## 1 Introduction  
Free-space optical communication (FSOC) has the characteristics of wide bandwidth, strong resistance to radio frequency interference, and high security, which has attracted extensive attention of researchers in this field. However, because the laser beam is easily affected by factors such as atmospheric turbulence [1, 2], foggy weather [3, 4, 5, 6, 7, 8], and pointing errors [9, 10, 11] in outdoor transmission, the transmission distance of outdoor FSOC is severely limited. Many mitigation techniques have been proposed to combat the degradation of signal quality, such as adaptive optics [12, 13, 14], spatial diversity [15, 16], and aperture averaging [17]. However, the high system complexity and high cost of these mitigation techniques make it difficult to be widely applied to outdoor FSOC systems. Therefore, in order to realize reliable long-distance transmission of outdoor FSOC, the research on multi-hop FSOC technology has received extensive attention. In multi-hop FSOC systems, decode-and-forward (DF) relay [18, 19] and amplify-and-forward (AF) relay [20, 21] are two commonly used relay technologies. The DF scheme decodes and re-encodes the received signal at each relay node and then forwards it to the next node. Both its system complexity and its cost are high, and it is not suitable for multi-hop FSOC systems. However, the AF scheme only amplifies the received signal at each relay node without demodulating and recoding, so it can achieve good communication performance in a simple system. At present, some researchers have analyzed the communication performance of AF relay systems under atmospheric turbulence [22; 23; 24; 25; 26]. For example, [PERSON] et al. analyzed the outage probability (OP) and average bit error rate (ABER) for multi-hop AF relay system under Gamma-Gamma distribution atmospheric turbulence model [22]. [PERSON] et al. further obtained the expressions of OP and ABER of multi-hop AF relay system with heterodyne detection under the combined effect of platform random jitter and Gamma-Gamma distribution atmospheric turbulence model [23]. Further, [PERSON] et al. studied the expression of OP, ABER, and average ergodic channel capacity for multi-hop AF relay system with intensity modulation/direct detection (IM/DD) under the combined effect of platform random jitter and Gamma-Gamma distribution atmospheric turbulence model [24]. In the simulated atmospheric turbulence device, [PERSON] et al. experimentally verified the 10 Gbps three-hop AF relay scheme, the paper confirms that the theoretical analysis results of the Gamma-Gamma distribution atmospheric turbulence model match the experimental results [25]. [PERSON] et al. gave a unified expression of OP and ABER for multi-hop AF relay system with heterodyne detection and IM/DD under the combined effect of platform random jitter and double generalized Gamma distribution atmospheric model [26]. Although some researchers have conducted theoretical studies on the performance of multi-hop AF FSOC systems in the past decade, the above research work still has shortcomings. Specifically: on the one hand, the above analysis of the multi-hop AF relay system only considers platform random jitter and does not consider the influence of the non-zero boresight error. The pointing error is cause by two factors: platform random jitter and non-zero boresight error, both of which will seriously affect the performance of the multi-hop AF relay FSOC system; on the other hand, the above research work is based on the assumption of sunny weather conditions, without considering the impact of foggy weather on the communication performance of the multi-hop AF relay FSOC system. In the actual outdoor FSOC system, the foggy weather will seriously attenuate the optical signal power and reduce the performance of the communication system. Previous studies on fog have assumed that fog is a deterministic path attenuation related to visibility. In fact, fog attenuation is a random state, and the Gamma distribution model can better describe fog attenuation [27].