# Theory of AdaDelSPGD Algorithm in Fiber Laser-Phased Array Multiplex Communication Systems
## 1 Introduction
However, the heterodyne method requires that the co-axis of the reference light and the signal light of each path must be strictly parallel and perpendicular to the detector array. As the amount of system bunding increases, the number of control modules increases correspondingly, which leads to an increase in the system's complexity and a deterioration in the system's stability. Researchers at HRL Laboratories in the United States have achieved a 7-cell, 1-watt coherent synthetic output of fiber lasers using multi-dither active phase control techniques [23,24]. [PERSON] et al., US Air Force Research Laboratory, reported the coherent synthesis of 16 narrow line-width fiber lasers, with a total output power of 1.4 kW [25]. The Kirtland AIR Force Research Laboratory implemented a nine-channel high power fiber laser phase-locked beam without reference beam using the multi-dithering method [26]. Meanwhile, Thales Research Technology Laboratory obtained coherent beam synthesis of 64 independent amplifying fibers by using four-wave shear interferometer [27]. Compared with the heterodyne method, the multi-dither method has no reference light. The structure of the system is simple and easy to implement. However, the phase noise frequency of the system increases with the increase in the number of beam combination elements. In order to effectively control the phase of each unit beam, the carrier frequency of the phase modulation part of the system must be increased. The bandwidth of the phase control system is higher than the noise frequency of the coherent synthesis system, which undoubtedly makes the design of the phase control demodulation circuit more complex. Considering the complexity of the system and the control effect, the optimization algorithm has been widely used in adaptive optics. The more commonly used adaptive stochastic optimization algorithms are mountain climbing, genetic algorithm, simulatedannealing and stochastic parallel gradient descent (SPGD) [28; 29]. By comparing the application of various optimization algorithms in coherent laser synthesis, the convergence speed of mountain climbing and genetic algorithm is too slow. The convergence speed of the simulated annealing algorithm is slightly lower than that of the SPGD algorithm. Meanwhile, the SPGD algorithm not only has a fast convergence speed, but also has a good tolerance in dealing with power fluctuations [30]. Therefore, the SPGD algorithm has been widely used in recent years. Researchers at the University of Maryland in the United States achieved the coherent synthesis of a seven-unit milliwatt fiber laser using SPGD algorithm phase control technology [31]. [PERSON] et al., Massachusetts Institute of Technology, USA, reported a 48-channel milliwatt-level coherent fiber laser synthesis scheme [32]. In 2011, researchers from the MIT Lincoln Experiment used the SPGD algorithm to control the phase of each unit beam, and realized the coherent synthetic output of eight channels of a 0.5 kW commercial fiber laser amplifier [33]. [PERSON]'s team realized coherent beam synthesis of 21 channels at a distance of 7 km using SPGD algorithm [34; 35]. [PERSON] et al. used SPGD algorithm to achieve two km laser aiming [36]. Then they realized a large-scale fiber laser array over 2.1 km [37]. In addition, China's National University of Defense Technology (CNNDT) applied SPGD algorithm to achieve a coherent beam synthesis system of 60 fiber lasers [38]. In 2021, [PERSON] et al. implemented 19 laser-phased array using SPGD algorithm [39; 40]. Then, in 2022, they use a new evaluation function to optimize the SPGD algorithm. [PERSON] et al. used SPGD algorithm to realize the coherent synthesis of 64 elements in phased array [41]. Several modification techniques have been used to improve the convergence of SPGD algorithm. [PERSON] et al. proposed a decoupled stochastic parallel gradient descent algorithm with adaptive change gain coefficients.