# Parameter Optimization Based BPNN of Atmosphere Continuous-Variable Quantum Key Distribution
## Appendix A The Atmospheric Transmittance Analysis  
In this appendix, we will demonstrate the formulas for the aforementioned parameters in Section 2.1. Under the circumstance of beam wandering (see Figure 1a), an approximate analytical representative of transmission efficiency can be expressed as  
\[T^{2}=T_{0}^{2}\text{exp}[-(\frac{r}{R})^{\lambda}], \tag{1}\]  
where \(r\) denotes the distance of beam-deflection, \(\lambda\) and \(R\) are the shape and scale parameter, respectively, which can be derived  
\[\lambda= 8\frac{a^{2}}{W^{2}}\frac{\text{exp}[-4\frac{a^{2}}{W^{2}}]I_{1} (4\frac{a^{2}}{W^{2}})}{1-\text{exp}[-4\frac{a^{2}}{W^{2}}]I_{0}(4\frac{a^{2}} {W^{2}})} \tag{2}\] \[\times[\text{ln}(\frac{2T_{0}^{2}}{1-\text{exp}[-4\frac{a^{2}}{W^ {2}}]I_{0}(4\frac{a^{2}}{W^{2}})})]^{-1},\]  
\[R=a[\text{ln}(\frac{2T_{0}^{2}}{1-\text{exp}[-4\frac{a^{2}}{W^{2}}]I_{0}(4 \frac{a^{2}}{W^{2}})})]^{-(1/\lambda)}, \tag{3}\]  
and \(T_{0}\) denotes the transmission coefficient maximum for the specified beam-spot radius value \(W\); one can be given by  
\[T_{0}^{2}=1-\text{exp}[-2\frac{a^{2}}{W^{2}}], \tag{4}\]  
where \(a\) denotes the aperture radius. Obviously, the transmittance efficiency \(T^{2}\) is dominated by the ratio \(a/W\) and reduced at \(T_{0}\). Designed for simplicity, the aperture radius \(a\) is normalized, so that the distribution of \(T^{2}\) is acquired. For strong turbulence, the maximal transmittance for a centered beam \(T_{0}\) can be given by  
\[ \begin{split} T_{0}=& 1-I_{0}\big{(}a^{2}\frac{W_{1}^{2}-W_{ 2}^{2}}{W_{1}^{2}W_{2}^{2}}\big{)}\text{exp}\big{(}-a^{2}\frac{W_{1}^{2}+W_{2}^ {2}}{W_{1}^{2}W_{2}^{2}}\big{)}\\ &-2\Big{\{}1-\text{exp}\Big{[}-\frac{a^{2}}{2}(\frac{1}{W_{1}}- \frac{1}{W_{2}})\Big{]}\Big{\}}\\ &\times\text{exp}\Big{[}-\Big{\{}\frac{(W_{1}+W_{2})^{2}}{|W_{1} ^{2}-W_{2}^{2}|}\Big{\}}^{\lambda(1/W_{1}-1/W_{2})}\Big{]},\end{split}  \tag{10}\]  
which is a function of the two eigenvalues \(W_{i}^{2},i=1,2\). The shape \(\lambda\) and scale \(R\) functions are given by  
\[ \begin{split}\lambda(\xi)=& 2a^{2}\xi^{2}\frac{ \text{exp}[-a^{2}\xi^{2}]I_{1}(a^{2}\xi^{2})}{1-\text{exp}[-a^{2}\xi^{2}]I_{0}( a^{2}\xi^{2})}\\ &\times\Big{[}\text{ln}(2\frac{1-\text{exp}[-0.5a^{2}\xi^{2}]}{1 -\text{exp}[-a^{2}\xi^{2}]I_{0}(a^{2}\xi^{2})})\Big{]}^{-1},\end{split}  \tag{11}\]  
\[R(\xi)=\Big{[}\text{ln}\Big{(}2\frac{1-\text{exp}[-0.5a^{2}\xi^{2}]}{1-\text{ exp}[-a^{2}\xi^{2}]I_{0}(a^{2}\xi^{2})}\Big{)}\Big{]}^{-(1/\lambda(\xi))}. \tag{12}\]  
In the case of \(\varphi=\Phi-\phi_{0}\), the effective squared spot radius is expressed as  
\[ \begin{split} W_{eff}^{2}(\varphi)=& 4a^{2}\bigg{\{}\mathcal{W} \Big{(}\frac{4a^{2}}{W_{1}W_{2}}\text{exp}\Big{[}\frac{a^{2}}{W_{1}^{2}}(1+2 \cos^{2}(\varphi))\Big{]}\\ &\times\text{exp}\Big{[}\frac{a^{2}}{W_{2}^{2}}(1+2\sin^{2}( \varphi))\Big{]}\Big{)}\bigg{\}}.\end{split}  \tag{13}\]  
On the basic of the above equations, the probability distribution of atmospheric transmittance can be calculated by the Monte Carlo method. It is evident that the transmittance \(T\) is decided by five parameters \(\{x_{0},y_{0},\Theta_{1},\Theta_{2},\varphi\}\), where \(W_{i}\) is log-normally distributed and it can be acquired by \(W_{i}^{2}=W_{0}^{2}\text{exp}\Theta_{i}\). Parameter \(\varphi\) is uniformly distributed and irrelevant to the other parameters. The relevance of \(\{x_{0},y_{0},\Theta_{1},\Theta_{2}\}\) is represented by covariance matrix [24; 25]  
\[\Xi=\left( \begin{array}{cccc}\langle x_{0}^{2}\rangle&0&0&0\\ 0&\langle x_{0}^{2}\rangle&0&0\\ 0&0&\langle\Theta_{1}^{2}\rangle&\langle\Theta_{1}\Theta_{2}\rangle\\ 0&0&\langle\Theta_{1}\Theta_{2}\rangle&\langle\Theta_{2}^{2}\rangle\end{array}  \right). \tag{14}\]  
For weak turbulence, the elements of the covariance matrix \(\Xi\) can be given by [24]  
\[