Performance Analysis of an Optical System for FSO Communications Utilizing Combined Stochastic Gradient Descent Optimization Algorithm

Abstract

Wavefront aberrations caused by thermal flows or arising from the quality of optical components can significantly impair wireless communication links. Such aberrations may result in an increased error rate in the received signal, leading to data loss in laser communication applications. In this study, we explored a newly developed combined stochastic gradient descent optimization algorithm aimed at compensating for optical distortions. The algorithm we developed exhibits linear time and space complexity and demonstrates low sensitivity to variations in input parameters. Furthermore, its implementation is relatively straightforward and does not necessitate an in-depth understanding of the underlying system, in contrast to the Stochastic Parallel Gradient Descent (SPGD) method. In addition, a developed switch-mode approach allows us to use a stochastic component of the algorithm as a rapid, rough-tuning mechanism, while the gradient descent component is used as a slower, more precise fine-tuning method. This dual-mode operation proves particularly advantageous in scenarios where there are no rapid dynamic wavefront distortions. The results demonstrated that the proposed algorithm significantly enhanced the total collected power of the beam passing through the 10 μm diaphragm that simulated a 10 μm fiber core, increasing it from 0.33 mW to 2.3 mW. Furthermore, the residual root mean square (RMS) aberration was reduced from 0.63 μm to 0.12 μm, which suggests a potential improvement in coupling efficiency from 0.1 to 0.6.

1. Introduction

Free space optics (FSO) communication has emerged as a viable alternative to traditional radio frequency communication, primarily due to its enhanced security features and high data transmission rates. In recent years, FSO communication has garnered significant attention, becoming increasingly popular across a diverse range of applications, from secure communications between parties [1] to data exchanges between aerial vehicles and ground stations [2]. However, a persistent challenge in the realm of quantum communications is the effective distribution of quantum states over long distances, particularly for entangled states—multi-particle states that exhibit correlated properties regardless of the distance separating the individual particles [3].

The distribution of photonic states can be achieved through either fiber optics or free-space transmission. The latter method offers the distinct advantage of requiring minimal infrastructure beyond the sending and receiving stations, which can be portable [4,5]. This characteristic facilitates the integration of free-space optical systems into existing quantum networks [6,7,8,9]. Additionally, free-space transmission of light is generally more efficient than light propagation through fiber, making it an attractive option for quantum communication.

One of FSO communication applications is quantum key distribution (QKD)—a technique that allows us to establish a secret key between two parties through a quantum channel. Although QKD holds significant promise as a countermeasure to the imminent risks posed by quantum computers to contemporary encryption protocols, the availability of efficient and robust systems that function effectively in free space remains constrained. Joseph Meyer et al., in [10], introduced a classical light analogy of QKD that utilizes spatial modes of light, which has the potential to achieve a higher bit-per-photon rate compared to the more prevalent method of polarization state encoding. Actually, a variety of QKD protocols have been proposed and executed in both optical fiber and free-space communication contexts. One of the principal challenges encountered when implementing QKD protocols over free-space links is the occurrence of free-space losses. Mitali Sisodia et al., in [11], presented an in-depth examination and comparative analysis of the performance of QKD protocols based on single photons and entangled photons. The quantum bit error rate and secure key rate were assessed for terrestrial free-space quantum communication, considering different types of free-space losses, such as geometrical losses, atmospheric losses, and imperfections inherent to the devices employed.

Nevertheless, a significant obstacle to effective free-space light transmission is the distortion of wavefronts caused by atmospheric turbulence and aerosol scattering, particularly over extended transmission distances [12,13]. To establish a reliable quantum communication link, it is essential that fiber coupling efficiency [14] exceeds 50%, and the root mean square error of the residual wavefront surface remains below 0.5 radians [15]. Achieving these parameters necessitates that the quality of the focused laser beam approaches the diffraction limit.

To address these challenges, devices capable of dynamically altering their reflective surfaces can be employed, including micro-electromechanical systems (MEMS), spatial light modulators [16], and piezoelectric mirrors [17,18,19,20]. Numerous research studies have been dedicated to exploring the application of such devices in free-space optical communication systems [21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40].

In laser communication systems, wavefront compensation mechanisms can be implemented at two key locations: the transmitter side and the receiver side [33,41,42,43].

The first component, the transmitter side, is critical because the laser beam must traverse an atmospheric channel, where its wavefront is subject to degradation due to atmospheric turbulence. This turbulence disrupts the spatial modes of the beam, hindering the achievement of optimal communication performance. In this context, a pre-compensation system can be employed not for data recovery but rather to prevent data loss before it occurs.

The second component where wavefront compensation can be integrated is the receiver side. The primary challenge here lies in the necessity to focus the laser beam, which has traversed a free-space channel, into an optical fiber. As the beam travels from the receiver aperture to the fiber, it is not only modulated by atmospheric turbulence but can also be further degraded by thermal flows, the quality of optical components, and potential misalignments within the optical setup. To effectively focus the incoming laser beam into the optical fiber, it is essential to accurately measure and compensate for all aberrations introduced by the optics on the receiver side. Failure to do so will impede the ability to achieve effective coupling of the laser beam into the optical fiber. This manuscript addresses this specific challenge.

Several studies have contributed to understanding and mitigating the effects of turbulence on FSO communication systems. Wang et al. [23] examined the influence of turbulence on the quality of FSO communication and proposed an optical system design capable of maintaining a stable connection by compensating for 35 Zernike modes. Fischer et al. [44] outlined various optical system designs suitable for implementation in FSO communication applications. Vorontsov et al. [22] emphasized the importance of wavefront distortion compensation in enhancing FSO communication performance. Wang et al. [15] demonstrated the adverse effects of turbulence on FSO system performance using the power-in-bucket method. Weyrauch et al. [45] and Han et al. [46] tested an FSO communication system employing a wavefront-sensorless approach [47], which, in contrast to conventional systems, reduces complexity, size, and cost while achieving commendable performance, with a Strehl ratio of 0.78 following the application of the Stochastic Parallel Gradient Descent (SPGD) algorithm. Additionally, Liu et al. [48] proposed a system utilizing fast steering mirrors to compensate for laser beam jitter.

Various algorithms are employed in active optics applications, particularly within the realm of FSO communication [49]. Among these, Stochastic Parallel Gradient Descent (SPGD) algorithm, Genetic Algorithm (GA), Simulated Annealing (SA), Algorithm of Pattern Extraction (Alopex), and others have emerged as the most widely utilized [50]. These algorithms are capable of concurrently searching for optimal values across multiple dimensions of parameters, thereby facilitating a more rapid convergence of the algorithms. Additionally, they possess certain stochastic characteristics that enable them to partially evade local extrema. Consequently, any of the aforementioned algorithms could serve as a viable candidate for the control of active optics systems. However, these algorithms exhibit variations in terms of convergence rates, correction capabilities, and ease of implementation, among other factors. The advantages of those algorithms include (1) its efficiency in high-dimensional optimization which are common in optical communication systems, (2) robustness against noise which is also important in free space optical communication, (3) rapid convergence, adaptability to changing conditions in the communication channel, and (4) capacity for parallel processing, allowing multiple parameters to be optimized simultaneously, which can significantly speed up the optimization process.

Despite all advantages, the above-mentioned algorithms have known limitations, namely local minima, computational overhead, parameters sensitivity, implementation complexity and limited exploration. Some of those algorithms can become trapped in local minima, which may prevent them from finding the global optimum solution. Its performance can be sensitive to the choice of parameters such as learning rate and step size. Poor choices can lead to slow convergence or oscillations around the minimum. The need for multiple evaluations of the merit function (especially in parallel settings) can lead to significant computational overhead, particularly if the cost function is ex-pensive to compute. Implementing some algorithm like SPGD effectively may require a good understanding of the underlying system and careful tuning of parameters, which can complicate the design and implementation process. The stochastic nature of some of those algorithms may limit its ability to explore the parameter space thoroughly, especially if the stochastic component is not optimally calibrated, potentially missing better solutions.

In this paper, we present an approach aimed at addressing these challenges through the development of a combined stochastic-gradient descent optimization algorithm specifically designed for compensating optical distortions in laser beams. We developed and tested an algorithm to ensure its efficacy and reliability. Our research investigates the potential energy collection of a laser beam by a fiber optic system, establishing a correlation between residual wavefront error and coupling efficiency. We further assess the efficacy of the combined stochastic gradient descent algorithm in the context of laser beam focusing through a diaphragm, simulating a fiber core diameter of 10 μm.

The organization of the manuscript is as follows. In Section 2, we detail the principal optical scheme that might be used to test the algorithm, highlighting the key parameters of all essential components. This section is succeeded by a description and flowchart illustrating the combined stochastic gradient descent algorithm that we developed. Section 3 describes the significant numerical results obtained during the testing of the algorithm, particularly the total power collected both prior to and following the implementation of wavefront correction, along with an analysis of the RMS of the wavefront. Following this, a discussion subsection analyzes the numerical estimations performed with Zemax and explores the specific design considerations of the phase modulation device (i.e., piezoelectric mirror) that might be implemented to achieve improved wavefront correction results. Section 4 concludes the manuscript.

2. Materials and Methods

2.1. Principal Optical Setup Scheme

The principal optical scheme for numerical algorithm testing is presented in Figure 1a.

A laser diode (LD) at a wavelength λ = 1.55 µm, coupled to a single-mode fiber (SMF) of numerical aperture NA = 0.14 and mode field diameter MFD = 10.4 µm (refer to Appendix A.1 for details) is collimated with an achromatic lens (L1) with focal length f = 400 mm into a beam of diameter 45 mm. The beam is propagated along the optical bench for a distance of 2 m and reflected from the phase modulator (PM) [51,52] of diameter 50 mm with 42 control elements (refer to Appendix A.2 for PM parameters and principle of operation). The beam is then collected by the receiving telescope consisted of an achromatic lens (L2) of diameter 50.8 mm and focal length f = 150 mm, which focuses it into the fiber-simulator diaphragm (FSD) of diameter 10 µ (refer to Appendix A.3 for diaphragm and photodiode parameters) that emulates the real single-mode fiber at the receiver side of the optical communication system. The power passed through a diaphragm (i.e., collected by a fiber in real setup) is measured by a photodiode DET01CFC (PD), that is placed after the FSD. Its output is read using analog-to-digital converter (ADC) linked to the control personal computer (PC) using a USB 2.0 connection. The signal received is processed with our proprietary BeamShaper software v2.1 that calculates the merit function and uses it to define the control signals to apply to a PM [53,54,55]. Figure 1b demonstrates the idea of the replacement of an SMF of diameter 10 µm with a diaphragm of the same diameter. FSD (and SMF, consequently) is placed at a focal plane f of a coupling lens, i.e., L2.

2.2. Combined Stochastic Gradient Descent Algorithm

The combined algorithm employed in this research leverages the strengths of both stochastic algorithms [12] and gradient descent methods [13,14]. This algorithm operates in two distinct modes. Initially, the stochastic component rapidly maximizes the merit function to reach an optimal threshold, a process that will be elaborated upon later in this paper. Subsequently, the gradient descent component is activated to further refine and enhance the results obtained.

In instances where the merit function value stagnates or even declines, the stochastic algorithm is re-engaged to reinitiate the optimization process. The rationale behind this switch-mode approach is that the stochastic component serves as a rapid, rough-tuning mechanism, while the gradient descent component functions as a slower, more precise fine-tuning method. This dual-mode operation proves particularly advantageous in scenarios where there are no rapid dynamic wavefront distortions. A flowchart illustrating the structure and operation of the combined algorithm is provided in Figure 2.

Firstly, the stochastic stage of the algorithm was run (the left part of Figure 2).

Initially, we compute the vector ${∆X}_t$ of random control signals, which are uniformly distributed within the interval [0; 1] and scaled by a predefined amplitude of randomization A_R (Equation (1)):

$${∆X}_t={\epsilon }_t·A_R,$$

(1)

where ${∆X}_t$ represents the resultant vector of control signals, ${\epsilon }_t$ is a vector of random numbers, A_R denotes the amplitude of control signal randomization, t indicates the iteration number.

To introduce sparsity into the vector ${∆X}_t$, we apply a Bernoulli distribution [56], setting a portion of the vector to zero according to a predefined ratio Z_R within the interval (0; 1). The operation is presented in Equation (2).

$${∆X}_t={∆X}_t\odot Bernoulli,$$

(2)

where the function Bernoulli returns either 0 or 1 based on the Bernoulli distribution. Here, the $\odot$ sign denotes element-wise multiplication, since ${∆X}_t$ is a vector of control signals and $Bernoulli$ is a binary mask vector composed of 0 and 1. As is known [56], a random variable X follows a Bernoulli distribution with parameter p if its probability function takes the form (Equation (3)):

$$P\left(X=x\right)=\left\{\begin{array}{c}p,\\ q=1-p, \end{array}\right.\begin{array}{c} for x=1\\ for x=0\end{array},$$

(3)

Following this, we calculate two temporary vectors of control signals as in Equation (4):

$$\begin{array}{c}{∆X}_{t+}=X_t+{∆X}_t,\\ {∆X}_{t-}=X_t-{∆X}_t,\end{array}$$

(4)

where $X_t$ is the vector of control signals currently applied to the PM. Initially, this vector may consist of either zero values or bias values, depending on the state of the PM.

The calculated increment and decrement vectors are sequentially applied to the control elements of the PM, and the corresponding merit functions $f_{i+}\left(x\right)$ and $f_{i-}\left(x\right)$ are computed. The merit function is defined as the signal value obtained from the photodiode PD.

The correlation parameter is then calculated as follows (Equation (5)):

$${\eta }_t={\displaystyle \frac{f_{i+}\left(x\right)-f_{i-}\left(x\right)}{f_{i+}\left(x\right)+f_{i-}\left(x\right)}},$$

(5)

Subsequently, the resultant vector of control signals is updated according to Equation (6):

$$X_t=X_t+{∆X}_t\odot {\eta }_t.$$

(6)

Finally, the merit function value $f_i\left(x\right)$ is evaluated. If this value falls below a predefined threshold, the stochastic procedure is repeated. Conversely, if the threshold is met or exceeded, the rough-tuning stage concludes, allowing for a transition to the fine-tuning phase utilizing a gradient descent algorithm. The specific power threshold was found to be approximately 2 mW (it will be explained in more details in the next section of the manuscript).

Two variations of the gradient descent algorithm are implemented. The first variation operates slowly but offers greater precision, allowing for controlled adjustments to each element of the PM individually. The second variation employs Zernike-based control [57,58,59,60], whereby control values are not manually set but are instead computed based on the Zernike modes that need to be reproduced.

The initial step involves recalculating and setting a new vector of control signals $X_t=X_t+{∆X}_t$. If the merit function $f_i\left(x\right)$ decreases, the direction is reversed by changing the sign of ${∆X}_t=-{∆X}_t$ and this step is repeated. If the merit function $f_i\left(x\right)$ increases, adjustments continue in the same direction until a decrease is observed.

Once a decrease in the merit function $f_i\left(x\right)$ is detected, the direction is reversed once more, and the process moves to the next control element of the PM. The merit function $f_i\left(x\right)$ is assessed at each step, and if it falls below the threshold, the stochastic algorithm is reactivated.

The pseudocode of the proposed algorithm is given in Algorithm 1 below.

Algorithm 1. Combined stochastic gradient descent algorithm

Parameters: ${∆X}_t$—the vector of random control signals, uniformly distributed within the interval [0; 1]; AR—an amplitude of randomization; ${\epsilon }_t$—a vector of random numbers; t—an iteration number; $Bernoulli$—a function that returns either 0 or 1 based on the Bernoulli distribution; $X_t$—the vector of control signals currently applied to the PM; $f_{i+}\left(x\right)$$, f_{i-}\left(x\right)$—merit function values for calculated increment and decrement vectors; ${\eta }_t$—the correlation parameter; $f_i\left(x\right)$—resultant merit function value; Threshold—a threshold total power value that indicates which stage of the algorithm to activate. DelayCounterThreshold—parameter that indicates whether enough iterations were performed in order to switch to a gradient descent stage. Algorithm pseudocode:  #1. $\mathrm{Calculate} \mathrm{a} \mathrm{vector} {∆X}_t={\epsilon }_t·A_R$;  #2. $\mathrm{Zero} \mathrm{part} \mathrm{of} \mathrm{a} \mathrm{vector} {∆X}_t={∆X}_t\odot Bernoulli$;  #3. $\mathrm{Calculate} \mathrm{a} \mathrm{vector} {∆X}_{t+}=X_t+{∆X}_t$;  #4. $\mathrm{Set} \mathrm{vector} {∆X}_{t+}$ signals to the PM;  #5. $\mathrm{Calculate} \mathrm{a} \mathrm{merit} \mathrm{function} f_{i+}\left(x\right)$;  #6. $\mathrm{Calculate} \mathrm{a} \mathrm{vector} {∆X}_{t-}=X_t-{∆X}_t$;  #7. $\mathrm{Set} \mathrm{vector} {∆X}_{t-}$ signals to the PM;  #8. $\mathrm{Calculate} \mathrm{a} \mathrm{merit} \mathrm{function} f_{i-}\left(x\right)$;  #9. $\mathrm{Calculate} \mathrm{a} \mathrm{correlation} \mathrm{parameter} {\eta }_t$;  #10. $\mathrm{Calculate} \mathrm{a} \mathrm{vector} X_t=X_t+{∆X}_t\odot {\eta }_t$;  #11. $\mathrm{Set} \mathrm{vector} X_t$ signals to the PM;  #12. $\mathrm{Calculate} \mathrm{a} \mathrm{merit} \mathrm{function} f_i\left(x\right)$;  #13. $\mathbf{IF} \mathbf{(}{f}_i\left(x\right)<Threshold$)  #14.   GOTO: #1  #15. ELSE IF (DelayCounter < DelayCounterThreshold)  #16.   DelayCounter++;  #17. ELSE  #18.   // Initiate gradient descent stage  #19. $\mathrm{Calculate} \mathrm{a} \mathrm{vector} X_t=X_t+{∆X}_t$;  #20. $\mathrm{Set} \mathrm{vector} X_t$ signals to the PM;  #21. $\mathrm{Calculate} \mathrm{a} \mathrm{merit} \mathrm{function} f_i\left(x\right)$;  #22. $\mathbf{ IF} \mathbf{(}\mathrm{IsBetter}(f_i\left(x\right)$))  #23.  $\mathrm{Calculate} \mathrm{a} \mathrm{vector} {∆X}_{t+}=X_t+{∆X}_t$;  #24.  $\mathrm{Set} \mathrm{vector} {∆X}_{t+}$ signals to the PM;  #25.  $\mathrm{Calculate} \mathrm{a} \mathrm{merit} \mathrm{function} f_i\left(x\right)$;  #26.   $\mathbf{IF} \mathbf{\left(}\mathrm{IsBetter}\right(f_i\left(x\right)$))  #27.   GOTO: #23  #28.  ELSE  #29.   $\mathrm{Calculate} \mathrm{a} \mathrm{vector} {∆X}_{t-}=X_t-{∆X}_t$;  #30.   $\mathrm{Set} \mathrm{vector} {∆X}_{t-}$ signals to the PM;  #31.   $\mathrm{Reverse} \mathrm{the} \mathrm{signal} \mathrm{signs} {∆X}_t=-{∆X}_t$;  #32.   $\mathrm{Calculate} \mathrm{a} \mathrm{merit} \mathrm{function} f_i\left(x\right)$;  #33.   $\mathbf{IF} \mathbf{(}\mathrm{IsBetter}(f_i\left(x\right)$))  #34.      GOTO: #19  #35.    ELSE  #36.       GOTO: #1  #37. ELSE  #38.  $\mathrm{Reverse} \mathrm{the} \mathrm{signal} \mathrm{signs} {∆X}_t=-{∆X}_t$;  #39.  GOTO: #23;

Regarding the specific features and novelty of the algorithm, the developed combined stochastic gradient descent algorithm exhibits linear $O(N·T)$ time (or computational) complexity and linear $O\left(N\right)$ space complexity, where N is the number of samples (number of control elements) and T is the number of iterations required for convergence. It also demonstrates low sensitivity to variations in input parameters. Furthermore, its implementation is relatively straightforward and does not necessitate an in-depth understanding of the underlying system, in contrast to the SPGD method. A developed switch-mode approach allows us to use a stochastic component of the algorithm as a rapid, rough-tuning mechanism, while the gradient descent component as a slower, more precise fine-tuning method. This dual-mode operation proves particularly advantageous in scenarios where there are no rapid dynamic wavefront distortions.

3. Results and Conclusions

3.1. Principal Scheme Calibration

Before evaluating the performance of the developed algorithm, a comprehensive system calibration has to be conducted (refer to Appendix A.4 for detailed methodology) alongside an analysis of the total power that could potentially be captured by the photodiode.

In the initial phase of the calibration process, the diaphragm (FSD) needs to be removed, and the PM has to be substituted with a high-quality etalon plate to measure the total power. Following this, the FSD should be installed and a subsequent power measurement should be conducted. Finally, the PM should be installed, and additional power measurements should be performed to ensure consistency across different configurations.

The most important part for calibration was a photodiode. For calibration purposes, a simple optical setup might be assembled. The laser beam from a fiber-coupled diode laser should be collimated and directed to a set of neutral density filters. The filters are precisely set in the holders, and each of four holders has a few filters with different transmittance levels so that the desired transmittance can be selected. The attenuated beam fell on the photodiode, and a collected signal was analyzed. By changing the filters, a set of measurements can be performed, and the raw data set containing the transmittance values of the filters and the corresponding signal values from the photodiode can be obtained (Figure 3a).

The resultant set of points is perfectly approximated by the linear function (Figure 3b).

3.2. Measurements and Correction Results

Due to the thermal fluctuations present around the optical bench, each measurement phase has to be carried out over several hours to ensure stability and accuracy. The resulting graph depicting the power levels over time is presented in Figure 4.

The chart clearly illustrates the influence of thermal fluctuations on the wavefront of the optical beam, as well as their consequent effects on the total power collected.

Figure 5 provides an assessment of the wavefront aberrations caused by various optical components, including the etalon plate, PM, and lenses. These aberrations contribute cumulatively to the overall wavefront error budget of the system.

The green bar in Figure 5 represents the RMS error of an optical configuration in which an etalon plate has been utilized in place of a PM, and the photodiode has been substituted with a calibrated wavefront sensor (Shack–Hartmann type) [61,62,63,64,65,66]. In contrast, the yellow bar indicates the mean RMS of the entire system, which includes the contributions from the collimating lens, when the etalon plate is replaced with a PM. It is evident that this configuration results in an increase in RMS error, reaching up to 0.4 µm. The final red bar illustrates the additional RMS increase resulting from the aberrations introduced by the coupling lens. Furthermore, the dashed blue bar indicates the extra aberrations that arise over time due to thermal fluctuations within the optical setup.

Upon completion of the preliminary setup and calibration processes, we proceeded with the optimization procedure. The chart presented in Figure 6 depicts the power collected by the photodiode alongside the curve representing the residual wavefront error. The results in Figure 6 represent average values derived from approximately 20 repetitions, accompanied by a standard deviation of approximately 0.1.

The figure illustrates several distinct intervals during which various phases of the combined optimization algorithm are executed. Initially, the stochastic phase is implemented, as it facilitates rapid convergence and effectively compensates for a majority of wavefront errors, thereby significantly enhancing the total power collected by the photodiode. Once the power threshold is reached and the wavefront stabilizes over multiple iterations of the algorithm, the gradient descent phase is activated. This phase allows for further improvements in total power; however, due to its inherently slower nature, it is susceptible to failure if wavefront aberrations increase. Consequently, should this occur and the total collected power drop below a predetermined threshold (approximately 2 mW), the stochastic phase is reinitiated. To enhance the clarity of the data trends, we employed a moving average algorithm to smooth the raw data.

Figure 7 presents RMS for each Zernike mode, both prior to and following correction with the PM.

The final estimation results are provided in

Table 1. Before and after results.

Parameter	Before Optimization	After Optimization
Power	0.33 mW	2.3 mW
Wavefront RMS	0.63 μm	0.12 μm
CE	0.1	0.6

.

3.3. Algorithm and Error Analysis

During the numerical experimental trials, the optimal step size for the algorithm, specific to the given setup, was found to be equal to 15 V for the stochastic stage and 5 V for the gradient descent stage, which correspond to 5% and ~2% of the control signals range. We employed a linear adaptation rate (AR) for the step size throughout the optimization process. Notably, the Zero Rate (ZR)—a parameter designed to regulate the algorithm’s behavior in scenarios where no significant improvement is observed—was not utilized in our work. This omission was primarily due to the continuous activity of the algorithm, which was necessitated by the presence of heat flows.

Furthermore, since we implemented a combined algorithm, the gradient descent phase was initiated once the specified threshold was reached. The inherent characteristics of this consequential gradient descent algorithm resulted in a gradual fine-tuning of the merit function, contingent upon the conditions being favorable. We found that the algorithm may be terminated when the change in power level falls within ±3%, which could serve as a viable stopping criterion.

In our approach, we systematically decreased the step size in a linear fashion as the power value increased, and conversely, we increased the step size when the power value diminished. We conducted numerous tests, revealing that a range of 50 to 70 step sizes proved sufficient to attain optimal results during the stochastic phase of the algorithm. It is important to note, however, that the experiments did not account for real turbulence conditions, which may affect the performance of the algorithm in practical applications.

There are a few possible sources of errors during measurements. First of all, the noise produced by the photodiode which directly affects the measurement results. The estimated amplitude of power signal deviations was found about ±0.04 mW. The next one is the hysteresis of a PM, which is around 12% normally for such types of correctors. But since the control loop is closed (a PM performs based on the signal obtained by the photodiode), this might not be considered as a direct source of error. The next possible source of errors is an instability of a laser diode having 1% to 3% of a power instability and 1% RMS noise. The last possible source of errors is misalignments of optical setup. But since the initial calibration of the setup was performed without a diaphragm, and the PM has the capacity to actively control a laser beam and guide it through a pinhole, it is also less likely to be considered as a source of errors.

By assessing the resultant correction efficiency, we can conduct a comparative analysis of the developed algorithm against established methods, specifically the Stochastic Parallel Gradient Descent (SPGD) algorithm, Genetic Algorithm (GA), Simulated Annealing (SA), and the Algorithm of Pattern Extraction (Alopex). An examination of the Zernike coefficients reveals that the outcomes produced by the newly developed Combined Stochastic Gradient Descent (CSGD) algorithm align more closely with those of the SPGD algorithm. The CSGD algorithm exhibits superior efficiency, particularly in the presence of low-intensity heat flows, due to the implementation of an effective consequential gradient descent stage that meticulously calibrates the control signals, thereby minimizing disruptions to the merit function.

When evaluating the Mean Radius (MR), the resultant MR values across the aforementioned algorithms are relatively comparable. For an initial Strehl ratio of 0.1, the MR value obtained with the CSGD algorithm is approximately 0.53, while the corresponding values for the SPGD, SA, GA, and Alopex algorithms are approximately 0.52, 0.54, 0.55, and 0.53, respectively, as referenced in [51]. At present, it remains challenging to compare the number of iterations due to the absence of experiments conducted under real turbulence conditions. Given the quasi-static nature of the aberrations, the convergence rate is notably high, requiring around 50 iterations. Considering the stochastic characteristics inherent in the CSGD algorithm, it can be inferred that the number of iterations would likely fall within the range of 400 to 500, in contrast to the 464 iterations observed for the SPGD algorithm.

The next point of discussion pertains to the total power collected by the photodiode, as illustrated in Figure 4. The data reveal that the installation of a fiber-simulator diaphragm (FSD) within the optical path results in a dramatic reduction in the power level, decreasing to approximately 22% of the power level achieved without the FSD in place. This significant drop is attributed to the far-field energy distribution characteristics of the coupling lens.

To further investigate this phenomenon, we conducted a simulation using Zemax optical design software. The results indicate that, for the coupling lens used, only 23–24% of the total energy is capable of passing through the utilized diaphragm, as depicted in Figure 8a. Figure 8b illustrates the energy distribution associated with a diffraction-limited lens, while Figure 8c presents the simulated encircled energy curve, which indicates that up to 92% of the power could potentially be collected after traversing a diaphragm when employing a diffraction-limited lens.

As we did not utilize an actual fiber for coupling, we estimated the potential coupling efficiency (CE) by examining its dependence on the RMS of the wavefront, as previously documented in the literature [15]. According to this theoretical framework, the CE improved from less than 0.1 to approximately 0.6 during the correction process facilitated by the developed algorithm.

One notable enhancement implemented during the testing of the algorithm involved grouping control elements into larger assemblies based on the distribution of Zernike modes that required compensation. As illustrated in Figure 7, axis-symmetrical aberrations, particularly defocus and spherical aberrations, are predominant. To effectively and robustly compensate for these specific Zernike modes, it is essential to maintain axial symmetry among the control elements. Consequently, we can programmatically combine the corresponding rings of control elements in a PM to form larger quasi-elements. This method of combining control elements is well-established and is predominantly utilized in systems featuring spatial light modulators and MEMS mirrors, which typically contain a high number of control elements. However, this approach is also applicable to our system. Specifically, such combinations enable the reduction in the number of control elements from 42 to 6, which may enhance the convergence speed of the gradient descent algorithm.

The developed algorithm has $O(N·T)$ time (computational) complexity and $O\left(N\right)$ space complexity, where N is the number of samples (number of control elements) and T is the number if iterations required for convergence.

The external control elements of any phase correction device are essential to the overall efficiency and performance of the system. Their functionality significantly influences the effectiveness of the phase correction process, thereby impacting the system’s operational capabilities. To maximize the PM’s ability to correct for symmetrical Zernike modes—namely modes #3, #8, #15, and #24—it is imperative that the control elements within the outer ring are illuminated by radiation covering more than 70% of their surface area [67]. Failure to achieve this coverage may result in residual aberrations at the edges of the beam, thus diminishing the focusing efficiency of the entire system. Given that beam diameter can fluctuate due to various factors, the outer ring of control elements should be intrinsically larger (Figure 9).

Preliminary estimates suggest that this design modification might enhance correction efficiency by approximately 5–9%. Looking ahead, we plan to extend this research by testing the system and algorithm under real laboratory conditions as well as under conditions of atmospheric turbulence. This will involve artificial turbulence, induced through methods such as a fan heater or a precise turbulence generator [68,69]. To address the limitation of utilization of a diaphragm and a photodiode, we intend to substitute it with an actual optical fiber and subsequently replicate the research to evaluate the impact of this modification on the results.

4. Conclusions

In conclusion, we have developed and examined a combined stochastic gradient descent algorithm specifically designed for the focusing of a laser beam through a diaphragm that simulates a fiber core with a diameter of 10 μm. The algorithm we developed exhibits linear time and space complexity and demonstrates low sensitivity to variations in input parameters. Furthermore, its implementation is relatively straightforward and does not necessitate an in-depth understanding of the underlying system, in contrast to the SPGD method. In addition, a developed switch-mode approach allows us to use a stochastic component of the algorithm as a rapid, rough-tuning mechanism, while the gradient descent component is used as a slower, more precise fine-tuning method. This dual-mode operation proves particularly advantageous in scenarios where there are no rapid dynamic wavefront distortions. The algorithm demonstrated a significant capacity to enhance the total power collected by the beam as it passed through the 10 μm diaphragm, increasing the power from 0.33 mW to 2.3 mW. Additionally, the RMS of the wavefront was effectively reduced from 0.63 μm to 0.12 μm, which has the potential to elevate the coupling efficiency from 0.1 to 0.6.