## 2 System Model
### Channel Model
#### 2.2.1 Channel Gains  
With the consideration of atmospheric loss, turbulence-induced fading, and geometric and misalignment losses, the FSO channel gain can be modeled as follows [27; 28]:  
\[h=h_{p}h_{a}h_{g}, \tag{2}\]  
where \(h_{p}\) represents the loss of beam energy due to absorption and scattering in the transmission link. \(h_{a}\) and \(h_{g}\) correspond to the influence of atmospheric turbulence and the matching distortion error of the IRS, respectively. In addition to the reflection effect, IRSs can also absorb or scatter part of the beam power. Let \(\xi\) stand for the reflection efficiency. For the FSO system operating at \(\lambda=1550\) nm, the value of \(\xi\) typically ranges from 0.7 to 1 [29]. The absorption at the IRS could be considered part of the atmospheric loss, \(h_{p}\). Specifically, \(h_{p}\) can be modeled as [30]  
\[h_{p}=\xi 10^{-\kappa d_{\alpha}/10}, \tag{3}\]  
where \(\kappa\) represents the weather-dependent attenuation coefficient of the FSO link, and \(d_{\alpha 2e}\) represents the end-to-end distance between the LED and the PD. The gamma-gamma distribution stands out as a key channel model for FSO links, given its capacity to effectively simulate diverse atmospheric turbulence scenarios. Its close match with empirical data at different turbulence levels further solidifies its suitability as a pivotal channel model for FSO communications [31; 32]. The probability density function (PDF) of the optical intensity fading is expressed as [33]  
\[\Pr(h_{a})=\frac{2(\alpha\beta)^{\frac{\alpha+\beta}{2}}}{\Gamma(\alpha) \Gamma(\beta)}h_{a}^{\frac{\alpha+\beta}{2}-1}K_{\alpha-\beta}\left(2\sqrt{ \alpha\beta h_{a}}\right)\,h_{a}>0, \tag{4}\]  
where \(\Gamma(\cdot)\) is the standard Gamma function, and \(K_{\
u}(\cdot)\) denotes the second kind of the modified Bessel function. The scintillation parameters, \(\alpha\) and \(\beta\), satisfy  
\[\alpha =\left[\exp\left(\frac{0.49\sigma_{R}^{2}}{\left(1+1.11\sigma_{R} ^{12/5}\right)^{7/6}}\right)-1\right]^{-1}, \tag{5}\] \[\beta =\left[\exp\left(\frac{0.51\sigma_{R}^{2}}{\left(1+0.69\sigma_{R} ^{12/5}\right)^{5/6}}\right)-1\right]^{-1}, \tag{6}\]  
where \(\sigma_{R}^{2}\) stands for the Rytov variance. The parameter \(\sigma_{R}^{2}\) serves as a pivotal descriptor, quantifying the intensity of turbulence across a spectrum of scenarios. In specific terms, turbulence is categorized as weak when \(\sigma_{R}^{2}\leq 0.3\), moderate within the range of 0.3 to 5, and strong for values surpassing 5. In scenarios of weak turbulence disturbances, the gamma-gamma distribution shares similarities with the lognormal distribution, both adept at characterizing FSO links operating across unobstructed atmospheric conditions [34]. The matching distortion error arises from the divergence of the beam over the propagation distance, as well as the pointing error between the laser beam and the center of the PD, which is caused by building sway. Moreover, the introduction of random jitter frombuildings equipped with intelligent reflective surfaces heightens the unpredictability of the optical pathway. A two-dimensional topological model of the light path based on the optical intelligent reflecting surface proposed by [PERSON] et al. [28] is shown in Figure 2.  
For illustration convenience, we denote the angles between the laser beam, IRS plane, and PD and the horizontal direction as \(\theta_{L},\theta_{IRS}\), and \(\theta_{P}\), respectively. In addition, we denote the half-length of IRS and PD as \(r_{irs}\) and \(r_{p}\). The angle between the laser source and the PD is \(\psi=\theta_{L}+\theta_{P}-2\theta_{IRS}\). The beam width at distance \(d\) is defined as [35]  
\[w(d)=w_{0}\sqrt{1+\left(1+\frac{2w_{0}^{2}}{\rho^{2}(d)}\right)\left(\frac{ \lambda d}{\pi w_{0}^{2}}\right)^{2}}, \tag{7}\]  
where \(w_{0}\) denotes the beam waist radius. \(\rho\) is the coherence length, defined as \(\rho(d)=\left(0.55C_{n}^{2}k^{2}d\right)^{-3/5}\), where \(C_{n}^{2}\) is the medium refraction coefficient, and \(k\) is the wave number.