.3 launch; (d) T.123 launch.

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Based on (1), we calculated the scintillation index at the receiver under different transmission conditions. Fig. 8(a) shows the variation of the scintillation index under different transmission conditions in weak turbulence conditions at night ($C_n^2$ = 1 × 10⁻¹⁶ m^−2/3). When the transmitter was working alone, the maximum scintillation index was 0.107, with a minimum mean value of 0.015. However, when all three transmitters worked simultaneously, the maximum scintillation index was only 0.007, with a mean value of 0.005, representing a 67% reduction compared to the single-transmitter operation. Fig. 8(b) shows the variation of the scintillation index under different transmission conditions in moderate turbulence conditions during the day ($C_n^2$ = 0.8 × 10⁻¹⁴ m^−2/3). For single-transmitter operation, the maximum scintillation index was 0.44, with a minimum mean value of 0.21; whereas for simultaneous operation of all three transmitters, the maximum scintillation index was only 0.12, with a mean value of 0.085, representing a 60% reduction compared to single-transmitter operation. The above experimental results are basically consistent with the simulation results, indicating that the use of diversity transmission technology can effectively mitigate atmospheric scintillation, allowing the coupling power to maintain a higher value and concentration, thereby achieving high-quality communication. Additionally, we performed a Gamma fit on the performance metrics gathered from both single-transmitter operation and the simultaneous operation of all three transmitters. The resulting probability density distribution, illustrated in Fig. 9, confirms that the coupled power values adhere to the Gamma-Gamma model distribution.

Fig. 8.

(a) scintillation index under weak turbulence; (b) scintillation index under moderate turbulence.

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Fig. 9.

The probability density distribution of the received signal.

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C. MISO Communication Performance Test

To comprehensively evaluate the practical communication capabilities of the proposed multi-aperture transmission system, we conducted a series of error-free communication tests under real atmospheric turbulence conditions. The experimental setup utilized a 10 Gbit/s OOK-modulated signal generated by an SFP+ module, amplified through an EDFA, and transmitted via the multi-aperture collimator array. At the receiver, the AFC output was connected to a bit error rate tester (BERT) and SFP+ (sensitivity≥_25 dBm) for real-time performance monitoring and demodulation. Due to inconsistent patch cord lengths in Transmitter 3 compared to Transmitters 1 and 2, which introduced significant optical path differences, we exclusively utilized Transmitters 1 and 2 for the communication tests to ensure signal synchronization.

Fig. 10(a) compares the BER performance of single transmitter and dual transmitters over 30 s test intervals. The results indicate that the BER of a single transmitter is approximately at the 10⁻² level, while the BER decreases to the 10⁻⁴ level when using dual transmitters. Furthermore, we recorded the long-term communication BER at different times of the day, as shown in Table I. At 2 P.M (UTC + 8), corresponding to $C_n^2$ of approximately 1 × 10⁻¹³ m^−2/3, the diversity transmission did not significantly improve the communication quality, with the BER remaining at 10⁻¹ level.

Fig. 10.

(a) BER under different transmitting conditions; (b) the process of video transmission.

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TABLE I BER Statistical Results

At 6 P.M (UTC + 8), corresponding to $C_n^2$ of approximately 2 × 10⁻¹⁴ m^−2/3, BER reduced by 98.8% through spatial diversity compared to single-transmitter transmission. At 9 P.M (UTC + 8), corresponding to $C_n^2$ of approximately 1 × 10⁻¹⁶ m^−2/3, the dual-transmitters achieved error-free operation, while the single-transmitter exhibited sporadic errors. Additionally, under weak turbulence conditions at night, we conducted a transmission test of a high-definition video, as shown in Fig. 10(b). The above results show that the multi-aperture diversity transmission technology has reduced the BER of the communication system to a certain extent and improved its stability. Due to the limitations of the experimental conditions, we have not yet conducted further communication performance verification with more transmitters, and we will conduct further analysis in the future.