## 2 System Model and Theoretical Analysis
### Theoretical Basis of the CFSOC  
In a CFSOC system, the ME and BER are effective indicators to evaluate the performance. Assuming that the local oscillator (LO) is a plane wave and the intensity of the received optical signal (OS) is uniform, based on the theory of coherent detection, the total optical power of the combined beam in CFSOC is given by [32]:  
\[I=\int_{S}\Bigl{\{}A_{O}^{2}+A_{S}^{2}+2A_{O}A_{S}\cos[2\pi(f_{S}-\ f_{O})+ \Delta\varphi]\Bigr{\}}ds, \tag{3}\]  
where \(\Delta\varphi=\varphi_{S}-\varphi_{O}\) is the phase difference between OS and LO, \(f_{S}\) and \(f_{O}\) are their frequencies, and \(A_{S}\) and \(A_{O}\) represent the optical amplitudes of OS and LO, respectively. Generally, a signal symbol transmission time is less than 1 ns in the CFSOC system, while the Greenwood frequency (GF) is on the millisecond scale. According to [PERSON]'s turbulence hypothesis, the phase aberrations \(\Delta\varphi\) caused by atmospheric turbulence can be considered frozen during the detection time, and given by the expression:  
\[\Delta\varphi=\varphi(r)+\varphi(t), \tag{4}\]  
where \(\varphi(r)\) represents the spatial part of phase aberrations caused by atmospheric turbulence, which is time-independent, and \(\varphi(t)\) denotes the temporal part of phase in the optical  
Figure 2: Schematic of SLAO system.  
signal, which is space coordinate-independent. When in homodyne detection, \(f_{S}\)\(=\)\(f_{O}\), then ME can be defined by:  
\[ME=\left[\int_{S}A_{O}A_{S}\cos(\Delta\varphi)ds\right]^{2}/\left(\int_{S}A_{S}^ {2}ds\int_{S}A_{O}^{2}ds\right). \tag{5}\]  
According to Equation (5), ME is approximate to the Strehl ratio (SR) of the far-field images, defined by the ratio of far-field encircled energy to the diffraction limited encircled energy [33]. We only consider the spatial and temporal errors, then after \(A_{O}\) compensation, the mathematical expectation of the residual wavefront phase aberrations based on a CSDM can be expressed as [34; 35; 36]:  
\[E\left(\sigma_{\varphi}^{2}\right)=E\left(\sigma_{fit}^{2}+\sigma_{time}^{2} \right)=\left[\alpha_{F}\left(\frac{d}{r_{0}}\right)^{5/3}+\kappa\left(\frac{f _{G}}{f_{\text{3 dB}}}\right)^{5/3}\right]\left(rad\right), \tag{6}\]  
where \(\sigma_{fit}^{2}\) is the spatial characteristic that represents the wavefront fitting error caused by the limited number of CSDM actuators, \(\sigma_{time}^{2}\) is the temporal error due to the contradiction between fast-changing atmospheric turbulence and the finite closed-loop control bandwidth (CLCB) of the SLAO system, \(\alpha_{F}\) is the fitting error coefficient, \(r_{0}\) denotes the atmospheric coherent length, \(d\) is the equivalent interval of the actuators interval projected on the entrance pupil of the receiving antenna, \(\kappa\) is a constant which equals 1 for the plane wave, \(f_{\text{3 dB}}\) denotes the CLCB and \(f_{G}\) represents the GF.  
Thus, the relationship between ME and the average residual wavefront variance can be expressed as:  
\[ME\ \propto\ SR=\exp\left\{-\left[\alpha_{f}\left(\frac{d}{r_{0}}\right)^{5/3}+ \kappa\left(\frac{f_{G}}{f_{\text{3 dB}}}\right)^{5/3}\right]\right\}. \tag{7}\]  
The BER of the synchronous binary phase shift keying (BPSK) coherent detection can be expressed as:  
\[BER=\frac{1}{2}\textit{erfc}\left(\sqrt{ME\cdot 2N_{p}\delta}\right), \tag{8}\]  
where the function _erfc_ is the complementary error function, \(\delta\) is the quantum efficiency of the receiver detector and \(N_{p}\) respects the number of photons received within a signal bit.