## 1 Introduction  
Atmospheric turbulence is typically classified by its intensity into strong turbulence, weak turbulence, and moderate turbulence. Strong turbulence refers to turbulence with high intensity accompanied by significant fluctuations. In regions of strong turbulence, a beam experiences significant phase and logarithmic amplitude fluctuations induced by atmospheric disturbances. The turbulent atmosphere fragments the beam's wavefront into numerous small speckles, creating a discontinuous phase surface marked by numerous phase singularities. This phenomenon leads to pronounced fluctuations in light intensity, causing flickering. During such conditions, the wavefront sensor struggles to accurately measure the complete distribution of the wavefront phase, resulting in a notable reduction in the efficacy of conjugate phase correction [1, 2]. In contrast, wavefront sensor-less adaptive optics do not require a wavefront sensor to measure the distortion information of the signal light. Instead, it utilizes the control signal as an optimization parameter. Furthermore, parameters, such as imaging clarity, received optical energy, and other system performance indicators, can directly serve as objective functions for optimization algorithms. Leveragingthese algorithms can yield nearly optimal correction outcomes. Hence, this technology is suitable for rectifying wavefront aberrations in free-space laser communication systems operating amidst strong turbulence conditions [3,4]. In a free-space optical communication system, when the selected signal wavelength has poor penetration or severe attenuation in the atmospheric channel, the system employs beacon light with a wavelength different from signal light, in conjunction with the Acquisition, Pointing, and Tracking (APT) system, to achieve capture, aiming, and tracking, thereby establishing the communication link [5,6]. However, because of the different wavelengths of signal light and beacon light, atmospheric dispersion and beam diffraction effects may lead to differences in wavefront phase aberrations after atmospheric turbulence transmission [7,8]. When the correction system adjusts the signal light by measuring the beam quality of the beacon light emitted from the same optical path as the signal light, such as the Strehl Ratio (SR), an optimization parameter for wavefront correction algorithms, a divergence emerges between the beam SR value captured by the CCD camera, which measures the beacon light, and the actual SR value of the signal light. This discrepancy ultimately leads to a certain degree of residual aberration within the transmission even after the correction system has been applied. In recent years, numerous studies have delved into the disparities in wavefront aberrations between signaling light and beacon light within dual-wavelength free-space optical communication systems. In 2007, [PERSON] discovered that the wavelength of a beam, alongside atmospheric turbulence and receiving aperture characteristics, influences the performance of adaptive optics systems [9]. Subsequently, in 2008, [PERSON] identified variations in wavefront aberrations among different wavelengths transmitted through atmospheric turbulence, underscoring their significant impact on the correction of higher-order aberrations by adaptive optics systems [10]. In 2015, [PERSON] conducted an indoor space laser communication experimental link featuring the co-optical transmission of signal and beacon lights. Using a Shack-Hartmann wavefront sensor at the receiving end, [PERSON] measured the wavefront phase aberrations of optical signals at wavelengths of 530 nm and 1060 nm [11]. Additionally, in 2020, [PERSON] scrutinized the polarization chromatic aberration between 1300 nm and 589 nm beams arising from the surface coating of the optical original in the Cassegrain optical antenna.