## 1 Introduction  
Free space optical (FSO) communication technology is a kind of communication technology that transmits in the atmosphere and uses laser as a medium. In recent years, it has been widely studied by the research community due to its large transmission bandwidth, strong confidentiality, low implementation cost, and having no spectrum license [1]. FSO has made great achievements in the application of metropolitan area network expansion, cellular system backhaul and military secure communication [2]. However, the advantages of FSO communications are not without cost. The atmospheric turbulence encountered by the beam along the propagation path and the aiming error between the communication equipment will cause the optical signal to flicker and distort in amplitude and phase, which will degrade the communication performance of the FSO system [3, 4]. The gas molecules and particles in the atmospheric channel (such as rain, fog, dust) will have an atmospheric absorption effect on the laser, which will attenuate the power of the laser signal, that is, the path loss [5]. For many years, in order to better study the effect of atmospheric turbulence state on the communication quality of FSO systems, the research community has been trying to develop general models that can be applied to any type of atmospheric turbulence conditions. In 2011, [PERSON] et al. proposed a generalized model called Malaga (M) distribution, which provides a good fit with experimental data and hasgood advantages in atmospheric channel modeling [6]. In order to overcome the above problems and limitations, multiple input multiple output (MIMO) technology has become the main technology of FSO because it can improve the transmission capacity and signal quality of the system [7; 8; 9]. As a novel optical multiple input multiple output technology, optical spatial modulation (OSM) uses digital modulation constellation and laser index to transmit and carry information at the same time, and also creates a mapping relationship between input bit information and laser to carry and transmit information [10]. In 2014, [PERSON] et al. combined OSM with Pulse Position Modulation (PPM) and proposed a Spatial Pulse Position Modulation (SPPM) system suitable for FSO communication [11]. The research shows that OSM has both signal domain and spatial domain, which greatly improves the spectral efficiency of the system. OSM not only effectively avoids the problem of strong interference between channels and difficult synchronization between antennas in MIMO technology, but also reduces the link cost and the complexity of signal detection at the receiver [12]. In 2019, [PERSON] et al. proposed a differential optical spatial modulation and analyzed its BER performance under Gamma-Gamma and negative exponential channel models [13]. However, OSM only activates one laser at a time to send information, which also causes a waste of space resources. In addition, OSM must require the number of lasers in the system to be set to the integer power of 2, which greatly increases the cost of the system and limits the application of the system. In order to overcome these shortcomings, optical generalized spatial modulation (OGSM) has been further proposed by the research community [14; 15; 16]. In 2019, [PERSON]'s team improved the minimum mean square error (MMSE) algorithm by using the balance factor to solve the problem of high complexity of the Maximum Likelihood (ML) decoding algorithm and modified the weights according to the characteristics of the OGSM signal activating a certain number of lasers [14]. However, they did not study the performance problem of OSM system in atmospheric channel. In 2020, [PERSON] et al., based on pulse position modulation and pulse amplitude modulation suitable for intensity modulation/direct detection (IM/DD), focused on how to improve the rate of traditional OSM [15]. However, they did not study the power efficiency of OSM, the computational complexity at the receiver or the issue of how to effectively improve its ABER performance.