Objective High-power lasers are being used in an increasingly wide range of applications. Coherent beam combining (CBC) of multiple lasers represents the most effective method for obtaining high-quality and high-power laser output. Fast and accurate phase control of each laser is crucial for achieving laser CBC, and the stochastic parallel gradient descent (SPGD) algorithm is widely used because of its advantages of parallel control and simple structure. However, the traditional SPGD algorithm faces challenges where the convergence speed and effectiveness are compromised due to fixed gain coefficients and disturbance amplitudes. Specifically, when the gain coefficient or disturbance amplitude is small, the algorithm tends to converge slowly. Conversely, setting a large gain coefficient or disturbance amplitude may cause the algorithm to fall into local optimal solutions. Moreover, as the number of composite beams increases, the required iteration steps for algorithm convergence will significantly rise, failing to meet the iteration time requirements for large-scale CBC systems. Therefore, optimizing the SPGD algorithm is essential to enhance its convergence speed, stability, and scalability. This optimization is a necessary step for realizing large-scale laser CBC.

Methods In order to improve the issue that the traditional SPGD algorithm converges slowly and tends to fall into local optimal solutions when it is applied to large-scale laser CBC, a staged adaptive gain SPGD (Staged SPGD) algorithm is proposed. This algorithm adaptively adjusts the gain coefficient based on the performance evaluation function, using different strategies to enhance the convergence speed. Additionally, a control voltage update strategy with a gradient update factor is introduced to effectively update the control voltage, accelerate convergence, and mitigate instances where the algorithm gets stuck in local extreme values.

Results and Discussions The performance of the traditional SPGD algorithm and the proposed algorithm in the 19-laser CBC system is analyzed. The results show that the fixed gain coefficients of the traditional SPGD algorithm significantly impact convergence, with improper gain settings potentially leading to slow convergence or local optima (Fig.3). In contrast, the proposed algorithm significantly improves convergence performance by adaptively adjusting the gain coefficient and reduces sensitivity to the initial gain (Fig.4). Additionally, the random perturbation voltage affects convergence performance; The SPGD algorithm is more sensitive to changes in perturbation voltage, while the proposed algorithm exhibits better robustness within a certain range (Fig.5). The gradient update factor C also significantly impacts the convergence speed of the proposed algorithm, and setting C within a reasonable range (0.1 to 0.4) can substantially enhance algorithm performance (Fig.6). To validate the efficacy of the proposed algorithm's dual optimization strategy, independent simulation experiments were conducted on 19-laser CBC system under different improved operations (Fig.7). The results indicate that the proposed dual optimization strategy can effectively improve the convergence speed and stability of the SPGD algorithm when applied to the laser CBC system. The superiority of the proposed algorithm is assessed by comparing the convergence curve of the algorithm (Fig.8) and the number of iterative steps required for the convergence of the algorithm (Fig.9). The results show that when the evaluation function J of the proposed algorithm converges in 193 761 and 91 laser CBC systems, the number of iteration steps required is always smaller than other schemes, which has obvious advantages in convergence speed, and its advantages gradually increase with the increase of the number of beams. By calculating the time required for algorithm iteration (Tab.3), it can be seen that the proposed algorithm has improved overall efficiency. By analyzing the distribution of iteration steps when each algorithm converges (Fig.10), it can be concluded that the proposed algorithm has strong stability. Applying phase noise of different frequencies and amplitudes to both the Staged SPGD algorithm and the traditional SPGD algorithm (Fig.11) reveals that as the phase noise frequency, amplitude, and number of beams increase, the average J value of the system decreases, and convergence performance worsens. However, under the same phase noise condition, the average J of the system is higher after phase control with the Staged SPGD algorithm, which indicates that the proposed algorithm has the best control effect on the beam phase and better noise suppression performance.

Conclusions Targeting the slow convergence and observed susceptibility to local extremes when applying the traditional SPGD algorithm to large-scale laser CBC systems, a staged adaptive gain SPGD algorithm is proposed by improving the traditional SPGD algorithm. The proposed algorithm changes the fixed gain into the staged adaptive gain according to the performance evaluation function, and improves the real-time performance of the system. Additionally, the control voltage updating strategy with gradient updating factor is introduced to effectively update the control voltage and reduce the probability of getting stuck in the local optimal solution. The simulation results show that in the 19-laser array CBC system, compared with the traditional SPGD algorithm, the proposed algorithm improves convergence speed by 36.84%. The algorithm demonstrates better convergence in phase noise environments with varying frequencies and amplitudes, which greatly improves the stability of the algorithm, and the algorithm also has certain advantages in the iteration time. Furthermore, when directly applied to 37-laser, 61-laser, and 91-laser CBC systems, the Staged SPGD algorithm improves the convergence speed by 37.88%, 40.85%, and 41.10%, respectively, compared to the traditional SPGD algorithm. The improvement effect is more significant with the increase of the number of lasers, which indicates that the algorithm has certain advantages in convergence speed, stability and scalability, and has the potential to be extended to large-scale CBC systems.