Results and Discussions It can be seen from the simulation results of two-stage SPGD and conventional single-stage SPGD algorithms under 64- and 100-beam conditions that the convergence speed of two-stage SPGD algorithm is significantly faster than that of conventional SPGD algorithm (Fig. 4). When the same standard deviation is reached, the number of iterations of the two-stage algorithm is reduced by 56.6% (64 beams) and 67.9% (100 beams) compared with single-stage SPGD (Table 1). Simulation results show that the number of iterations of single-stage SPGD algorithm increases rapidly with the increase in beam number and reaches 9522 for 1000 beams, while the number of iterations of two-stage SPGD for 1000 beams is 107.25 (Fig. 5). Using multi-stage SPGD algorithm, the number of iterations is reduced by 98.87%. The optimal number of stages for different number of beams is given (Fig. 8). In the experiment, an optical phased array system is built, and a two-stage SPGD algorithm is used to lock the phase of 16 beams. When the closed loop is on, the optical signals of the first and second stages increased rapidly to a stable state (Fig. 10), which proves that the multi-stage SPGD algorithm can realize the phase locking of beams. And the phase locking performances using single-stage SPGD and two-stage SPGD are compared under the conditions of 8/12/16/32 beams respectively. The results show that two algorithms have almost the same effect when the number of beams is small, but with the increase of the number of beams, the convergence speed of the two-stage SPGD algorithm becomes faster than that of the single-stage SPGD algorithm. For 32 beams, the multi-stage algorithm reduces the number of iterations to 688.7, which is 59.6% of that of the single-stage algorithm (Fig. 12).

In the process of space laser communication networking, optical phased array (OPA) can replace the traditional mechanical turntable to realize the lightweight and miniaturization of laser terminals, and can quickly switch links among different terminals. The space laser communication network based on optical phased array will be the inevitable trend of the future development. In OPA technology, it is necessary to ensure the same phase of all the beams split from the same laser source. Therefore, the monitoring part is set to obtain the phase of beams to do the phase locking. Considering a large number of beams, it is not practical to calculate the phase directly, so it is suitable to use iterative algorithm to calculate the compensation. At present, there are many algorithms in the field of multivariable control, and stochastic parallel gradient descent (SPGD) algorithm has more advantages in convergence speed and effect, but with the increase of the number of OPA elements, the convergence speed of SPGD becomes significantly slower. There exist some optimized SPGD algorithms now, such as AdmSPGD, AdaDelSPGD and other schemes, which have optimized the step size, the gain parameter and other coefficients of SPGD, and improved the speed to a certain extent. In the experiments using SPGD algorithm, the largest number of beams in the array reported is written by the 107-channel fiber laser coherent synthesis based on SPGD algorithm. As the number of beams increases, the structure becomes more complex and the performance of the algorithm must be higher. Nowadays, OPA has been widely used in laser communication and lidar where there is always a large optical aperture and a large number of array elements are required. When the number of array elements reaches thousands, only optimizing the parameters of SPGD algorithm does not change the essence of control of all array elements based on one single evaluation parameter, which has limited performance improvement. Therefore, it is necessary to optimize the SPGD algorithm in a deeper level for the situation of large-scale array.

In this study, based on the principle and convergence of conventional SPGD algorithm, a multi-stage SPGD algorithm for OPA with a large number of elements is proposed. Different from optimizing the parameters of SPGD algorithm, the beams of multiple array elements are divided into several stages, and there are groups in each stage. The perturbation phase of all the stages is added to the beams, then the phase locking of beams at each stage is made, and finally the phase of all the beams is locked. The corresponding phase adjustment is carried out by monitoring the change of the combined beam power of each stage and the change of the total combined beam power.

Based on the conventional SPGD algorithm, a multi-stage SPGD algorithm with better performance is proposed, which can achieve faster convergence under the condition of large-scale OPA. The core of the multi-stage SPGD algorithm is to group the phased array beams and add multi-stage perturbation, which can realize the global phase control as well as the local phase compensation. In this paper, the principle and flow of multi-stage algorithm are introduced, and simulation is carried out. The results show that the multi-stage SPGD algorithm is more advantageous than the conventional one, and the larger the number of beams is, the more obvious the effect is. When the beam number is 1000, the multi-stage SPGD algorithm reduces the number of iterations by 98.87%, and the optimal number of stages of the multi-stage SPGD algorithm for different beam numbers is given. Finally, an OPA experimental system is built to verify the feasibility of the multi-stage SPGD algorithm. The experimental results show that the multi-stage SPGD algorithm can realize phase locking in the case of 8/12/16/32 beams, and has fewer iterations compared with the conventional single-stage algorithm. Especially, the number of iterations can be reduced to 59.6% of that of the single-stage algorithm for 32 beams. In the actual application, the elements of OPA can be divided to several groups to connect different terminals at the same time, and the multi-stage SPGD algorithm fits well with the requirements, which can adjust the phase of global and local beams. The research is of great significance to the development of phase control technology of large-scale OPAs.