## 4 Security Performance Analysis of Hybrid FSO/RF System  
### Security Outage Probability  
In the physical layer security theory, the security outage probability is the probability that the instantaneous security capacity \(C_{s}\) is lower than the target rate \(R_{s}\) (\(R_{s}>0\)). The SOP is usually considered an important parameter to evaluate the security performance of the wireless communication system. According to [32], the definition formula of SOP can be written as:  
\[\begin{split} SOP&=P_{r}\Big{\{}C_{s}\Big{(}\gamma^{ MRC},\gamma^{E}\Big{)}\leq R_{s}\Big{\}}\\ &\stackrel{{\Delta}}{{=}}P_{r}\Big{\{}\gamma^{MRC}\leq \Theta\gamma^{E}+\Theta-1\Big{\}}\\ &\stackrel{{\Delta}}{{=}}\int_{0}^{\infty}F_{\gamma^{ MRC}}\Big{(}\Theta\gamma^{E}+\Theta-1\Big{)}f_{\gamma^{E}}\Big{(}\gamma^{E}\Big{)}d\gamma^{E}, \end{split} \tag{17}\]  
where \(\Theta=\exp(R_{s})\geq 1\) and \(f_{\gamma^{E}}\big{(}\gamma^{E}\big{)}\) is the PDF of the eavesdropping channel. It should be noted that the eavesdropping channel can be either the RF link or FSO link, that is, \(\gamma^{E}=\gamma^{RF}\) or \(\gamma^{E}=\gamma^{FSO}\).  
According to the method provided in [32], assuming \(\gamma^{E}\rightarrow\infty\), the lower bound of the SOP of the hybrid system can be given:  
\[SOP_{L}=\int_{0}^{\infty}F_{\gamma^{MRC}}(\Theta\gamma)f_{\gamma^{E}}(\gamma)d\gamma. \tag{18}\]  
(1) RF link eavesdropping  
By substituting Equations (3) and (7) into Equation (18), the SOP of the hybrid system with the RF link eavesdropping can be given by  
\[SOP_{L}^{RF}(R_{s}) =A_{1}A_{m}^{E}\sum_{k=1}^{\beta}c_{k}\sum_{i=0}^{\infty}C_{1} \Theta^{D_{1}} \tag{19}\] \[\cdot\int_{0}^{\infty}(\gamma)^{C_{m}^{E}H_{m}^{E}+D_{1}-1}e^{-B_ {m}^{E}(\gamma)^{C_{m}^{E}}}\] \[\cdot G_{r+23r+2}^{3r2}\bigg{(}E_{1}\Theta\gamma\bigg{|}\begin{array} []{c}1-D_{\
u}\,1,\kappa_{1}\\ \kappa_{2}\,1-D_{\
u}-D_{1}\end{array}\bigg{)}d\gamma\]  
According to Equation (07.34.21.0012.01) in [44] and Equation (8.4.3.1) in [48], Equation (19) can be simplified as:  
\[SOP_{L}^{RF}(R_{s})=A_{1}A_{m}^{E}\sum_{k=1}^{\beta}c_{k}\sum_{i=0}^{\infty}C _{1}\Theta^{D_{1}}(E_{1}\Theta)^{-C_{m}^{E}H_{m}^{E}-D_{1}}H_{3r+2,r+3}^{3,3r} \bigg{(}\frac{B_{m}^{E}}{(E_{1}\Theta)^{C_{m}^{E}}}\bigg{|}\begin{array}{c} \kappa_{7}\\ \kappa_{8}\end{array}\bigg{)} \tag{20}\]  
where \(\kappa_{7}=(1-C_{m}^{E}\mu_{m}^{E}-D_{1}-\kappa_{2},C_{m}^{E}),(-C_{m}^{E}\mu_ {m}^{E},C_{m}^{E}),(1-C_{m}^{E}\mu_{m}^{E},C_{m}^{E})\) and \(\kappa_{8}=(0,1)\), \((-C_{m}^{E}\mu_{m}^{E},C_{m}^{E}),(-C_{m}^{E}\mu_{m}^{E}-D_{1},C_{m}^{E}),(1-C_ {m}^{E}\mu_{m}^{E}-D_{1}-\kappa_{1},C_{m}^{E})\).  
(2) FSO link eavesdropping  
Due to the close location of the FSO eavesdropping device and the FSO communication receiver, it can be considered that the FSO eavesdropping link and the FSO communication link have experienced the same turbulent environment, but the beam jitter on the plane of the two receivers is different. We modify all parameters (i.e., \(g\to g^{E},F\to F^{E},B\to B^{E}\), and \(c_{k}\to c_{i}^{E}\)) in Equation (1), and then bring them into Equation (18) with Equation (7). The SOP of the hybrid system with the FSO link eavesdropping can be obtained after a series of algebraic operations:  
\[SOP_{L}^{FSO}(R_{s})=\frac{\big{(}\xi^{E}\big{)}^{2}A^{E}A_{1}}{2^{r^{E}}}\sum _{k=1}^{\beta}c_{k}\sum_{i=0}^{\infty}C_{1}\Theta^{D_{1}}\sum_{k=1}^{\beta^{E }}b_{k^{E}}^{E}(E_{1}\Theta)^{-D_{1}}H_{3r+3,r+5}^{5,3r}\bigg{(}E_{2}\bigg{|} \begin{array}{c}\kappa_{9}\\ \kappa_{10}\end{array}\bigg{)} \tag{21}\]  
where \(E_{2}=B^{E}\Big{(}E_{1}\Theta\mu^{E,r^{E}}\Big{)}^{-\frac{1}{r^{E}}}\), \(\kappa_{9}=\Big{(}1-D_{1}-\kappa_{2},\frac{1}{r^{E}}\Big{)}\), \(\Big{(}0,\frac{1}{r^{E}}\Big{)}\),\(\Big{(}1,\frac{1}{r^{E}}\Big{)}\),\(\Big{(}1+\big{(}\xi^{E}\big{)}^{2},1\Big{)}\), and \(\kappa_{10}=\Big{(}\big{(}\xi^{E}\big{)}^{2},1\Big{)}\), \(\big{(}\alpha^{E},1\big{)},(k^{E},1\big{)},\Big{(}0,\frac{1}{r^{E}}\Big{)}, \Big{(}-D_{1},\frac{1}{r^{E}}\Big{)},\Big{(}1-D_{1}-\kappa_{1},\frac{1}{r^{E} }\Big{)}\).