## 4 Comparison with SPGD Algorithm  
SPGD is a typical algorithm in polarization control systems, and relevant investigations have been extensively carried out [18; 19; 20; 21; 27]. In this section, we simulate the convergence performance of the SPGD algorithm for comparison with the RMS-Prop algorithm.  
Similar to the RMS-Prop algorithm, the SPGD algorithm is one kind of gradient optimization algorithm, and random perturbations \(\delta\)u with Bernoulli distribution are also imposed to control signals when the SPGD algorithm works. Then, the cost function under positive and negative perturbations, J\({}^{+}\) and J\({}^{-}\), are obtained, respectively. The iteration of control signal u can be expressed as follows:  
\[u_{i+1}=u_{i}+\gamma\delta u_{i}\big{(}J^{+}-J^{-}\big{)} \tag{13}\]  
where i is the iteration number, and \(\gamma\) is the gain factor.  
Cost function J, perturbation amplitude \(\delta\)u, and gain factor \(\gamma\) are the three main factors that have influence on the convergence performance of the SPGD algorithm. The impact of the three parameters on the algorithm are studied. The simulation results are shown in six sub-pictures (Figure 5a-f). Among them, the three pictures on the left (Figure 5a,c,e) are the results of DOP, and the three pictures on the right (Figure 5b,d,f) are the results of PER. Sub-pictures Figure 5a,b show the convergence performance of the SPGD algorithm with different cost functions. We set \(\delta\)u = 0.05 and \(\gamma\) = 15 according to previous experience of correlational study [18; 27]. J\({}_{1}\)-J\({}_{3}\) correspond to Formulas (10)-(12). Apparently, J\({}_{2}\) has a better convergence speed and accuracy than J\({}_{1}\), while J\({}_{3}\) fails to converge in the SPGD algorithm. We use J\({}_{2}\) as the cost function in our following simulations. Sub-pictures Figure 5c,d are the simulation results of different perturbation amplitudes. \(\gamma\) is still set to 15, and \(\delta\)u ranges from 0.04 to 0.3. When \(\delta\)u = 0.04, 0.08, and 0.12, DOP converges to 0.95 through 167, 46, and 22 iterations. When \(\delta\)u = 0.16, the convergence value of PER begins to deteriorate. When \(\delta\)u = 0.3, it is clearer from both figures that the convergence value is low and unstable. Sub-pictures Figure 5e,f show the simulation results of different gain factors \(\gamma\) from 10-100. The convergence speed is relatively slow when \(\gamma\) is 10 or 20, converging to 0.99 with 50 and 32 steps, which is less than 20 steps for \(\gamma\) of 40 and 60. When \(\gamma\) is greater than 60, it is difficult for the SPGD algorithm to converge.  
When utilizing the DOP of output laser (J\({}_{2}\)) as the cost function, and setting the parameter \(\delta\)u to 0.12 and \(\gamma\) to 60, the SPGD algorithm will achieve the optimal convergence effect. We compare the SPGD algorithm with the RMS-Prop algorithm, and their optimal convergence effect of PER is shown in Figure 6. Although the convergence speed of the RMS-Prop algorithm is actually not slow, the SPGD algorithm converges faster obviously. When it comes to convergence accuracy, the RMS-Prop algorithm is superior to that of the SPGD algorithm. When the iteration number reaches a certain value (85 in our simulation), the convergence PER of RMS-Prop algorithm is greater than the SPGD algorithm. After three hundred iterations, the lead is extended to \(\sim\)10 dB (68.9 dB for the RMS-Prop algorithm and 58.4 dB for the SPGD algorithm. Thus, despite that the RMS-Prop algorithm still has room for improvement in the convergence speed, it is promising to improve the PER of active polarization control system, as well as increasing the power of linearly polarized fiber lasers.