## 2 Materials and Methods
### B. Channel Model  
The composite channel model at the \(i\)-th hop link is the product of random fog, pointing error, and atmospheric turbulence which is defined as:  
\[h_{i}=h_{f,i}h_{p,i}h_{a,i} \tag{3}\]  
where \(h_{f,i}\), \(h_{p,i}\), and \(h_{a,i}\) represent the random fog, pointing error and atmospheric turbulence of the \(i\)-th hop link, respectively. Since the time scale (milliseconds) of the above fading process is much larger than the bit interval (nanoseconds-microseconds), we assume that the composite channel is a slow fading channel. In the following, we will give the statistical models of \(h_{f,i}\), \(h_{p,i}\), and \(h_{a,i}\). (1) Random Fog Statistical Model  
In the multi-hop FSOC system, fog, as an extremely severe weather condition, will seriously attenuate the optical power and thus affect the system performance. The PDF of the random fog in the \(i\)-th hop link is as follows [27]:  
\[f_{h_{f,i}}(h_{f,i})=\frac{z_{i}^{k_{i}}}{\Gamma(k_{i})}\left[\ln\left(\frac{1 }{h_{f,i}}\right)\right]^{k_{i}-1}h_{f,i}^{z_{i}-1},0<h_{f,i}<1, \tag{4}\]  
where \(\Gamma(\bullet)\) represents the gamma function. \(z_{i}=4.343/\beta_{i}^{\text{fog}}d_{i}\), \(d_{i}\) represents the length of the \(i\)-th hop link. \(k_{i}>0\) and \(\beta_{i}^{\text{fog}}>0\) represent the shape parameter and scale parameter of the random fog, respectively. \(\left\{k_{i}=2.32,\beta_{i}^{\text{fog}}=13.12\right\}\) represents light fog, \(\left\{k_{i}=5.49,\beta_{i}^{\text{fog}}=12.06\right\}\) represents moderate fog, and \(\left\{k_{i}=6,\beta_{i}^{\text{fog}}=23\right\}\) represents thick fog. Figure 1: Schematic diagram of a multi-hop FSOC system. Atmospheric Turbulence Statistical Model  
Laser beams propagating outdoors are affected by atmospheric turbulence, which causes random fluctuations in light intensity and affects the performance of multi-hop FSOC systems. Here, we adopt the Gamma-Gamma distribution atmospheric turbulence model. The PDF of atmospheric turbulence in the \(i\)-th hop link is as follows [31]:  
\[f_{h_{a,i}}(h_{a,i})=\frac{2(\alpha_{i}\beta_{i})^{(\alpha_{i}+\beta_{i})/2}}{ \Gamma(\alpha_{i})\Gamma(\beta_{i})}(h_{a,i})^{(\alpha_{i}+\beta_{i})/2-1}K_{ \alpha_{i}-\beta_{i}}\Big{(}2\sqrt{\alpha_{i}\beta_{i}h_{a,i}}\Big{)},h_{a,i}>0, \tag{5}\]  
where \(K_{\
u}(\bullet)\) represents the v-order modified Bessel function of the second kind, \(\alpha_{i}\) and \(\beta_{i}\) represent the large-scale scattering coefficient and the small-scale scattering coefficient of the \(i\)-th hop link, respectively. The expressions of \(\alpha_{i}\) and \(\beta_{i}\) are as follows [32]:  
\[\alpha_{i}=\left[\exp\left(\frac{0.49\sigma_{R,i}^{2}}{\left(1+1.11\sigma_{R,i }^{12/5}\right)^{7/6}}\right)-1\right]^{-1}, \tag{6}\]  
\[\beta_{i}=\left[\exp\left(\frac{0.51\sigma_{R,i}^{2}}{\left(1+0.69\sigma_{R,i }^{12/5}\right)^{5/6}}\right)-1\right]^{-1}, \tag{7}\]  
Since the transmission distance of the beam under the composite channel is short, the Rytov variance of the \(i\)-th hop link adopts the spherical wave model: \(\sigma_{R,i}^{2}=1.23C_{n}^{2}\kappa^{7/6}d_{i}^{11/6}\). \(C_{n}^{2}\) is the atmospheric refractive index structural constant, \(\kappa=2\pi/\lambda\) is the space wave number, and \(\lambda\) is the wavelength of the signal light. Generalized Pointing Error Statistical Model  
In a multi-hop FSOC system, the buildings sway randomly due to physical effects such as dynamic wind loads and thermal expansion. It causes random variations in boresight error and jitter errors at the receiver, seriously degrading system performance. We assume that the radial displacement at the receiver follows Beckmann distribution. According to Ref. [33], the modified Rayleigh distribution can accurately approximate the Beckmann distribution.