# Performance Analysis of the Shore-to-Reef Atmospheric Continuous-Variable Quantum Key Distribution
## 4 Extinction Effect of Sea Salt Particles  
Sea salt particles are mainly produced by the tremendous transpiration of the ocean surface and the breakdown of the spoondrift [19]. The radius, relative humidity, and concentration of sea salt particles in ocean atmospheric channel cause absorption and scattering to beam and then affect the performance of CVQKD protocol. In this section, we establish a model of sea salt aerosol based on the characteristics of sea salt particles and Mie scattering theory. We definite \(x=\frac{2\pi a}{\lambda}\), where \(a\) is the radius of particle and \(\lambda\) is the wave length. When \(x\ll 1\), we can use Rayleigh scattering to describe the scattering. When \(x\approx 1\), we can use Mie scattering theory to solve the problem of scattering. The scattering of light in the atmosphere is mainly Mie scattering. Some suspended particles in nature for atmosphere can be regarded as spherical particles and Mie scattering theory is used for research. Mie scattering considers that the size of particles and the refractive index of materials have a decisive influence on the scattering and absorption of light [23; 24; 25]. In the analysis, sea salt particles are supposed to be spherical particles. When a plane wave is incident on a single spherical particle, based on the single scattering theory, the case of an isotropic homogeneous spherical particle is considered. The electric field components of the incident wave in the vertical and parallel scattering planes are connected with the corresponding components of the incident beam via the scattering matrix [19]:  
\[\left( \begin{array}{c}E_{V}^{S}\\ E_{H}^{S}\end{array} \right)=\frac{\textit{exp}(ikr)}{-ikr}\left( \begin{array}{ cc}S_{1}&S_{3}\\ S_{4}&S_{4}\end{array} \right)\left( \begin{array}{c}E_{V}^{i}\\ E_{H}^{i}\end{array} \right), \tag{23}\]  
Figure 3: The refraction structure at altitudes of \(h=0\) km, \(h=1\) km, \(h=10\) km, and \(h=20\) km with various wind speeds. where \(k=2\pi/\lambda\), \(\lambda\) is incident wave length and \(S_{i}\) is scattering matrix. The scattering matrix value is determined by the particle shape, scale, and refractive index. Due to the symmetry of spherical particles, \(S_{3}=0\), \(S_{4}=0\). After a complex solution \(S_{1}\), \(S_{2}\) are expressed as follows:  
\[ \begin{array}{l}S_{1}=\sum_{n=1}^{\infty}\frac{2n+1}{n(n+1)}\big{[}a_{n}\pi_ {n}+b_{n}\tau_{n}\big{]}\\ S_{2}=\sum_{n=1}^{\infty}\frac{2n+1}{n(n+1)}\big{[}b_{n}\pi_{n}+a_{n}\tau_{n} \big{]}.\end{array}  \tag{24}\]  
\(\pi_{n}=\frac{p_{n}^{(1)}(\cos 0)}{sin\theta}=\frac{dP_{n}(\cos 0)}{d(\cos 0)} \text{,}\tau_{n}=\frac{dP_{n}^{(1)}(\cos 0)}{d(\cos \theta)}\), is the coefficient of scattering angle, where \(P_{n}^{(1)}(X)\) is the first kind of Legendre function. The coefficient of the Mie scattering \(a_{n}\),\(b_{n}\) can be expressed:  
\[ \begin{array}{l}a_{n}=\frac{\psi_{n}(x)\psi_{n}^{\prime}(y)-m\psi_{n}^{ \prime}(x)\psi_{n}(y)}{\xi_{n}(x)\psi_{n}^{\prime}(y)-m\psi_{n}^{\prime}(x) \psi_{n}(y)}\\ b_{n}=\frac{m\psi_{n}(x)\psi_{n}(y)-\psi_{n}(x)\psi_{n}(y)}{m\xi_{n}(x)\psi_{n }^{\prime}(y)-\xi_{n}(x)\psi_{n}(y)},\end{array}  \tag{25}\]  
where \(\psi_{n}(X)=\sqrt{\frac{\pi X}{2}}J_{n+\frac{1}{2}}(X)\), \(\xi_{n}(X)=\sqrt{\frac{\pi X}{2}}H_{n+\frac{1}{2}}(X)\), \(J_{n+\frac{1}{2}}(X)\) is the first kind of semi-integral order Bessel function and Hankel functions of the second kind of semi-integral order, respectively. \(\psi_{n}^{\prime}(X),\xi_{n}^{\prime}(X)\) is the derivative of \(\psi_{n}(X),\xi_{n}(X)\), respectively. \(m\) is the refractive index of sea salt, \(m=m_{re}+in_{ie}\). The refractive index of wet sea salt particles is determined by both dry particles and condensed liquid water [19]:  
\[ \begin{array}{l}m_{re}=m_{ru}+(m_{rr}-m_{ru})\big{[}\frac{r(H)}{r_{0}}\big{]} ^{-3}\\ m_{ie}=m_{inv}+(m_{inv}-m_{inv})\big{[}\frac{r(H)}{r_{0}}\big{]}^{-3},\end{array}  \tag{26}\]  
\(m_{ru},m_{inv},m_{rr},m_{ir}\) is the real and imaginary parts of refractive index of water and dry sea salt particles, respectively.