## 2 System Model
### RF Interference Link  
The jammer harvests the energy and then transmits the signal. Considering that the transmitted jamming signal will only be effective for the eavesdropper, we assume \(\rho_{i}(\phi_{i})=1\), \(\phi_{i}=\theta_{i}+\varphi_{i}\), which maximizes the SNR of the interfering link [21]. The interference signal is reflected by the RIS to interfere with the eavesdropper. According to [22], the instantaneous SNR of the interference link is:  
\[\gamma=\sum_{i=1}^{N}{(x_{i}y_{i})^{2}\frac{P_{I}}{N_{0}}}=\sum_{i=1}^{N}{W_{i }^{2}\overline{\gamma}} \tag{7}\]  
where \(W_{i}=x_{i}y_{i}\) and \(x_{i},y_{i}\) correspond to two independent random variables that observe to the Nakagami-m distribution, N is the number of RIS reflecting elements, \(N_{0}\) is the variance of additive white Gaussian noise, and \(\overline{\gamma}\) is the average SNR of the interference link. Using the first term of Laguerre expansion, the PDF of \(W=\sum_{i=1}^{N}{W_{i}}\) is given as:  
\[f_{W}(z)=\frac{z^{a}}{b^{a+1}\Gamma(a+1)}\exp\Bigl{(}-\frac{z}{b}\Bigr{)} \tag{8}\]  
where a and b are mean and variance of W, respectively, and given as follows:  
\[a=\frac{(N+1)\Gamma(m+0.5)^{4}-m^{2}\Gamma(m)^{4}}{m^{2}\Gamma(m)^{4}-\Gamma( m+0.5)^{4}}\text{ and }b=m\Omega\left(\frac{\Gamma(m)^{2}\Gamma(m+1)^{2}-\Gamma(m+0.5)^{4}}{\Gamma(m+0.5 )^{2}\Gamma(m+1)^{2}}\right)\]where \(\Gamma(\cdot)\) is the Gamma function, \(m\in\{1,2,\ldots,\infty\}\) is the fading severity parameter, and \(\Omega\) is the mean fading power. Changing the variable of (8), the PDF of \(\gamma\) under Nakagami-m distribution is obtained as:  
\[f_{\gamma}(\gamma)\approx\frac{\gamma^{(a-1)/2}}{2\overline{\gamma}^{(a+1)/2}b^ {a+1}\Gamma(a+1)}\exp\left(-b^{-1}\sqrt{\frac{\gamma}{\gamma}}\right) \tag{9}\]  
The CDF of (9) can be obtained as:  
\[F_{\gamma}(\gamma)\approx\frac{1}{\Gamma(a+1)}G_{1,2}^{1,1}[b^{-1}\sqrt{\frac {\gamma}{\gamma}}|a+1,0] \tag{10}\]  
The fading of the J to E link is evident due to the no line of sight resulting from obstructions like bushes. Combining (10) and [23], the CDF of the instantaneous SNR of the whole interference link is obtained as follows:  
\[\begin{array}{l}F_{\gamma_{S|R^{\prime}E}}(\gamma)=1-\frac{1}{\Gamma(a+1) \Gamma\left(M_{j}-1\right)}\sum\limits_{k=0}^{m_{SE}-1}\frac{1}{k!}\left( \frac{m_{SE}}{\Omega_{SE}\left(M_{j}-1\right)}\right)^{k}\sum\limits_{p=0}^{k }\left(\begin{array}{c}k\\ p\end{array}\right)\Omega_{JR^{\prime}}^{p}\Omega_{R^{\prime}E}^{p}\gamma^{k} \\ \times\exp\left(-\frac{m_{SE\gamma}}{\Omega_{SE}\left(M_{j}-1\right)}\right)G_ {1,1}^{1,1}\left[\frac{m_{SE}\Omega_{JR^{\prime}}\Omega_{R^{\prime}E}\gamma}{ \Omega_{SE}\left(M_{j}-1\right)}\right.\left.-M_{j}-p+2\right.\\ 0\end{array}\right]G_{1,2}^{1,1}\left[\frac{\sqrt{\gamma}}{b\sqrt{\gamma}}|1+a,0\right]\end{array} \tag{11}\]  
The corresponding PDF of the instantaneous SNR can be obtained by derivation and simplification of Equation (11) as:  
\[\begin{array}{l}f_{\gamma_{S|R^{\prime}E}}(\gamma)=\frac{1}{\Gamma(m_{SE}) \Gamma\left(M_{j}-1\right)\Gamma\left(a+1\right)}\left(\frac{m_{SE}}{\Omega_{ SE}\left(M_{j}-1\right)}\right)^{m_{SE}}\sum\limits_{p=0}^{m_{SE}}\left( \begin{array}{c}m_{SE}\\ p\end{array}\right)\Omega_{JR^{\prime}}^{p}\Omega_{R^{\prime}E}^{p}\gamma^{m_{ SE}-1}\exp\left(-\frac{m_{SE\gamma}}{\Omega_{SE}\left(M_{j}-1\right)}\right)\\ \times\Omega_{1,2}^{1,1}[\frac{\sqrt{\gamma}}{b\sqrt{\gamma}}|a+1,0]\left.G_ {1,1}^{1,1}\left[\frac{m_{SE}\Omega_{JR^{\prime}}\Omega_{R^{\prime}E}\gamma}{ \Omega_{SE}\left(M_{j}-1\right)}\right.\left.-M_{j}-p+2\right.\\ 0\end{array}\right]-\frac{2\pi}{\sqrt{\gamma}\gamma}\sum\limits_{k=0}^{m_{SE}- 1}\frac{1}{k!}\left(\frac{m_{SE}}{\Omega_{SE}\left(M_{j}-1\right)}\right) \sum\limits_{p=0}^{k}\left(\begin{array}{c}k\\ p\end{array}\right) \tag{12}\]