## Appendix A
 \tag{14}\]  
With the integral simplification of \(\mathbf{r_{d}}\) again, Equation (13) can be written as Equation (12):  
\[ \begin{array}{ll}W_{aij}(\mathbf{\rho}_{1},\mathbf{\rho}_{2},z)=&\frac{k^{2}A_{i}A_{j}B _{ij}}{4\epsilon^{2}\Lambda_{aij}\sqrt{Q(z)_{aij}^{2}+T(z)_{aij}^{2}}}\exp \left[M_{aij}(z)(\rho_{1}-\rho_{2})^{2}\right]\exp\left[I_{aij}(z)(\rho_{1}+ \rho_{2})^{2}\right]\\ &\times\exp\left[G_{aij}(z)\left(\rho_{1}^{2}-\rho_{2}^{2}\right)\right]\exp \left[iL_{aij}(z)(\rho_{1}-\rho_{2})^{2}\right]\\ &\times\exp\left[iY_{aij}(z)(\rho_{1}+\rho_{2})^{2}\right]\exp\left[iN_{aij}(z) \left(\rho_{1}^{2}-\rho_{2}^{2}\right)\right]\end{array}  \tag{15}\]  
where\[\left\{ \begin{array}{l}H(z)_{aij}=\frac{Q(z)_{aij}\left[\frac{1}{4}\left[\frac{1}{2 }\left(\frac{M_{2}}{2}-\frac{b^{2}}{2e^{2}A_{aij}}\right)^{2}-T(z)_{aij}^{2} \right]+\frac{1}{2}T(z)_{aij}\left(\frac{T(z)_{aij}t^{2}}{2e^{2}A_{aij}}-\frac{ M_{3}T(z)}{2}\right)\right]}{Q(z)_{aij}^{2}+T(z)_{aij}^{2}}\\ G(z)_{aij}=\frac{Q(z)_{aij}\frac{1}{4}\frac{1}{2}T(z)_{aij}+\frac{1}{2}T(z)_{aij} \left(\frac{M_{2}k}{4e^{2}A_{aij}}+\frac{b^{3}}{4e^{2}A_{aij}}\right)}{Q(z)_{aij }^{2}+T(z)_{aij}^{2}}\\ I(z)_{aij}=\frac{-\frac{1}{2}Q(z)_{aij}\frac{1}{4e^{2}}}{Q(z)_{aij}^{2}+T(z)_{aij }^{2}}\\ Y(z)_{aij}=\frac{-\frac{1}{2}T(z)_{aij}\frac{1}{4e^{2}}}{Q(z)_{aij}^{2}+T(z)_{aij }^{2}}\\ R(z)_{aij}=\frac{T(z)_{aij}\frac{1}{4}\frac{1}{2}T(z)_{aij}-\frac{1}{2}Q(z)_{aij }\left(\frac{M_{2}k}{4e^{2}}-\frac{b^{3}}{4e^{2}A_{aij}}\right)}{Q(z)_{aij}^{2 }+T(z)_{aij}^{2}}\\ L(z)_{aij}=\frac{-\frac{1}{2}T(z)_{aij}\left[T(z)_{aij}^{2}-\left(\frac{M_{2}} {2}-\frac{b^{2}}{2e^{2}A_{aij}}\right)^{2}\right]-\frac{1}{2}Q(z)_{aij}\left( \frac{T(z)_{aij}t^{2}}{2e^{2}A_{aij}}-\frac{M_{3}T(z)}{2}\right)}{Q(z)_{aij}^ {2}+T(z)_{aij}^{2}}\\ M_{aij}(z)=-\frac{k^{2}}{4e^{2}A_{aij}}-\frac{M_{1}}{2}+H_{aij}(z)\\ N_{aij}(z)=R_{aij}(z)-\frac{k}{2\pi}\end{array} \right. \tag{10}\]  
Subsequently, the analytic process of IF power is also derived. Using the relationship \(\boldsymbol{\rho}_{1}\boldsymbol{\rho}_{2}=\rho_{1}\rho_{2}\text{cos}( \boldsymbol{\rho}_{1}\text{-}\boldsymbol{\rho}_{2})\) and rearranging the integration, Equation (10) can be written as:  
\[ \begin{array}{l}P_{1 Fij}=\int_{0}^{2\pi}\int_{0}^{\infty}\int_{0}^{\infty} \text{Re}\Bigg{\{}\frac{k^{4}A_{i}^{2}A_{j}^{2}B_{i}^{2}}{16^{4}A_{iij}A_{iij} \sqrt{Q(z)_{ij}^{2}+T(z)_{ij}^{2}}\sqrt{Q(z)_{ij}^{2}+T(z)_{ij}^{2}}}\\ \exp\Big{[}g_{ij}\big{(}\rho_{1}^{2}+\rho_{2}^{2}\big{)}+s_{ij}\big{(}\rho_{1} ^{2}-\rho_{2}^{2}\big{)}+im_{ij}\big{(}\rho_{1}^{2}+\rho_{2}^{2}\big{)}+ie_{ij} \big{(}\rho_{1}^{2}-\rho_{2}^{2}\big{)}\Big{]}\rho_{1}\rho_{2}d\rho_{1}d\rho_{2} \\ \times\int_{0}^{2\pi}\exp\Big{[}\big{(}2q_{ij}\rho_{1}\rho_{2}+in_{ij}2\rho_{1 }\rho_{2}\big{)}\big{(}\cos\rho_{1}\cos\rho_{2}+\sin\varphi_{1}\sin\varphi_{2} \big{)}\Big{]}\Big{\}}d\varphi_{1}d\varphi_{2}\end{array}  \tag{11}\]  
where  
\[\left\{ \begin{array}{l}g_{ij}=M_{1 ij}(z)+M_{5 ij}(z)+I_{Lij}(z)+I_{Sij}(z)- \frac{1}{w^{2}}\\ q_{ij}=L_{Lij}(z)+I_{Sij}(z)-M_{Lij}(z)-M_{Sij}(z)\\ s_{ij}=G_{Lij}(z)+G_{Sij}(z)\\ m_{ij}=L_{Lij}(z)+L_{Sij}(z)+Y_{Lij}(z)+Y_{Sij}(z)\\ n_{ij}=Y_{Lij}(z)-Y_{Sij}(z)-L_{Lij}(z)+L_{Sij}(z)\\ e_{ij}=N_{Lij}(z)-N_{Sij}(z)\end{array} \right. \tag{12}\]  
Using the identities written as formulas in Ref. [24]:  
\[\int_{0}^{2\pi}\exp(m\cos x+n\sin x)\exp(ir\cos x)dx=2\pi I_{0}\Big{(}\sqrt{E+ iF}\Big{)},m^{2}+\left(r+n\right)^{2}>0 \tag{13}\]  
where \(E=m^{2}+n^{2}\text{-}r^{2}\), \(F=2 mr\). Using Equation (13), Equation (11) can be simplified as:  
\[ \begin{array}{l}P_{1 Fij}=\int_{0}^{\infty}\int_{0}^{\infty}\text{Re}\Bigg{\{} \frac{n^{2}k^{4}A_{i}^{2}A_{j}^{2}B_{ij}^{2}}{4e^{4}A_{Lij}\Delta_{Sij}\sqrt{Q(z) _{ij}^{2}+T(z)_{ij}^{2}}}\\ \exp\big{[}(g_{ij}+s_{ij}+i\left(m_{ij}+e_{ij}\right))\rho_{1}^{2}\big{]}\exp \big{[}(g_{ij}-s_{ij}+i\left(m_{ij}-e_{ij}\right))\rho_{2}^{2}\big{]}\\ I_{0}\big{(}2q_{ij}\rho_{1}\rho_{2}+i2n_{ij}\rho_{1}\rho_{2}\big{)}\rho_{1}\rho_{2 }d\rho_{1}d\rho_{2}\end{array}