# Theory of AdaDelSPGD Algorithm in Fiber Laser-Phased Array Multiplex Communication Systems
## 1 Introduction  
In recent years, due to their strong anti-interference ability and high confidentiality, lasers have shown good application prospects in free-space optical communication, optical detection and ranging (LIDAR), image projection, laser radar, optical storage, and other fields [1, 2, 3]. IPG Photonics uses cascade pumping to achieve a 10,000 W fiber laser output [4]. The literature shows that the output power of IPG Photonics' single-fiber single-mode fiber laser reaches 20 kW, and that of a multi-mode fiber laser reaches 50 kW [5, 6]. With the development of high-power fiber lasers, many countries have shifted the research and development of tactical laser weapons to high-power fiber laser weapons. The Zeus laser mine-clearing system developed by SPATA, Raytheon's Naval Laser Weapon System (Laws), and the Prototype fiber laser weapon developed by Euromissile have all been approved by the military [7, 8, 9]. At present, the maximum output power of an individual single-mode continuous fiber laser can reach 20 kW, but this is far from the requirement for high-energy lasers in some industries and in national defense. Thus, the output power of an individual single-mode fiber laser is limited [10]. [PERSON] et al., from the Livermore National Laboratory in the United States, analyzed in detail the factors limiting the improvement of the output powerof fiber lasers. According to their calculations, the theoretical output power limit of a single-mode single-frequency fiber laser is approximately 1.86 kW, and the theoretical output power limit of a single-mode wide-spectrum fiber laser is approximately 36.6 kW [11]. Therefore, it is not feasible to obtain a laser output satisfying the tactical high-energy laser system requirements only through a single fiber laser [12]. Through coherent combination technology, the output power of laser beam combinations has improved significantly in the past 20 years, resulting in outstanding performance in terms of output power, beam quality, and overall efficiency of various devices based on laser combination [13,14]. The laser coherent synthesis method has successfully solved the nonlinear effect of single fiber output power and the limitation of thermal damage [15]. The key difficulty of laser coherent combination is to realize the in-phase output of the whole laser array. At present, there are two main solutions of phase locking technology: passive phase locking [16], and active phase locking [17]. Researchers at the Abbe Photonics Center, University of Jena, Germany, achieved a coherent synthetic output of a four-unit high-power femtosecond pulse fiber laser through active phase compensation [18]. The average/peak output power of the system was 530 W/1.8 GW, and the system synthesis efficiency was up to 93%. In 2018, the research team reported a four-channel ultra-fast fiber laser coherent synthesis system with an average output power of 3.5 kW [19]. This system is the ultrafast fiber laser coherent synthesis system with the largest average output power reported. With the development of adaptive optics, researchers have developed several active phase control methods for multi-channel coherent synthesis. Heterodyne, multi-jitter, and the optimization algorithm are the most widely used active phase control techniques. [PERSON] et al. achieved coherent fiber laser synthesis with a total output power of 8 W in four units using the heterodyne method [20]. [PERSON] et al., MIT, used two commercial 10 W ytterbium-doped fiber amplifiers to achieve a coherent synthetic output based on heterodyne [21]. The Lincoln Laboratory used heterodyne method to achieve 32-channel fiber laser phase-locking [22]. A coherent synthesis system based on heterodyne phase control requires a reference beam and an array of detectors with the same number of combined beams. The phase control modules of each beam in the coherent synthesis system work in parallel. This means that the heterodyne phase control system has a higher system control bandwidth and stability.