# Parameter Optimization Based BPNN of Atmosphere Continuous-Variable Quantum Key Distribution
## 2 Transmittance and Security Analysis
### Transmittance Analysis  
In a weak turbulent atmosphere, beam-wandering plays a primary role. Generally, they are induced by unstable adjustment of a radiation source and aperture truncation of the light at the receiving end [43]. The characterization of the beam wandering mechanism and channel models of fluctuation are demonstrated in [23]. Under the circumstance of beam wandering (see Figure 1a), an approximate analytical representative of transmission efficiency can be expressed as  
\[T^{2}=T_{0}^{2}\mathrm{exp}[-(\frac{r}{R})^{\lambda}], \tag{2}\]  
where \(r\) denotes the distance of beam-deflection. \(T_{0}\) denotes the transmission coefficient maximum for the specified beam-spot radius value \(W\), \(R\) and \(\lambda\) are scale and shape parameters. The above parameters will be explained in detail in \"Appendix A\". Designed for simplicity, the aperture radius \(a\) is normalized, so that the distribution of \(T^{2}\) is acquired. As shown in Figure A1, \(T^{2}\) decreases with the increase of beam deflection distance, while the maximum increases with the falling of beam spot radius \(W\). On the basic of Ref. [44], the fluctuation of beam-deflection distance \(r\) is subject to the Rice distribution [45] with the variance \(\sigma^{2}\) and the distant \(d\) from the center of the aperture. Under the circumstance of \(d=0\), this distribution is simplified to the log-negative Weibull distribution,  
\[P(T)=\frac{2R^{2}}{\sigma^{2}\lambda T}(2\mathrm{ln}\frac{T_{0}}{T})^{2/ \lambda-1}\mathrm{exp}[-\frac{R^{2}}{2\sigma^{2}}(2\mathrm{ln}\frac{T_{0}}{T} )^{2/\lambda}], \tag{3}\]  
for \(T\in[0,T_{0}]\) and \(P(T)=0\), otherwise. Then, the mean of the fading probability distribution can be calculated from \(\langle T\rangle=\int_{0}^{T_{0}}TP(T)dT\) and \(\langle\sqrt{T}\rangle=\int_{0}^{T_{0}}\sqrt{T}P(T)dT\). Figure 1: (**a**) in the beam wandering case, the fluctuation of transmittance results from the variation of the beam-deflection distance \(r\); (**b**) the elliptical beam profile is characterized with the half axis \(W_{1}\) and \(W_{2}\), where \(W_{1}\) rotates on the angle \(\Phi\) relating to the \(x\)-axis. Beam wandering is characterized by parameter \(r\), which represents the beam-centroid position with respect to the center of the aperture. \begin{table}
\begin{tabular}{c c c c c} \hline  & **Spring** & **Summer** & **Autumn** & **Winter** \\ \hline \(C_{n}^{2}\,(m^{-2/3}\times 10^{-15})\) & 2.03 & 2.12 & 5.56 & 7.46 \\ \hline \end{tabular}
\end{table} 
Table 1: The values (median) of \(C_{n}^{2}\) in four seasons. As for strong turbulence, the elliptic-beam model can well describe its characteristics. For now, the probability distribution for free space transmittance could be obtained by using the Glauber-Sudarshan P function on the basic of elliptic beam approximation [24; 46]. Compared with the log-normal model [23], the elliptic-beam model is more consistent with the experimental data [47]. Similarly, this model can also be employed to analyze atmospheric quantum channels under different weather circumstances [25]. As depicted in Figure 1b, the elliptic-beam model can describe any spot at the receiving aperture with five parameters, \(\mathbf{v}=(x_{0},y_{0},W_{1},W_{2})^{T}\) and \(\Phi\). Here, \((x_{0},y_{0})\) represents the position of the beam centroid and reflects the degree of beam wandering, while \((W_{1},W_{2})\) denotes the half axis of elliptical beam section, and \(\Phi\) is the angle between the half axis \(W_{1}\) and the \(x\)-axis. The above three parameters are used to describe the characteristic of beam broadening and distortion, and give a definition of all possible directions.