# Performance Analysis of UAV-IRS Relay Multi-Hop FSO/THz Link
## 2 System and Channel Models
### FSO Link
where the parameter \(g_{ij}^{2}=w_{eq,ji}{}^{2}/4\sigma_{s,ij}{}^{2}\) denotes the ratio of the radii of equivalent beamwidths at the receiver (RX), while \(A_{ij}\) signifies the fraction of power collected at the focal point \(r=0\), calculable via parameter \(u=\sqrt{\pi}a_{ij}/\sqrt{\pi}w_{d,ij}\). The term \(r\) is defined as the pointing error, quantified by the radial distance between the centers of the transmitted and received beams. The variance of the pointing error displacement at the RX is indicated by \(\sigma_{s,ij}\), with a circular detector beam characterized by a radius \(a_{ij}\). The maximum radius at a given distance d on the \(i\)-th link is denoted by \(w_{d,ij}\) and \(w_{eq,ij}\) represents the equivalent beam width, calculated as \(w_{eq,ji}{}^{2}=w_{d,ij}{}^{2}\times\sqrt{\pi}\cdot erf(u)/2u\cdot\exp(-u^{2})\). \(g_{ij}^{2}\) is influenced by the IRS-UAV link type and node fluctuation. G(\(\cdot\)) is the [PERSON]'s G function; because we installed the IRS on the UAV, the IRS-UAV relay node is analyzed as a whole for determining the aiming inaccuracy. Furthermore, the link types in this system can be split into three categories, S-IRS-UAV(SU), IRS-UAV-IRS-UAV(UU), and IRS-UAV-D(UD), and the pointing error parameters of different types of connections differ, and we assume the pointing error parameters of the same types of links to be the same. According to references [15; 16], the specific formula for parameter \(\sigma_{s,ij}{}^{2}\) is provided in Equation (3) and the turbulent parameters for large and small scales are denoted by \(\alpha_{i}\) and \(\beta_{i}\), respectively, as detailed in Equation (3):  
\[\sigma_{s,ij}^{2}=\left\{ \begin{array}{ll}\sigma_{i,j,u}^{2}+ \sigma_{i-1,j,q}^{2}&\text{if $i=1$}\\ 2\sigma_{i,u}^{2}+L_{FSO,i-1}\sigma_{i-1,jangle}^{2}&\text{if $i=2,\ldots,N$}\\ \sigma_{i,j,u}^{2}+\sigma_{i-1,j,q}^{2}+L_{FSO,j-1}\sigma_{i-1,jangle}^{2}& \text{if $i=N+1$}\end{array} \right. \tag{3}\] \[\alpha_{i}=\left[\exp\left(\frac{0.49\sigma_{\text{FSO},j}^{2}}{( 1+1.11\sigma_{\text{FSO},j}^{2/5})^{2/6}}\right)-1\right]^{-1}\] \[\beta_{i}=\left[\exp\left(\frac{0.51\sigma_{\text{FSO},j}^{2}}{( 1+0.69\sigma_{\text{FSO},j}^{2/5})^{5/6}}\right)-1\right]^{-1}.\]  
where \(je\){SU,UU,UD} and \(L_{FSO,i-1}\) is the \((i-1)\)-th link length, \(\sigma_{FSO,i}^{2}=1.23C_{n}^{2}k^{7/6}L_{FSO,i}^{11/6}\) is the Rytov variance, \(C_{n}^{2}\) is the atmospheric refractive index structure parameter, \(k=\frac{2\pi}{\lambda_{F}}\) is the wave number, and \(\lambda_{F}\) is the wavelength. The PDF describing the likelihood of link disruption due to AoA fluctuations has been detailed in reference [15], so the PDF of overall channel gain (denoted as \(h_{FSO,i}\)) can be approximated as Equation (4):  
\[f_{FSO,i}(h_{FSO,i})\simeq exp\left(-\frac{\psi_{AoA,i}^{2}}{2m_{i }\sigma_{i-1,jangle}^{2}}\right)\delta(h_{FSO,i})\ +\left[1-exp\left(\frac{-\psi_{AoA,i}^{2}}{2m_{i}\sigma_{i-1,jangle}^{2}} \right)\right] \tag{4}\] \[\times\frac{\alpha_{i}\beta_{i}\delta_{ij}^{2}}{\Lambda_{ij}h_{l, FSO,i}\Gamma(\alpha_{i})\Gamma(\beta_{i})}\times\mathrm{G}_{1,3}^{3,0}\left(\frac{ \alpha_{i}\beta_{i}}{A_{i}h_{l,i}}h_{l,FSO,i}\right| \begin{array}{c}g_{ij}^{2 }\\ g_{ij}^{2}-1,\alpha_{i}-1,\beta_{i}-1\end{array} \right),\]  
the variable \(m_{i}\), which varies according to the type of link, is characterized in detail within reference [16]:  
\[m_{i}=\left\{ \begin{array}{ll}1,&i=1,N+1\\ 2,&i=2,\ldots,N.\end{array} \right. \tag{5}\]  
In the scenario involving \(W\) optical links, each link passes through \(W\) reflective elements on the IRS-UAV to reach the next IRS-UAV relay node. This configuration results in an overall instantaneous SNR, as follows:  
\[\gamma_{SUMFSO,i}=\sum_{n=1}^{W}\ \gamma_{FSO,i}=\tilde{\gamma}_{FSO,j}\sum_{n=1} ^{W}h_{FSO,i}^{2}\, \tag{6}\]  
\(\tilde{\gamma}_{FSO,i}\) represents the average SNR and \(h_{FSO,i}\) signifies the total channel gain of the initial FSO link.