## 1 Introduction  
Free-space optical communication (FSO) technology is applied widely in satellite-ground, satellite-satellite and other communication links [1; 2] for its small size, low power cost, high reliability and security [3]. Considering the atmospheric channel, turbulence is one of the most important factors that influence system performance. The wavefront of the signal light at the receiving telescope will distort heavily, and the parameters, which are used to evaluate system performance, turn into random variables because of the randomness of turbulence. The Kolmogorov turbulence model is used extensively in considering the turbulence effect of the atmosphere for its simple mathematical structure and usability for numerical calculation [4; 5; 6][PERSON] However, with the increase of theoretical discussion and experimental research [7; 8; 9], it has been found that [PERSON]'s theory could not always describe the characteristics of actual turbulence, and the non-Kolmogorov turbulence model has been widely applied to evaluate the performance of FSO terminals. [PERSON] studied the influence of moderate-to-strong non-Kolmogorov turbulence on the imaging system based on the atmospheric turbulence modulation transfer function [10]. [PERSON] investigated the behavior of the coherence length in non-Kolmogorov satellite links [11]. Moreover, [PERSON] generalized the oceanic spatial power spectrum to the non-Kolmogorov turbulence regime based on temperature and salinity concentration [12]. Signal light received by FSO terminals is usually coupled into an Erbium-doped optical fiber amplifier (EDFA) for next processing, especially in a coherent communication system, in order to achieve a high date rate [13], so the coupling progress becomes one of the interests [14; 15; 16]. Therefore, coupling efficiency (CE) is one of the most important parameters to evaluate the performance of communication terminals and should be well discussed in the conditions of non-Kolmogorov turbulence. The mean value and variance of the CE are two essential parameters to evaluate system performance because some system indicators are directly associated with CE. Moreover, non-linear relationships widely exist between the CE and these indicators [17]. For example, the signal-to-noise ratio (SNR) is proportional to the square of the receiving power in the IM/DD system. The receiving power is linear to the CE, so the SNR is non-linear to the CE. Thus, the statistical model of CE should be built to analyze the mean value and fluctuation of these indicators affected by turbulence [18]. [PERSON] and [PERSON] discussed the CE of a Gauss beam transmitting though non-Kolmogorov turbulence and gave the expression of the CE. However, it was a complex double integral and not convenient for further discussion of the system performance for FSO terminals [19]. They also discussed a fiber CE based on non-Kolmogorov theory in a satellite-ground link, and showed the trends, but did not study its statistical characteristics [20]. [PERSON] and [PERSON] gave us an expression of the CE for a partially coherent laser beam propagating though non-Kolmogorov turbulence. This also ended up with an integral with no analytic solution [21]. Moreover, [PERSON] and [PERSON] discussed the CE on focal plane spot extension caused by turbulence. The results showed that CE was related to aperture, the wavelength of incident light, the radius of the receiving fiber, the atmosphere coherence length and the coupling system focus length, but they did not discuss the influence when different turbulence models were chosen [22]. [PERSON] and [PERSON] studied the CE of a Gaussian beam passing through weak fluctuation regimes. However, the results were based on Kolmogorov turbulence [23]. [PERSON] and [PERSON] analyzed the influence of the atmospheric turbulence on the SMF coupling efficiency over various turbulences. They verified that the adaptive optics system was one of the most effective methods to improve the FSO system performance.