# Research on a Decoupling Algorithm for the Dual-Deformable-Mirrors Correction System
## 3 Numerical Simulation Analysis  
To further validate the algorithm proposed in this article, experimental data were analyzed in the laboratory of Xi'an University of Technology. The transmitting end of the data measurement was positioned in Teaching building 5 of Xi'an University of Technology, while the receiving end was located in building 2 of Discipline, and the straight-line distance between the two buildings is 600 m. The slope data of wavefront distortion were measured using the Shack-Hartmann wavefront sensor. According to the wavefront reconstruction  
Figure 2: Flowchart of separation coefficient algorithm. principle [15], the wavefront reconstruction matrix was inverted and the product of the slope data measured by the Hartmann wavefront sensor was obtained (i.e., Equation (10) to obtain the initial Zernike coefficient of wavefront distortion, as depicted in Figure 3a. Given the Zernike coefficient value of wavefront distortion, the phase of wavefront distortion can be calculated by substituting it into Equation (1), as shown in Figure 3b, which displays the phase diagram of wavefront distortion. Next, a dual-deformable-mirrors wavefront correction system consisting of FSM (Woofer) and a 69-unit DM (Tweeter) is mainly used to correct the wavefront distortion data measured in the experiment. The FSM corrects the 2 nd-order aberration, the DM corrects the 30 th-order aberration, and the parameters of the PI controller are set to: \(k_{i,w}=0.999\), \(k_{p,w}=0.1\), \(k_{i,l}=0.999\), \(k_{p,l}=0.1\). The wavefront phase diagrams corrected by the FSM and DM, respectively, as well as the wavefront phase diagrams corrected by working together, are shown in Figure 4.  
Figure 4 presents the phase diagrams of the FSM and DM after correcting the aberration individually and in combination. Specifically, Figure 4a illustrates the wavefront phase diagram after correction with an FSM, while Figure 4b shows the residual wavefront phase diagram after correction with a DM. Additionally, Figure 4c depicts the residual wavefront phase diagram after both FSM and DM have been employed for correction. As shown in Figure 3a, the 2 nd order and 30 th-order aberrations have the largest Zernike coefficient values, indicating that they contribute significantly to the total wavefront aberration. Therefore, the FSM is used to correct the 2 nd-order aberration, while the DM corrects the 30 th-order aberration. Comparing Figure 3b with Figure 4b, it is evident that the tilted degree of the corrected wavefront distortion phase diagram is minimal, indicating that the residual wavefront after correction has a minimal proportion of tilted aberrations. However, when the DM was used to correct the 30 th-order aberration without compensating for the  
Figure 4: Wavefront phase diagrams. (**a**) Wavefront phase diagram after FSM correction. (**b**) Wavefront phase diagram after DM correction. (**c**) Wavefront phase diagram after both FSM and DM correction. Figure 3: Initial wavefront distortion. (**a**) Zernike coefficient. (**b**) Wavefront distortion phase diagram. 2 nd-order aberration, as shown in Figure 4b, there was a degree of tilt in the corrected wavefront phase diagram. As seen in Figure 4c, there was a significant change between the corrected wavefront phase diagram and the initial distorted wavefront phase diagram. By evaluating their PV and RMS values, it can be concluded that the proposed algorithm's wavefront correction ability is accurately reflected, demonstrating that the decoupling algorithm achieves independent correction of specified aberrations. In a dual-deformable-mirrors wavefront correction system, the deformation generated by two deformable mirrors is typically used to correct wavefront aberration. However, when the two deformable mirrors are coupled, their deformations can affect each other, leading to an unwanted coupling effect. To evaluate this effect, the coupling coefficient is employed as an important indicator.