## 1 Introduction
### Experimental Research  
The wavefront-free adaptive optics correction system is constructed indoors, with the experimental design principle illustrated in Figure 7. Both the signal light and the beacon light, emanating from a shared optical path, traverse through polarizer P1 before entering a 4F system comprising lenses L1 and L2 for beam expansion and collimation. Subsequently, the collimated beam proceeds to a spatial light modulator after being split by beam splitter BS1. Loaded with a phase screen, the spatial light modulator modulates the collimated light, and the modulated beam is reflected by mirror PM. Following reflection, the beam is again divided by BS2 and directed to the deformable mirror (DM). Here, the DM undertakes the initial correction of the distorted light, which is then converged through lens L3 and directed to the CCD. Furthermore, the CCD captures the intensity of the distorted beam and transmits this information to the computer. Utilizing optimization algorithms, the computer generates control signals for the deformable mirror, thereby executing multiple closed-loop corrections of the wavefront aberrations. The detailed equipment parameters employed in the experiment are outlined in Table 1.  
Figure 6: Signal light SR value variation curve with the number of iterations. (**a**) Without the wavefront correction coefficient. (**b**) With the wavefront correction coefficient.  
Figure 7: Experimental schematic diagram of wavefront sensor-less adaptive optics.  
In the experiments, we selected wavelengths of 1550 nm as the signal light and 632.8 nm as the beacon light, respectively. The turbulent phase screen based on the Zernike polynomial method was loaded into the liquid crystal spatial light modulator (LC-LSM) to cause wavefront distortion of the beam [23, 24, 25]. The SPGD algorithm based on the variable gain coefficient was used to correct the wavefront distortion. We used the average grey scale value of 10 \(\times\) 10 pixel area centered on the spot's center of mass, as captured by the CCD camera, to evaluate the calibration effect in terms of spot brightness. The size of the phase screen was D = 0.4 m. For the SPGD algorithm, we selected a perturbation amplitude value of \(\delta=0.006\) and a gain coefficient value of \(\gamma=1.6/(1+J)\). In order to simulate strong turbulence conditions, we set the atmospheric coherence length to \(r_{0}\) = 0.01 m [26, 27]. We conducted closed-loop correction experiments, and the results are shown in Figure 8a,b. By comparing the two figures, we can observe that the system without wavefront correction has a significant correction effect on the wavefront aberration in regions of strong turbulence. The SR value of the spot prior to correction stands at 0.112. Post-correction, it notably increases to 0.74, signifying a substantial enhancement in spot energy convergence. Upon employing the SPGD algorithm to generate the adaptive mirror voltage control signal, correction coefficients are incorporated to address the aberrations of the signal light. As depicted in Figure 8c, the results illustrate heightened spot energy compared with Figure 8b, where the dark spot is rectified. Notably, the beam's SR value escalates to 0.812, underscoring the effectiveness of this approach in further refining the system's correction accuracy without wavefront correction.