## 1 Introduction  
In recent years, due to its high capacity, license-free spectrum, fast network construction, and other characteristics, free space optical (FSO) communication is often considered as an ideal solution for the \"last mile of telecommunications\" and local area network (LAN) links between buildings as well as a potential next-generation wireless communication technology [1, 2]. However, the performance of FSO communication is greatly affected by the atmospheric environment. For example, atmospheric turbulence will lead to scintillation effects, beam wandering and dispersion, bad weather (fog, rain, clouds, and snow) will lead to the attenuation of optical power, and the thermal expansion of the building causes sway, which makes the beam misalignment between the transmitter and the receiver. These all lead to the deteriorating performance of the FSO communication system [3]. Therefore, scholars from various countries have proposed various solutions to the problems faced by FSO communication systems. For example, the alignment of satellites and ground stations can be realized by acquisition, tracking, pointing (ATP) technology in satellite ground laser communication system [4]. Adaptive optics can be used to compensate for the negative impact of turbulence on the performance of the FSO system [5]. For the FSO system operating in bad weather, the beam with the wavelength of the atmospheric window can be used as the carrier, and a beam with excellent performance can also be selected to transmit information [6]. For the more complex and fast time-varying communication environment, the hybrid FSO and radio frequency (RF) link can be used to mitigate the impact of this environment on the system, so as to improve the performance of the communication system [7]. In order to realize the long-distance transmission and wide coverage of free space optical communication, the multi-hop transmission mode is considered as an effective solution [8; 9; 10]. The performance of the FSO system may be further improved by adopting some common communication technologies, such as diversity technology (time diversity, space diversity, and wavelength diversity) [11; 12], modulation technology (on-off keying modulation, pulse position modulation, and subcarrier modulation) [13; 14; 15]. Currently, the FSO link has the advantage of large transmission capacity and the RF link is less affected by the environment. The hybrid FSO/RF system, which combines the advantages of the FSO and RF links, can greatly improve the communication performance; therefore, it has been widely studied by researchers. The hybrid system can generally be divided into hard switching and soft switching. In the hard switching system, the RF link, as an alternative, can only be enabled when the FSO link is blocked and channel-state-information (CSI) needs to be provided [16; 17; 18]. [PERSON] et al. [19] deduced the closed-form expressions of throughput and outage probability of the hybrid system (RF link was used as a backup link at this time) considering the presence and absence of hybrid automatic retransmission request (HARQ). In the soft switching system, the FSO link and RF link transmit the same information at the same time, and the received signal is processed by different combination schemes at the receiver, which does not need to provide CSI [20; 21; 22]. [PERSON] et al. [23] proposed a hybrid system based on the selective combination scheme and derived new closed-form expressions for the average bit error rate (ABER) and outage probability of the system under turbulent conditions. When the transmission distance exceeds 1 km, the performance of the FSO communication system will deteriorate sharply due to environmental impact [24]. Therefore, for the needs of long-distance transmission and coverage expansion, increasingly scholars have conducted research on multi-hop relay networks based on FSO. The multi-hop relay network is divided into serial multi-hop relay and parallel multi-hop relay.