## 1 Introduction  
The explosive proliferation of wireless communications poses a shortage of frequency band resources [1]. FSO communications have been increasingly attractive due to the large capacity brought by unexploited spectrum as well as high security [2]. Nevertheless, despite their promising prospects, atmospheric turbulence has long been the principal encumbrance clogging the development of the subject. Spatially, inhomogeneities of temperature, humidity, momentum and other atmospheric components lead to a space-varying atmospheric refractive index along the transmission link causing phase shifting. Meanwhile, the existence of atmosphere turbulence significantly enhances the vertical and horizontal exchange of the above atmospheric components which is greater than the exchange intensity of molecular motion. Such influence is generalized as atmospheric turbulence. Displayed in the dynamic variation of the amplitude and phase of the received laser signal, the influence gradually becomes prominent when distance between communication terminals goes beyond 1 km [3]. According to [4], various models of the power spectrum were established to describe statistical features including the Kolmogorov power-law spectrum, the Tatarskii spectrum, the von Karman spectrum, and the modified atmospheric spectrum. In this study, we focus on a ground-to-ground scenario where communications happen between base stations. The study conducts theoretical pre-research for the subsequent use of laser communication base stations instead of microwave relay base stations so as to achieve 40\(\sim\)100 Gbps communication rate. As a basic method to improve signal-to-noise ratio (SNR) at the receiver so as to enhance service quality, enlarging SNR from the sender would be an effective method but it is not always the most agreeable solution. On the contrary, correcting phase shifting, compensating wavefront aberrations would benefit the sender in terms of energy saving. Therefore, AO aiming at extracting real-time wavefront distortion and dynamically compensating wavefront aberrations rises up as a prospective countermeasure which could reduce the likelihood of signal fading and mitigate intensity variations (scintillation) [5, 6]. Existing studies [7, 8, 9] have shown that AO techniques are of significant practical value in satellite-to-ground optical communication. Our findings are not only applicable to ground-to-ground FSO communications; as long as it involves dynamically compensating the incident light with a continuously changing phase, our algorithm can be effective with wavefront correctors with suitable stroke. For example, in optical modulation, beam shaping, astronomy, and microscopic imaging [10, 11, 12]. In general, an AO system consists of a wavefront reconstructor, a wavefront sensor, and a wavefront corrector [13], where the wavefront sensor generates real-time feedback of wavefront aberrations and the wavefront reconstructor computes commands instructing an appropriate deformation of the corrector. The Shark-Hartmann wavefront sensor (SHWS) which gauges atmospherically induced wavefront aberrations across an aperture is widely used in diverse studies [14, 15, 16, 17]. In addition, embedded development boards free from the operating system with better instantaneity and a flexible size would be better suited for a wavefront reconstructor than computers in engineering applications. As for the wavefront corrector, DMs are the most frequently employed optical hardware, among which, micromachined membrane DMs (MMDM) [18] taking advantage of low hysteresis and negligible power consumption are applicable in high-precision scenarios. By contrast, piezoelectric DMs (PDM) [19] benefit from large strokes and high response frequency, satisfying the demand of large-scale aberration compensation. Apart from the above two DMs, unimorph DMs [20] and liquid DMs [21] are indispensable devices used in many AO architectures as well.