# Theory of AdaDelSPGD Algorithm in Fiber Laser-Phased Array Multiplex Communication Systems
## 2 Theoretical Analysis and Simulation of the Improved SPGD Algorithm
### SPGD Algorithm and Simulation  
The coherent combination process of n-channel fiber laser-phased array multiplex communication system using the traditional SPGD algorithm can be described by several steps. Firstly, the performance metric \(J=J(u_{1},u_{2},\cdots,u_{N})\) is used as the algorithm's optimization object, which is a function of the phase control voltage \(u=(u_{1},u_{2},\cdots,u_{N})\) of each sub-beam. Then, the phase controller applies a random microperturbation voltage satisfying Bernoulli distribution to each phase modulator. Unique extreme values are obtained by calculating changes \(\delta J^{(n)}=J^{(n)}_{+}-J^{(n)}_{-}\) in performance indicators. The iteration formula of phase-controlled voltage update is:  
\[u^{(n+1)}=u^{(n)}-\gamma\delta J^{(n)}u^{(n)} \tag{1}\]  
Figure 1: The scheme of the N-channel fiber laser-phased array multiplex communication system. MA: microwave amplifier; IM: intensity modulator; PM: phase modulator; VG: V-shaped groove; BS: beam splitting mirror; OFC: optical fiber collimator; PD1: Photo detector of phase locking; PD2: Photo detector of communication receiving system; EDFA: Erbium Doped Fiber Amplifier; Subsystem1: Laser transmitting system; Subsystem2: Communication modulation system; Subsystem3: Phase locking system; Subsystem4: Communication receiving system.  
where the superscript \(n\) is the number of iterations, and \(\gamma\) denotes the fixed gain coefficient of algorithm. When the performance index of the coherent synthesis system is optimized to the maximum, \(\gamma<0\); otherwise, \(\gamma>0\). If the fixed gain coefficient is too small and the iteration step is short, the convergence speed of the algorithm is slow, and it is difficult to realize real-time phase compensation. However, if the fixed gain coefficient is too large, the iteration step may cause the system performance evaluation function to fluctuate too much during the algorithm iteration, making the algorithm unstable.  
In order to verify the effectiveness of SPGD algorithm in fiber laser-phased array multiplex communication systems, simulation analysis is carried out in free space environment. A 1064 nm laser beam with a waist of 5 mm was arranged in a honeycomb shape with an interval of 20 mm between each sub-beam. Using SPGD algorithm, PIB signal was used as the performance index of the system to control the phase of the transmitted beam. The initial phase value of the beam satisfied the Gaussian distribution between \([-\pi,+\pi]\) under zero mean value.  
The performance measures obtained by SPGD algorithm in different gain systems (the average value of 100 independent runs) are shown in Figure 2. In addition, the convergences of the SPGD algorithm in three-channel and seven-channel laser coherent synthesis are shown in Figure 2a,b, respectively.  
As can be seen from the Figure 2, in the case of three-channel, the larger the iteration coefficient is, the faster the convergence rate is. But in the case of seven-channel, the expected value cannot be reached when the iteration coefficient is 0.9, so the iteration step size of the SPGD algorithm should not be too small or too large. If the step length is small, the speed is slow. If the step length of iteration is large, it may eventually fail to reach the maximum value of the target. Thus, we propose the AdaDelSPGD algorithm, hoping to improve the convergence speed and finally make the system stable at the target position at the same time.