## 4 Experimental Verification  
### Experimental Setup  
We established the experimental system shown in Figure 9 to validate our CE calculation model and theoretical analysis. Figure 9a shows a schematic diagram of the system and Figure 9b shows a photograph of the system. We established the calculation model in the case of parallel light incidence, in order to obtain ideal experimental results, we kept the optical power emitted by the light source as stable as possible throughout the experiment. We expanded the beam and used an aperture stop to intercept the center part of the beam with a relatively uniform power to weaken any variations caused by changes in the laser beam direction, which also made the power distribution of the beam more uniform. The DM is used to generate different single aberrations. In order to reduce the experimental error and improve the accuracy of the aberration fitting, we calibrated the DM fitting aberration by interferometer before the experiment. In the experiment, we use a Hartmann wavefront sensor (WFS) to measure the DM to generate aberrations in real time, and based on the measured values of WFS, estimate the wavefront aberration at the entrance of the coupling lens.  
Figure 8: Residual control range of adaptive optics (AO) system under different coupling thresholds.  
The working principle of the experimental optical system is as follows: 808 nm and 1550 nm lasers are emitted from single-mode optical fibers by a combination of optical fiber combiners. After passing through mirrors, M1 and aperture stop S1, they are expanded by the beam expander group L1 and output as the light source of the system. After being reflected by the FSM, the collimated beam is further expanded by L2 and L3; after being reflected by DM, it enters the beam splitter BS1. Next, 80% of the 808 nm light is reflected by BS1 and enters the WFS after being compressed by L6 L7. The remaining 20% of the 808 nm light and all the 1550 nm light pass through BS1 and are compressed by L4 L5 into a beam of 10 mm diameter before reaching the coupling lens. The 1550 nm light is coupled into the SMF after being focused by the coupling lens, while the 808 nm light is filtered out. We used image sensor and WFS to measure the intensity and wavefront phase of the initial 808 nm beam after being expanded and corrected, and used it as the basis for evaluating the quality of the 1550 nm coupled beam. The results are shown in Figure 10, the standard deviation of the power fluctuation of the beam is 15.78 gray value, and the RMS value of the wavefront phase is 0.0239\(\lambda\) (\(\lambda\) = 808 nm).  
The model and parameters of the equipment used in the experiment are as follows: The 1550 nm laser is PS-NLL210816 from TeraXion, the 808 nm laser is 808-030-SM01 from SFOLT, the FSM is a FSM-300 from Newport, the DM is a DM145-25 from ALPAO with 145 actuators in its 30 mm aperture. The Hartmann WFS has a lens array consisting of a 2.3 mm diameter 15 * 15 microlens array. The coupling lens has an effective aperture of 20 mm and a focal length of 50 mm. The SMF placed at the receiving end is a SMF-28e which has a mode-field diameter of 10.4 um at 1550 nm.  
Figure 10: Measurement results of initial 808 nm beam: (**a**) Beam image; (**b**) beam intensity diagram; (**c**) light spot array of wavefront sensor (WFS); (**d**) fitted wavefront phase.  
Figure 9: (**a**) Schematic diagram of experimental system; (**b**) photograph of experimental system. M: Mirror; L: Lens; S: Aperture stop; FSM: Fast-steering mirror; BS: Beam splitters; DM: Deformable mirror; OPM: Optical power meter; SMF: Single-mode fiber.