## 3 NadamSPGD Algorithm in SLAO
### Conventional SPGD in SLAO
The SPGD reaches the index after at least 166 iterations, and at most 242 iterations in the worst case, averaging at

Figure 4: Wavefront correction results under different iterations: (**a**) residual wavefront aberration and the corresponding PSF after 100 iterations; (**b**) residual wavefront aberration and the corresponding PSF after 200 iterations; (**c**) residual wavefront aberration and the corresponding PSF after 300 iterations; (**d**) residual wavefront aberration and the corresponding PSF after 400 iterations. Figure 5: Comparison of the optimization curves between SPGD and NadamSPGD: (**a**) ME curves; (**b**) system BER curves. 193 iterations. NadamSPGD converges after at least 81 iterations, and at most 163 iterations, with an average of 112 iterations, which is 58.03% of SPGD. In addition, for the standard deviation, SPGD fluctuates greatly, as they merely depend on the random disturbance at each iteration and the current gradient. The NadamSPGD considers both the current gradient and historical gradients during the iteration process; thus, it reduces the impact of randomness. In summary, NadamSPGD not only converges faster than SPGD, but also has better robustness to the randomness of disturbances. Figure 5b shows that the system BER is substantially improved after the wavefront aberration is compensated with NadamSPGD. The BER dropped from approximately \(10^{-3}\) to \(10^{-10}\) after 149 iterations. Due to the inherent limitation of SPGD, the value of BER cannot be suppressed below \(10^{-10}\) until 262 iterations. The amplitude of random perturbation voltages has a great influence on the performance of gradient-based algorithms. To verify the robustness of the two algorithms to perturbations, we evaluate SPGD and NadamSPGD under the same settings as previous simulations, except changing \(\Delta u\) from 0.01 to 1. The results are shown in Figure 6, from which we can see that SPGD is extremely sensitive to \(\Delta u\). When \(\Delta u\) is 0.01, the correction speed of SPGD is extremely slow (Figure 6a), and when \(\Delta u\) increases to 0.1, the ME curve based on SPGD only reaches 0.3308 after 400 iterations (Figure 6b). When \(\Delta u\) increases to 1, SPGD prematurely converges to the local optimum (Figure 6c). However, the change in \(\Delta u\) has almost no effect on the correction performance of NadamSPGD, and it still works well with \(\Delta u\in[0.01\,\ 1]\) (shown in Figure 6d-f). The results demonstrate that NadamSPGD has robustness in a large range of \(\Delta u\) values, which makes it easy to apply to practical applications of SLAO. According to the Kolmogorov turbulence model, the ratio of the receiving antenna aperture \(D\) to the atmospheric coherence length \(r_{0}\) (\(D/r_{0}\)) can characterize the intensity of the atmospheric turbulence. In order to further explore the correction performance of the

Figure 6: Comparison of SPGD and NadamSPGD under different \(\Delta u\) values: (**a**) SPGD curve when \(\Delta u=0.01\); (**b**) SPGD curve when \(\Delta u=0.1\); (**c**) SPGD curve when \(\Delta u=1\); (**d**) NadamSPGD curve when \(\Delta u=0.01\); (**e**) NadamSPGD curve when \(\Delta u=0.1\); (**f**) NadamSPGD curve when \(\Delta u=1\). two algorithms under different turbulence intensities, we analyze the relationship between ME and the iteration number under different \(D/r_{0}\) in Figure 7.

From Figure 7, it is obvious that NadamSPGD can effectively correct turbulence with different intensities. As the intensity of turbulence increases, the correction speed gradually slows down. When \(D/r_{0}=5\), NadamSPGD achieves an ME of 0.8 after 38 iterations, while SPGD needs 55 iterations to achieve the equal correction effect. When \(D/r_{0}\) increases to 15, NadamSPGD converges after 291 iterations, while SPGD requires 594 iterations. However, the gap between the number of iterations required for two algorithms becomes larger with the increasing turbulence intensity.