## 3 NadamSPGD Algorithm in SLAO
### Conventional SPGD in SLAO
Finally, the beam reaches the lens L2, and the speckle images are captured by the HSC. The computer is used to process the captured images, execute the algorithms and control the CSDM through the driving circuits to compensate the aberrations. In our experiments, the performances of SPGD and NadamSPGD are evaluated under the same initial conditions for a fair comparison. A set of CSDM initial control voltages are generated according to Zernike coefficients to generate the initial wavefront aberration, and then the control voltages are algorithmically calculated to gradually flatten the CSDM itself. The flattening process of CSDM can simulate the correction process of atmospheric turbulence. The energy concentration rate of speckle images is calculated to reflect the performance of SLAO. The optimal setting for SPGD is \(\Delta u=0.5\) and \(\gamma=2\), and the optimal setting for NadamSPGD is \(\Delta u=0.5\), \(\alpha=0.1\), \(\mu=0.999\), \(\upsilon=0.99\) and \(\varepsilon=10^{-8}\). The experimental results are shown in Figure 11, from which we find that SPGD corrects slowly. After 300 iterations, the initial ME reaches 0.5331 from 0.1142, which significantly affects the system performance. However, NadamSPGD dynamically adjusts the gain factor according to the gradient estimation to achieve a rapid convergence. The ME using NadamSPGD reaches 0.8828 after 300 iterations, which is 1.66 times faster than SPGD. ## 5 Conclusions

In this paper, a novel NadamSPGD algorithm combining Nadam and SPGD is proposed to compensate wavefront aberrations more effectively in CFSOC. The theoretical analysis and numerical simulations demonstrate that the negative influence of varying degrees of atmospheric turbulence on ME and BER of the CFSOC system can be suppressed by NadamSPGD. Specifically, by integrating the NAG momentum and adaptive gain coefficients into the conventional SPGD, the proposed algorithm can not only accelerate the correction speed by approximately 50%, but also improve the robustness of parameters over a large range (\(\Delta u\in[0.01\), \(1]\)) without noticeably increasing the complexity of the

\[\hat{n}^{(k)}=n^{(k)}/\Big{(}1-v^{k}\Big{)}, \tag{14}\]6 Comparison of correction results using SPGD and NadamSPGD on the SLAO experiment platform: (**a**) initial far-field image before correction; (**b**) SPGD correction result after 300 iterations; (**c**) SPGD residual wavefront ME as a function of iterations; (**d**) initial far-field image before correction; (**e**) NadamSPGD correction result after 300 iterations; (**f**) NadamSPGD residual wavefront ME as a function of iterations. algorithm. Simultaneously, the stronger the turbulence intensity, the more obvious are the advantages of NadamSPGD. In addition, NadamSPGD exhibits improved dynamic capabilities as the Greenwood frequency increases, and therefore, it is more suitable for real-time SLAO systems. Finally, the effectiveness of the proposed algorithm is evaluated on our SLAO experiment platform, and the experimental results indicate that NadamSPGD converges much faster. In conclusion, NadamSPGD is more effective for SLAO to improve the communication quality of CFSOC systems, and is a good substitute for SPGD. From the findings of this paper, researchers can design SLAO systems with excellent performance in CFSOC based on NadamSPGD. The proposed algorithm may shed light on the application of SLAO-based wavefront correction technology, such as astronomical observation, fiber laser coherent synthesis and biological microscopic imaging. In the future, we will build a high-performance processing platform based on FPGA and GPU, and apply the NadamSPGD to dynamic aberration correction experiments. Conceptualization, L.X. and H.Z. ; methodology, J.W. ; software, L.X. ; validation, L.Y. and H.Z. ; data curation, H.Z. ; writing--original draft preparation, L.X. ; writing--review and editing, H.Z. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable.