## 1 Introduction  
With the rise of new applications such as the Internet of Things (IoT) and virtual reality (VR), there is an increasing demand for higher data rates and capacities in sixth-generation (6G) communication technology. Free space optical (FSO) communication is an emerging technology that offers unique advantages in terms of high speed, anti-interference capabilities, and efficient spectrum resource utilization, making it an effective supplement for 6G technology [1, 2]. Unlike fiber-optic communication, which transmits signals through optical fibers, FSO utilizes collimated beams to transmit signals through the atmosphere [3]. In scenarios where laying cables is challenging, FSO can significantly reduce costs and eliminate the difficulties associated with cable installation [4]. However, atmospheric absorption, scattering, and turbulence can adversely affect FSO transmission beams, making them vulnerable to unstable weather conditions [5].  
Multiple-input, multiple-output (MIMO) technology is one of the effective ways to combat turbulence fading. By introducing spatial diversity, the performance of FSO communications in atmospheric turbulence can be improved [6, 7]. In addition, as a signal processing technology used at the transmitter end, precoding technology has been widely adopted in FSO communications to combat turbulent fading [8, 9]. When the transmitter is equipped with multiple antennas, data streams from multiple users can be spatially multiplexed onto independent sub-channels through precoding. By optimizing the precoder as a variable, performance requirements such as rate and energy efficiency in the communication system can be achieved [10].  
Photon-counting technology is an effective method for detecting low-power optical signals in FSO communications [11; 12]. This advantage becomes even more prominent in power-constrained IoT or optical communication scenarios under strong atmospheric turbulence. In such scenarios, it would be difficult to effectively detect extremely weak optical signals using traditional optical communication receivers. Photon-counting receivers, based on the photoelectric effect and high-sensitivity detectors, can effectively detect and receive low-power optical signals. However, due to quantum effects, the photon-counting process involves randomness, resulting in the generation of Poisson shot noise associated with the signal. Specifically, shot noise is modeled as a Poisson counting process (PCP) characterized by two independent parameters: the desired signal and background radiation [13]. These characteristics make it difficult to apply signal processing schemes developed based on additive white Gaussian noise (AWGN) to photon-counting systems, so it is necessary to develop new technologies to meet this challenge.  
Recently, the applicability of intelligent reflective surfaces (IRSs) has been introduced to FSO scenarios to address the problem of line-of-sight (LOS) link blockage due to buildings and other obstacles [14]. Optical IRS can be implemented through either a reflector array-based design or a meta-surface-based design [15]. Considering the low cost and easy portability of IRSs, it has been applied in various scenarios in communication systems to improve communication performance.