## 3 NadamSPGD Algorithm in SLAO
### Conventional SPGD in SLAO
The results show that the advantages of NadamSPGD become more significant with the increase in turbulence intensity. Based on the theoretical analysis in Section 2.3, we know that the GF of the atmosphere and the CLCB of SLAO severely affect the performance of the CFSOC system. When \(d=0.06\), \(\alpha_{F}=0.28\), \(r_{0}\)= 0.15 and \(\kappa=1\), the relationships between the ME, BER and CLCB under different GFs were obtained and are shown in \[\hat{n}^{(k)}=n^{(k)}/\Big{(}1-v^{k}\Big{)}, \tag{14}\]0

As Figure 8 illustrates, under different GFs, the higher the CLCB of the SLAO is, the higher the ME and lower the BER that will be achieved. Thus, a higher CLCB is necessary to guarantee the communication quality with the increase in GF. Since the resonant frequency of CSDM is very high (>5 kHz), its response time is extremely short, and its impact on CLCB is negligible compared to the computational delay introduced by algorithms [42]. As

\[\hat{n}^{(k)}=n^{(k)}/\Big{(}1-v^{k}\Big{)}, \tag{14}\]1 Relationships between ME, BER and CLCB under different GFs: (**a**) relationship between ME and CLCB under different GFs; (**b**) relationship between BER and CLCB under different GFs. \[\hat{n}^{(k)}=n^{(k)}/\Big{(}1-v^{k}\Big{)}, \tag{14}\]2 Comparison of SPGD and NadamSPGD under different turbulence intensities: (**a**) correction curves using SPGD and NadamSPGD when \(D/r_{0}=5\); (**b**) correction curves using SFGD and NadamSPGD when \(D/r_{0}=15\); (**c**) number of iterations required for ME reaching to 0.8 of the two algorithms and the difference between them under different \(D/r_{0}\). a result, the computational delay of the algorithm used for correction is the dominant factor in determining the CLCB. Generally, a more efficient algorithm means fewer iterations and higher CLCB. To further show the ability of NadamSPGD in the time domain, we assume that the number of iterations is inversely proportional to CLCB and the previous simulation results of NadamSPGD have achieved 100 Hz CLCB, considering the processing capacity of the FPGA and GPU-based high-performance processing platform. Thereafter, the ME and BER at different GFs can be calculated according to Equations (7) and (8), and the results are shown in \[\hat{n}^{(k)}=n^{(k)}/\Big{(}1-v^{k}\Big{)}, \tag{14}\]3

Figure 9 illustrates that as the GF increases, the dynamic ability of SLAO based on the two algorithms degrades. However, the attenuation of SPGD is more serious than NadamSPGD, meaning that the proposed algorithm has better dynamic performance and is more suitable for real-time SLAO systems. ### Experiment

In this study, we also compare the two algorithms based on our SLAO experimental platform to analyze and demonstrate the performance improvement of NadamSPGD in the actual systems. Figure 10 presents the setup and photograph of the SLAO experimental platform. As shown in Figure 10, the SLAO experimental platform is constructed based on an auto-collimating optical system. An optical fiber-coupled laser source with a wavelength of 635 nm is used to emit the laser beam. The laser beam is collimated by lens L1, then

\[\hat{n}^{(k)}=n^{(k)}/\Big{(}1-v^{k}\Big{)}, \tag{14}\]4 Setup and photograph of the SLAO experiment platform. (**a**) Optical layout of the platform. The optical signal is colored in orange and the electronic signal is colored in blue for easy distinction. (**b**) Photograph of the platform. The focal lengths of L1–L4 lenses are 10 mm, 10 mm, 5 mm and 15 mm, respectively. \[\hat{n}^{(k)}=n^{(k)}/\Big{(}1-v^{k}\Big{)}, \tag{14}\]5 Comparison of ME and BER between SPGD and NadamSPGD under different GFs. (**a**) ME of the two algorithms under different GFs. (**b**) BER of the two algorithms under different GFs. reflected by the BS and further expanded by L3 and L4 to match the size of the 97-element CSDM. A field stop is added between lenses L3 and L4 to suppress the stray light. After correcting by the CSDM, the beam passes through the L4, L3 and BS again, and is narrowed and split.