## 1 Introduction  
Recently, with the popularity of low Earth orbit (LEO) satellites and high-altitude platforms (HAPs), satellite-aerial-ground integrated networks (SAGINs) have been widely envisioned as an attractive solution for sixth-generation (6G) and beyond-6G (B6G) communications [1]. Due to the congested spectrum and limited bandwidth, traditional radio frequency (RF) links are unable to meet the massive bandwidth demand in next-generation communications [2]. Free-space optical (FSO) communication has become an effective alternative due to its large unlicensed spectrum and high data rates [3]. In SAGIN, the Terabits/s FSO links can provide high data rates and affordable transmission, while the RF links can achieve ubiquitous coverage. Despite this advantage, the hybrid FSO/RF system is sensitive to complex weather conditions (e.g., fog, rain, and snow) and is greatly affected by attenuation caused by long-distance transmission [4]. Relay technology is an effective way to relieve the impact of attenuation, where the decode-and-forward (DF) relaying technology can enhance the overall signal-to-noise ratio (SNR) while the amplify-and-forward (AF) relaying technology owns low processing complexity [5]. The link connecting the ground station (GS) to the satellite is called the feeder link, while the link between the satellite and ground users is referred to as the user link [2, 6]. Considering the Terabits/s feeder link between GS and satellite, a single FSO feeder link is usually adopted due to its immense available bandwidth, along with the advantages of being immune to interference and high security [7]. As for the user link from satellite to ground users, a HAP is often deployed as a relay node to relieve atmospheric attenuation and increase each session time between satellite and ground users [8]. Due to the different beam characteristics of FSO and RF links, the FSO link can only cover less than a few kilometers, while the RF link can range up to several hundred kilometers. Therefore, we can combine the large capacity of the FSO link with the wide coverage of the RF link to improve the throughput and reliability of access areas while achieving ubiquitous coverage [9]. Attracted by the enormous potential of hybrid FSO/RF systems, its implementation has been widely studied in terrestrial networks [10; 11]. The authors of [10] proposed a switching scheme for a hybrid FSO/RF network with a selective DF-relaying protocol, assuming a maximal ratio combining (MRC) scheme at the destination. In [11], the authors analyzed the performance of dual-hop multiuser relay systems with mixed FSO/RF links, in which nonorthogonal multiple access (NOMA) technology was adopted to improve the system ergodic sum rate. Considering the limited range of terrestrial networks, the hybrid FSO/RF system has recently been extended to the HAP-based SAGIN [8; 12; 13; 14]. The authors of [7] investigated the performance of dual-hop hybrid FSO/RF links for HAP-based SAGIN, and the optimal switching threshold value between FSO and RF links was also obtained. The comprehensive performance of a dual-hop SAGIN-based hybrid FSO/RF switching system was investigated in [12], where the authors derived the closed-form expressions for the optimum switching threshold. Considering the multiple-hop mixed RF/FSO systems, the authors of [13] proposed two practical cases for multicast networks and analyzed the tradeoff between outage probability and energy efficiency. In [14], the authors analyzed the performance of a multiuser terrestrial-satellite system with mixed FSO/RF links, where the outdated channel state information (CSI) in the RF link was also considered. Due to the advantages of low complexity and ease of implementation, the hard-switching scheme is widely adopted in the existing hybrid FSO/RF systems [15; 16; 17].