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Satellite constellation networks play a pivotal role in making Beyond 5G services universally accessible to people worldwide. Inter-satellite laser communication is a crucial component of satellite communication networks. To utilize fiber amplifiers, it is imperative to couple the free-space optical beam into a single-mode fiber. However, vibrations and non-coaxial errors significantly impact coupling efficiency, thereby affecting communication quality. We have developed a parameter-free fiber coupling method based on Gaussian approximation to rectify non-coaxial errors and mitigate incident angle deviations induced by vibrations, ensuring consistently high coupling efficiency. This method is fast, stable, and parameter-free, eliminating the need for manual intervention. Furthermore, experimental results demonstrate the method’s effectiveness across the entire correction range of the correction device. In environments with strong vibrations, the communication link experiences a bit error rate of 6.734 E-2, which is reduced to zero bit errors after implementing this method in a closed loop.
© 2024 Optica Publishing Group
1. INTRODUCTION
Optical fiber coupling is a key technology in a wide range of fields, such as high-power lasers, free-space optical communication, stellar interferometry, lidar, and wavefront sensing [1–6]. With the assistance of low-noise optical amplifiers, such as erbium-doped fiber amplifiers (EDFAs), free-space optical communications are anticipated to play a significant role in the future of Beyond 5G networks [7,8]. It is crucial to overcome factors such as turbulence to enhance and stabilize coupling efficiency. For instance, in satellite-to-ground laser communication, the use of an adaptive optical (AO) system is necessary to compensate for phase distortions in the signal light. However, traditional deformable-mirror-based AO systems can be costly and bulky [9–11].
As the backbone of satellite constellation networks, inter-satellite laser communication differs from satellite-to-ground laser communication. While it experiences minimal disruption from atmospheric turbulence, it faces significant challenges from vibrations. These vibrations primarily stem from the transmitting satellite platform but can also originate from the communication terminal [12]. Satellite platform vibrations primarily impact laser beam acquisition probability [12–14], leading to increased tracking noise from electro-optic trackers and vibrations generated by internal satellite mechanical mechanisms, both of which significantly degrade network performance [15–17]. While optimizing the coupling efficiency and bit error rate (BER) of the communication system can be achieved through adjusting the relative aperture of the coupling lens, it is important to note that there exists an upper limit for the optimal value of the relative aperture due to the influence of random jitter on the average BER optimization capability [18].
Solutions to satellite vibrations include (1) increasing transmitter power, (2) employing a more sophisticated pointing system, and (3) implementing a model that adapts system parameters, such as power, bandwidth, and telescope gain, to the level of vibration [15,16]. While increasing transmit power on satellites is often impractical and employing a more complex pointing system typically results in increased weight, volume, and cost, a model that adjusts system parameters to match the level of vibration emerges as the most attractive solution. For instance, in cases where unstable optical beam tracking causes significant jitter in received optical power, a low-noise amplifier (LNA) can be configured in automatic power control (APC) mode. This configuration effectively stabilizes the amplification of the signal spike while suppressing the noise from wide-spectrum amplifier spontaneous emission (ASE) [19]. However, it is impossible to completely eliminate the optical power jitters caused by beam tracking errors only relying on a LNA. Furthermore, in the presence of static errors at the receiving end, a LNA with an APC scheme may prove ineffective due to extremely weak coupling power. These static errors can arise from thermal and mechanical distortions or from non-coaxial alignment of the fiber and tracking camera during assembly or rocket launch [20]. Consequently, relying solely on automatic optical amplifiers is insufficient. Space optical communications demand additional technologies, such as adaptive coupling methods, to effectively address these challenges [21].
A fast-steering mirror (FSM) is commonly employed to simultaneously fulfill the fiber-coupling process and control the acquisition, pointing, and tracking (APT) systems at the signal-receiving end. Nevertheless, it is worth noting that the error in nonideal coaxial optical paths can be substantial, and the associated structures are often intricate [22]. In response to these challenges, alternative devices have been developed, including the adaptive fiber coupler (AFC) and the piezoelectric ceramic tube (PCT). These innovations allow for the direct manipulation of the receiving fiber tip to optimize coupling power, resulting in more efficient and less complex solutions [22–24].
These devices offer the advantages of lightweight construction and high motion accuracy. However, current research lacks control algorithms specifically tailored to these devices. Demonstrations in research often utilize the nutation algorithm [22,25,26], the stochastic parallel gradient descent (SPGD) algorithm [24,27,28], and control algorithms based on deep learning optimization techniques [29–31]. While the nutation algorithm involves circular scanning in the focal plane, it struggles to effectively correct high-frequency vibrations. On the other hand, the SPGD algorithm and control algorithms based on deep learning optimization show promising performance but necessitate manual adjustment of hyper-parameters. Typically, these parameters are selected based on human experience or by trial and error. Given the time constraints in inter-satellite communication, swift data transmission is crucial. Therefore, a parameter-free control algorithm for vibration suppression is essential. We propose a model-based control algorithm that calculates the coupling model using system parameters. It significantly enhances coupling efficiency in a single iteration, effectively reducing bit error rates and improving communication stability. In this paper, we thoroughly simulate and verify the algorithm’s vibration suppression capabilities, showcasing its ability to lower bit error rates.
2. PRINCIPLES AND METHODS
A. Coupling Efficiency and Vibration Model
To achieve efficient coupling of a space laser to a single-mode fiber (SMF), mode field matching is necessary. This means matching the amplitude and phase distribution of the electromagnetic field of the laser beam coupled into the SMF with the laser propagating in the SMF [32].
The system is designed for inter-satellite situations for which the distance is much larger than the Fraunhofer distance (typically tens of kilometers) and turbulence can be ignored; hence the incident optical field in the aperture plane of the receiver can be considered a plane wave, which can be expressed by Eq. (1):
where $A$ is the amplitude of the received light corresponding to the amount on the coupling lens. $k$ is the wave number. $\theta$ denotes the pointing error. $P(r)$ is the aperture function, which is defined as in Eq. (2) [33]: where $P(r)$ is the aperture function and $D$ is the diameter of the aperture. It is necessary to mention that a telescope with an obscured center will cause the energy of the spot on the receiving surface to be obscured, leading to a decrease in fiber coupling efficiency due to mode mismatch [11]. This effect is not significant when receiving a plane wave with a small obscuration. However, if the system is integrated with a transceiver, the Gaussian-distributed laser will encounter obstruction in the center during transmission, potentially blocking the central part of the emitted laser with higher energy. Therefore, this paper only considers the case where the center is not obscured.The plane wave is focused by a coupling lens to form an Airy spot ${E_O}$ and coupled into the fiber. The distribution of the laser propagating in the SMF is approximately a Gaussian beam, which can be expressed as
Ideally, the tip of the fiber should be on the focal plane of the lens and its core should coincide with the Airy spot. In this ideal situation, the pointing error $\theta = 0$ and we assume the amplitude of the received light $A = {{1}}$; then the distribution of the focused optical beam on the fiber tip is an Airy pattern, of which the ${E_O}$ is given by Eq. (4) [34]:
Then the relationship between coupling efficiency and the optical system can be simplified as follows:
The formulas mentioned above provide guidance for designing optical systems to achieve the highest possible coupling efficiency under ideal conditions. It can be deduced that when $\psi = 1.12$, the coupling efficiency reaches its maximum $\eta = 81.45\%$. However, achieving high coupling efficiency is only possible under ideal conditions. In reality, there are vibrations from the transmitting satellite platform and the communications terminal. The frequency and amplitude characteristics of vibrations vary with different satellite platforms. Some studies directly use the Gaussian white noise to simulate vibrations for simplicity; however, Gaussian distributed vibrations of unlimited bandwidth are not physically plausible for a mechanical system [12]. A more realistic and widely used vibration model is called the Olympus S/C vibration model or SILEX model [13,20,35]. The European Space Agency (ESA) collected vibration data from a large communication satellite Olympus, and applied as a specification for the optical communication payload SILEX the following power spectral density (PSD) for the angular base motion:
B. Efficiency with Vibrations and the Parameter-Free Coupling Method
Suppose the plane wave laser can completely cover the receiving end after the coarse tracking system is completed; then there will be only angle misalignment due to static errors and angular jitters. Misalignment of microradian magnitude can significantly decrease the coupling efficiency [15–17,32], but can be completely compensated for by the radial offset of the fiber tip. The angular error $\theta$ of a plane wave relative to the optical axis of the focusing lens is equal to a static radial deviation $({\rho = \theta f})$ of the focused beam from the nominal axis of the fiber core in the lens’ rear focus plane. When static angular error $\theta$ is not zero, it is equivalent to the fiber tip being deviated from the center, and the distribution of the optical beam in the SMF is given by the Nakagami–Rice distribution [36]:
Substituting Eqs. (4) and (8) into Eq. (5), the coupling efficiency is expressed by
It can be seen that in inter-satellite laser communications, there is no random aberration caused by atmospheric turbulence. Once the optical parameters of the receiving system are determined, the model of coupling efficiency is determined. The only factor affecting the coupling efficiency is the angular jitter $\theta$, which is equivalent to the radial deviation $\rho$. Therefore, if the maximum coupling power of the actual coupling system is known, we can obtain the model between the coupling power and radial deviation through theoretical calculation.
However, the coupling efficiency model shown in Eq. (9) is too complex to solve. We propose a normalized Gaussian model to approximate the normalized coupling efficiency model, allowing us to derive the coupling power model with knowledge of the maximum coupling power. We found that the Gaussian model closely mirrors the form of the coupling efficiency and can strike a good balance between complexity and accuracy, achieving high fitting accuracy while maintaining a low computational cost. The formula for the normalized Gaussian model is as follows:
where ${\eta _G}$ is the normalized Gaussian coupling efficiency, and $\omega _G^2$ is the only parameter of the normalized Gaussian model that should be solved according to the system parameters.The normalized coupling efficiency curve ${\eta _N}$ to be fitted can be calculated by Eq. (9) or through the angular spectrum diffraction method. Assuming that the dynamic correction range of the AFC or PCT can cover the field of view in which the coupling efficiency is close to 0, then the normalized Gaussian coupling efficiency model can be obtained by calculating the normalized coupling efficiency ${\eta _N}({\rm middle})$ at the middle offset distance $\rho ({\rm middle})$:
Figure 1 illustrates the comparison between the actual normalized coupling efficiency and the normalized Gaussian model under various system parameters, which are detailed in Table 1. It is evident that, regardless of the specific system parameters or whether the coupling system is designed with optimal efficiency parameters, the normalized Gaussian model effectively approximates the actual normalized coupling efficiency. It should be noted that in Fig. 1(c), both the “real curve” and the “Gaussian model” curve are illustrated, with the “real curve” almost overlapping the “Gaussian model” curve.
Fig. 1. Comparison between the real normalized coupling efficiency and the normalized Gaussian model.
It is crucial to emphasize that the value of $\omega _G^2$ is determined through theoretical calculations without the need for any specific tests or calibration on experimental equipment. And calculating $\omega _G^2$ theoretically is straightforward: as seen from Eq. (11), the calculation of $\omega _G^2$ involves knowing the middle offset distance $\rho ({\rm middle})$ and the normalized coupling efficiency ${\eta _N}({\rm middle})$ at that distance. $\rho ({\rm middle})$ corresponds to the mode field radius of the SMF, denoted as ${\omega _0}$. This is because, with a fixed laser wavelength, achieving maximum coupling efficiency requires that the diameter and focal length of the coupling lens match the mode field radius of the SMF, resulting in a theoretical maximum coupling efficiency of 81.45%. Consequently, ${\omega _0}$ essentially determines the size of the Airy disk. Considering the Airy disk and the core of the SMF as circles, the radii of these circles can be approximated as ${\omega _0}$. When the two circles fully overlap, the coupling efficiency is at its maximum, and when the circles are externally tangent, the coupling efficiency is at its minimum. Therefore, it can be approximated that the coupling efficiency is at its minimum when the distance between the Airy disk center and the fiber core center is $2{\omega _0}$, making $\rho ({\rm middle})$ equal to ${\omega _0}$. Once $\rho ({\rm middle})$ is determined, ${\eta _N}({\rm middle})$ can be directly calculated using Eq. (9) or through numerical simulation employing the angular spectrum method. The required parameters for this calculation, including wavelength, aperture diameter, focal length, and mode field radius, are all system parameters and can be accurately obtained.
After the normalized Gaussian coupling efficiency model is obtained, the maximum coupling power ${P_{\max \_\rm scan}}$ of the system is obtained by a fast scanning, and the system coupling power model can be obtained:
Table 1. Specific System Parameters of Four Different Coupling Systems in Fig. 1
According to the radial deviation of the current coupling power, we can directly calculate the distance between the position of the current optical fiber tip and the center of the spot, so as to drive the fiber tip to move the corresponding distance directly:
The direction of the fiber tip’s movement, which represents the accurate gradient direction, is determined through perturbations. The magnitude of the perturbation is selected based on the offset distance corresponding to 99.8% of the maximum coupling efficiency in the coupling model, eliminating the need for artificial adjustments. For instance, with the system parameters from Table 1(a), the perturbation magnitude is 0.2 µm. This value ensures minimal fluctuation in coupling efficiency when near the optimal level and provides a clear gradient direction when far from optimal. With this perturbation magnitude, the maximum angular error (excluding the magnification of the telescope) is approximately 2.5e-4 deg, equating to a tracking accuracy of approximately 0.44 µrad (including the magnification of the telescope, assuming a magnification of 10 times).
Using this precise gradient direction and the distance between the current fiber tip position and the spot center, we can rapidly iterate towards the position of maximum coupling power, as depicted in Fig. 2. In the figure, an example is shown where the optical fiber tip is not in the center of the coupling spot. The deviation between the current fiber tip position and the spot center is measured by assessing the distance between the current coupling power and the theoretical maximum coupling power in the Gaussian model curve. With the precise gradient direction obtained through perturbations, the fiber tip can quickly converge to the spot center. Figure 2(a) illustrates the distance between the current fiber tip position and the spot center, while Fig. 2(b) shows the accurate gradient direction towards the spot center. It is important to note that in practical applications, precise coupling power values are often unavailable. Instead, a portion of the coupling power measured by a photodetector (PD) is used as a metric. Since there is a linear relationship between the coupling power ($P$) and the metric ($J$), this does not impact the coupling efficiency model. The metric $J$ can be directly used to calculate the offset distance, guiding the fiber tip’s movement.
Fig. 2. (a) Distance between the current fiber tip position and the spot center. (b) Accurate gradient direction towards the spot center.
The parameter-free fiber coupling method based on Gaussian approximation is outlined in Algorithm 1. This entire iterative process requires no manual parameter configuration or adjustments.
Algorithm 1. Procedure of the Parameter-Free Fiber Coupling Method Based on Gaussian Approximation
Additionally, we have included an example link budget for inter-satellite links to demonstrate how the algorithm operates in inter-satellite communications. In inter-satellite optical communication applications, optical telescopes function as antennas with apertures ranging from 5 to 15 cm. However, it is important to note that we cannot directly use these optical antennas as coupling lenses. For instance, a 10 cm diameter optical antenna would have an approximate focal length of 150 mm. If we attempt to directly couple this light into a fiber, the efficiency would be very low due to mode mismatch.
The typical solution involves placing a beam reducer after the optical antenna. This reduces the 10 cm beam to, for instance, a 1 cm beam, where a coupling lens with a 10 mm diameter and 45.24 mm focal length can achieve an 81.45% coupling efficiency. The reduced beam size also facilitates the incorporation of optical filters and beam splitters for the distinct coarse tracking system and the fiber coupling system.
When using a beam reducer, the incident angle on the optical antenna is magnified by the reducer. For instance, with a telescope diameter of 10 cm and a coupling lens diameter of only 10 mm, the incident light beam jitter will be magnified by 10 times. Although the coarse tracking system compensates for some significant incident angle deviations, there remain dynamic errors that cannot be corrected. Our algorithm addresses these uncompensated beam jitters by considering an amplified incident angle on the optical antenna when analyzing the AFC correction capability. This amplification factor is demonstrated in the experimental comparison in Section 3.C, where we magnify the SILEX vibration model by different multiples until it exceeds the correction range of the AFC. The correction range of the AFC can be tailored to specific needs, allowing customization of up to hundreds of microns. Typically, the correction range of a high-resonant-frequency AFC falls within the range of ${{\pm 10}}\;\unicode{x00B5}{\rm m}$ to ${{\pm 50}}\;\unicode{x00B5}{\rm m}$. The selection of the correction range depends on the capabilities of the coarse tracking system. Adjusting the correction range of the AFC allows for flexibility in accommodating different tracking accuracy requirements for the coarse tracking system.
3. SIMULATIONS AND EXPERIMENTAL VERIFICATIONS
A. Vibration Noise Generation and Fast Scanning
To analyze the impact of the vibration environment on coupling efficiency and optical communications, we simulate the level of satellite platform vibrations using the SILEX model, where the power spectral density (PSD) is described by Eq. (7). The vibration generation approach aims to ensure that the PSD of the simulated vibrations closely aligns with the spectral characteristics observed in orbit.
To achieve this, we initially generate a pseudo-random signal with a Gaussian distribution. Subsequently, we filter the Gaussian white noise using a designed low-pass filter to generate the satellite platform’s vibration noise, as illustrated in Fig. 3(a). The sampling frequency is set at 2 kHz, resulting in vibration amplitudes ranging from approximately ${-}{{60}}\,\,\unicode{x00B5} {\rm{rad}}$ to 60 µrad. The total PSD, as seen in Fig. 3(b), aligns with the spectrum model employed by the ESA for the optical communication payload SILEX. As can also be seen from Fig. 3(b), the energy in the SILEX model is mainly concentrated below 10 Hz, so the frequency of sampling vibrations in the experiment does not need to be too high.
Fig. 3. (a) Generated vibration noise. (b) PSD of the generated vibration compared with the PSD of the SILEX model.
To derive the coupling power model from the normalized Gaussian coupling efficiency model, it is essential to determine the maximum achievable coupling power through rapid scanning. Unlike atmospheric applications, inter-satellite communications are not subject to scintillation caused by turbulence. Consequently, the maximum achievable coupling power is solely determined by the communication distance and the coupling efficiency, with minimal fluctuations throughout the communication process.
In order to obtain the maximum achievable coupling power ${J_{{\max}}}$ without occupying communication time, the fast-scanning process has the following requirements:
- 1. The overall scanning time cannot be too long, as communication time is often limited, and the communication link needs to be established as soon as possible after the capture and tracking are completed.
- 2. Collect as many points as possible within the entire AFC correction range to obtain ${J_{{\max}}}$ with the greatest probability.
To address these requirements, our plan is as follows:
- 1. Divide the entire correction range of the AFC into a ${{128}} \times {{128}}$ grid and scan the whole range in a zigzag pattern starting from the upper left corner. The scanning frequency does not exceed the resonant frequency of the AFC. The scanning process consists of 64 period scans, where one period scan contains ${{2}} \times {{128}}$ grid points. There will be a brief pause at each grid point during the scan.
- 2. To maximize data points, we maintain the PD at a sampling frequency of 200 kHz in experiment throughout the scanning process. This enables us to capture the coupled optical power at positions when the fiber tip moves between grid points, albeit improving the scanning point density in one direction only. Consequently, the final dataset comprises not just ${{128}} \times {{128}}$ optical power values but rather the product of the PD sampling frequency and the scanning time of optical power values. This approach ensures that we can find the maximum achievable coupling optical power, ${J_{{\max}}}$, of the coupling system while also ensuring that the entire scan time is not too long and can be completed within 1 s.
We conducted 100 fast scanning simulations under vibrations using the Monte Carlo method. The angular spectrum diffraction method was employed to calculate the focused light spot and coupling power across the entire range of the AFC correction ability. Results indicate that vibrations have a negligible effect on the ability of fast scanning to determine the maximum achievable coupling power. The average coupling power from these 100 simulations reached 99.98% of the maximum coupling power. This suggests that, provided the sampling frequency is sufficiently high, vibrations have minimal impact on the accurate acquisition of the maximum coupling power. Moreover, the time required for fast scanning is negligible, ensuring it does not extend the time needed to establish the communication chain.
B. Coupling Efficiency Verification
We have conducted simulations and experimental verification of coupling efficiency in a vibrational environment, especially when static deviations may be present. The experimental setup comprises a platform composed of a laser, a collimating lens, a FSM equipped with a high-voltage amplifier (HVA), a computer for vibration control, a fiber coupling device, and a metric collection subsystem. The metric collection subsystem includes an optical fiber splitter, a PD, and a monitoring computer.
The fiber coupling device itself consists of a coupling lens, an AFC housing a SMF, a PD, a HVA, which amplifies the input voltage by 100 times, and a controller, as depicted in Fig. 4(a). The coupling lens is a cemented doublet with a focal length of 45.24 mm and a diameter of 10 mm, optimized for fiber coupling with a wavelength of 1550 nm. The detailed parameters of the coupling device align with those listed in Table 1(a). Figure 4(b) shows the correction range of the AFC, and in Fig. 4(c), results of the impedance analyzer indicate the intrinsic frequency of the AFC is near 10.6 kHz. In the following experiments, vibrations were applied in both the ${{X}}$ and ${{Y}}$ directions, with characteristics consistent with the SILEX model. The output frequency from the vibration generator to the FSM was approximately 190 Hz, and the closed-loop controller’s bandwidth was roughly 470 Hz. As the majority of power in the SILEX model is concentrated at frequencies below 10 Hz [13], the sampling frequency of 190 Hz for the vibration sufficiently meets the Nyquist sampling rate to accurately simulate the SILEX vibration model.
Fig. 4. (a) Setup for coupling efficiency verification. HVA: high-voltage amplifier. FSM: fast-steering mirror. SMF: single-mode fiber. AFC: adaptive fiber coupler. PD: photodetector. (b) Position deviation of the fiber tip as a function of the AFC’s driving voltage. (c) Results of the impedance analyzer.
Fig. 5. (a) Simulation results of normalized coupling efficiency with no static error, 50 µrad static error, and 100 µrad static error. (b) Experiment results of normalized coupling efficiency with no static error, 50 µrad static error, and 100 µrad static error.
The simulation and experimental results, as illustrated in Fig. 5, exhibit a high degree of consistency. We verified cases with no static error, a 50 µrad static error, and a 100 µrad static error in incident angle. It is evident that, within the range of 100 µrad static error, the coupling efficiency decreases as the static error increases, and the influence of vibration on coupling efficiency becomes more pronounced.
In both cases, the parameter-free fiber coupling method rapidly compensates for static errors and mitigates the impact of vibration, thereby stabilizing the coupling efficiency at the optimal value. In simulations, the average normalized coupling efficiencies for no static error, 50 µrad static error, and 100 µrad static error in incident angle are 95.61%, 78.12%, and 41.69%, respectively. After closed-loop control, these values improve to 99.13%, 99.12%, and 99.08%. In experiments, the average normalized coupling efficiencies for no static error, 50 µrad static error, and 100 µrad static error in incident angle are 91.40%, 77.13%, and 35.76%, respectively. Following closed-loop control, these values increase to 96.49%, 96.98%, and 97.41%.
Fig. 6. (a)–(c) Probability distributions of normalized coupling efficiency in simulations with no static error, 50 µrad static error, and 100 µrad static error. (d)–(f) Probability distributions of normalized coupling efficiency in experiments with no static error, 50 µrad static error, and 100 µrad static error.
The average efficiency of spatial light coupling into a single-mode optical fiber is a widely used metric in free-space optical communication. However, the BER is nonlinearly related to the coupling efficiency (CE). Consequently, relying solely on the average CE for estimating link performance metrics may be insufficient. Even with a high average CE, the collected signal can still exhibit significant fluctuations [9,37]. Therefore, it is of critical importance to analyze the probability distributions of the CE to accurately estimate BER performance, as illustrated in Fig. 6.
In both simulations and experiments, it is apparent that the normalized CE in the closed-loop configuration exhibits greater concentration. The variances of the open-loop normalized CE under different static errors in simulations are 1.42 E-3, 1.33 E-2, and 1.79 E-2, respectively. After implementing the closed loop, these values decrease to 7.30 E-5, 1.02 E-4, and 3.34 E-4. In experiments, the variances of the open-loop normalized coupling efficiency under different static errors are 2.47 E-3, 1.17 E-2, and 1.05 E-2, respectively. Following closed-loop control, these values reduce to 6.92 E-5, 9.48 E-5, and 2.50 E-4. The higher average and smaller variance of the normalized CE after closed-loop control indicate a lower BER.
C. Communication Tests under Vibrations
To validate the effectiveness of the parameter-free fiber coupling method in vibration suppression and its impact on BER improvement in the communication system, we utilized a bit error rate tester (BERT) in conjunction with an optical transceiver (type: MTRS-1S60-01) to replace the laser as the light source. This setup allowed us to transmit the received laser signal back to the BERT, forming a loop, and perform BER testing at a communication rate of 10.26 Gb/s. The experimental platform is depicted in Fig. 7.
In practical scenarios, the incident angle of light is typically magnified by a telescope before coupling. While a telescope is not part of our test bed, we achieve the same effect as telescope magnification by amplifying the amplitude of vibrations [33]. Additionally, the APT system can partially suppress vibrations, but its effectiveness depends on the specific system’s tracking accuracy. To cover a range of application scenarios and assess the communication performance of the parameter-free fiber coupling method, we amplified the original vibrations by factors of 1, 2, 4, and 6 in the communication test experiments. This allowed us to obtain data in different multiples of vibration environments. We conducted the first 1 min open-loop data acquisition and later 1 min closed-loop data acquisition under each level of amplification. This allowed us to collect data on the normalized coupling efficiency, the number of bit errors per second, the control voltages of the AFC, and the average BERs for both open-loop and closed-loop scenarios.
Fig. 8. (a)–(c) Normalized coupling efficiency, number of errors at every second, and control voltages of the AFC under 1-time amplification vibrations, respectively. (d)–(f) Normalized coupling efficiency, number of errors at every second, and control voltages of the AFC under 2-times amplification vibrations, respectively.
In Fig. 8, we present the experimental results after 1-time and 2-times amplifications of the original vibrations. Figures 8(a)–8(c) display the normalized CE, the number of errors per second, and the control voltages of the AFC under 1-time amplification vibrations, respectively. Figures 8(d)–8(f) show the normalized CE, the number of errors per second, and the control voltages of the AFC under 2-times amplification vibrations. Under 1-time and 2-times amplifications of the original vibrations, the average BERs in the open loop are 1.510 E-10 and 1.471 E-3, respectively. The average BERs in the closed loop are both 0. It is evident that as the vibration amplitude increases, the average CE in the open loop decreases, leading to larger fluctuations and increased open-loop BER. Comparing Fig. 8(c) with Fig. 8(f), it is apparent that as the vibration amplitude increases, the AFC in the closed loop also requires larger control voltages to effectively suppress vibrations. In both scenarios, the parameter-free fiber coupling method consistently suppresses vibrations and maintains a stable zero BER.
Figures 9(a)–9(c) display the normalized coupling efficiency, the number of errors per second, and the control voltages of the AFC under 4-times amplification of vibrations, respectively. Figures 9(d)–9(f) depict the normalized coupling efficiency, the number of errors per second, and the control voltages of the AFC under 6-times amplification of vibrations.
Fig. 9. (a)–(c) Normalized coupling efficiency, number of errors at every second, and control voltages of the AFC under 4-times amplification vibrations, respectively. (d)–(f) Normalized coupling efficiency, number of errors at every second, and control voltages of the AFC under 6-times amplification of vibrations, respectively.
Under 4-times and 6-times amplifications of the original vibrations, the average BERs in the open loop are 6.734 E-2 and 2.777 E-1, respectively. The average BERs in the closed loop are 0 and 5.467 E-4, respectively. Notably, the yellow bars in Figs. 9(b) and 9(e) represent a complete loss of communication ability. It is evident that increased vibration amplitudes lead to a higher incidence of communication failures. Moreover, under 6-times amplification of vibrations, the BER in the closed loop is not 0. This occurs because the offsets required to correct the vibrations have exceeded the correction range of the AFC. To protect the AFC device from potential damage, we limit the control voltages of the AFC to ${{\pm}}{4.9}\;{\rm{V}}$. As shown in Fig. 9(f), the control voltages reach this limit. Consequently, portions of the vibrations beyond the AFC’s correction ability are left uncorrected, resulting in a significant reduction in coupling efficiency and a higher number of error bits. While such extreme situations are rarely encountered in real application environments, this test demonstrates that the parameter-free fiber coupling method effectively suppresses vibrations within the entire correction range of the AFC, ultimately reducing the BER and enhancing communication stability. We also conducted a continuous 30 min coupling power test to verify the proposed algorithm and reached the same results. The coupling power stabilizes significantly after the algorithm is applied throughout the entire 30 min. This indicates that the proposed algorithm can achieve stable coupling power throughout the entire communication process.
As observed above, vibrations lead to a reduction in the average coupling power and significant fluctuations in coupling efficiency. To analyze this effect, we conducted power spectral density analysis on the normalized CE in both open loop and closed loop, as presented in Figs. 8(a)–8(d) and Figs. 9(a)–9(d). The results are summarized in Fig. 10. Following closed-loop operation, the low-frequency content decreases by two orders of magnitude, while the high-frequency content experiences a slight amplification. This pattern is characteristic of vibration correction systems employing an integrative controller. In this instance, the characteristic frequency of the fiber coupling system, corresponding to the point where the correction effectiveness diminishes, is approximately 100 Hz. More specifically, in Figs. 10(a)–10(d), we display the PSDs of normalized CE under 1-time, 2-times, 4-times, and 6-times amplifications of the original vibrations, respectively. The blue line represents the open loop, while the red line represents the closed loop. It is evident from the figures that as the vibration amplitude increases, the PSD under the open loop gradually rises at low frequencies. However, after implementing the closed loop, the PSD at low frequencies is effectively suppressed. The closed-loop PSDs in Figs. 10(a)–10(c) are essentially identical. When the vibration amplitude reaches 6 times the original value, and some of the vibration exceeds the correction range of the AFC, Fig. 10(d) shows that the closed-loop PSD is slightly higher than in the first three cases at low frequencies. Nevertheless, it still effectively suppresses low-frequency CE vibrations within approximately 100 Hz. These findings demonstrate that the algorithm proposed in this paper effectively enhances the stability of coupling efficiency in a vibration environment across the entire effective correction range of the correction device.
Fig. 10. Power spectral density analysis on the normalized CE under (a) 1-time, (b) 2-times, (c) 4-times, and (d) 6-times amplifications of the original vibrations, as shown in Figs. 8(a)–(d) and Figs. 9(a)–(d).
4. CONCLUSION
In this paper, we present a parameter-free fiber coupling method designed for application in fiber coupling systems for inter-satellite laser communications. This method aims to rectify non-coaxial errors and mitigate the influence of vibrations. First, we introduce the fundamental principles of fiber coupling and the SILEX vibration model. Second, we derive the coupling efficiency under vibration environments and propose the parameter-free fiber coupling method. Within this method, we employ a Gaussian model to approximate the real coupling efficiency model, and we determine the maximum coupling power with a fast scan. Consequently, we establish the coupling power model for satellite laser communication. The current coupling power dictates the position, and perturbations guide the gradient direction. Remarkably, this approach can swiftly find the optimal coupling position with minimal iterations. Third, we implement this method in simulations and experiments under varying vibration environments and static deviations. The results demonstrate that the method consistently converges quickly and reliably under different static deviations in a vibration environment. The method significantly improves the average coupling efficiency and greatly reduces the variance of coupling efficiency. In the final phase, we conduct real communication experiments under different vibration amplitudes. These experiments confirm that the method effectively suppresses vibrations within the maximum effective range of the correction device. Additionally, it leads to substantial enhancements in BER and communication stability, thereby validating the efficacy of our proposed method.
Most notably, this method eliminates the need for parameter adjustments, a critical advantage for satellite applications. It operates autonomously without requiring human intervention and avoids time-consuming parameter optimization processes. Consequently, it has the potential to expedite the establishment of satellite communication networks. We anticipate that this method will find valuable applications in inter-satellite communications.
Funding
Science Fund for Distinguished Young Scholars of Sichuan Province (2022JDJQ0042); National Natural Science Foundation of China (62005286, 62175241, U2141255).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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