## 1 Introduction
### Simulation Verification  
The wavefront correction system uses a charge-coupled device (CCD) camera to measure the spot energy, Strehl ratio, and other beam parameters of the laser beam, which are used as optimization indicators for the control program and converted into control signals for the deformable mirror [20; 21]. The working principle of the system is shown in Figure 4, which mainly includes the following three parts: wavefront controller, wavefront corrector (deformable mirror, DM), and imaging detector (CCD). The parallel light generated by the light source forms distorted beams carrying aberrations after undergoing atmospheric turbulence transmission. This distorted beam is projected onto the deformable mirror, which corrects the distortion for the first time and reflects the remaining distorted beam onto the CCD. Next, the wavefront controller drives the intelligent algorithm to recalculate the control instructions of the deformable mirror based on the system performance indicators collected by the CCD. The deformable mirror corrects the wavefront distortion of the beacon light by adjusting its surface morphology. This process is cyclic, and after each round of correction, the deformable mirror further adjusts its shape based on the newly obtained feedback information of the light spot, gradually approaching the ideal correction state and achieving multiple closed-loop corrections of the distorted beam.  
The control system adopts the stochastic parallel gradient descent (SPGD) algorithm. The algorithm first generates the basic voltage command for the deformable mirror and superimposes a random perturbation voltage signal that conforms to the Bernoulli distribution. On this basic voltage, forward and reverse voltage disturbances are applied separately, and then the change in the system objective function under each disturbance voltage is calculated. Finally, based on the calculation principles shown in Figure 5, the voltage signal that should be applied during the iteration cycle is determined. This process continues to loop until the system objective function reaches the preset optimization conditions.  
In conditions characterized by an atmospheric coherence length \(r_{0}=0.01\) m, a disturbance amplitude \(\delta=0.006\), a high gain coefficient of \(\gamma=1.6/(1+J)\) (where \(J\) denotes the value of the system's objective function), and 100 iterations, numerical simulations were conducted to investigate the performance of dual-wavelength wavefront-free sensing adaptive optics technology in correcting far-field spot SR under atmospheric turbulence. The simulations employed the multi-phase screen method to model laser beam propagation through a turbulent atmosphere, with phase screens generated using Zernike polynomials [22]. The specific parameters for the simulation included a signal wavelength of 1550 nm, a beacon wavelength of 632.8 nm, a transmission distance of 10 km, a phase screen interval of \(\Delta z=1500\) m, and a phase screen size of D = 0.4 m. The Zernike coefficient matrices for the beacon light, both pre- and post-correction, were integrated into the adaptive optics control system to rectify the wavefront aberration of the signal light. The resulting changes in SR for the signal light are depicted in Figure 6. Analysis of the figure reveals that over 100 iterations of correction, the SR value of the signal beam within the correction system lacking additional correction coefficients exhibits slow convergence, reaching a limit of 0.89, accompanied by notable jitter during the correction process. Conversely, the system augmented with correction coefficients realizes a higher SR value limit of 0.982, marking  
Figure 4: The working principle of the wavefront correction system.  
Figure 5: Schematic diagram of the correction of wavefront aberration differences without wavefront correction.  
an 11% enhancement over the initial convergence target of 0.89. Furthermore, it attains convergence more rapidly, stabilizing at 70 iterations with significantly reduced jitter.