# Performance Analysis of the Shore-to-Reef Atmospheric Continuous-Variable Quantum Key Distribution
## 3 Atmospheric Turbulence Effect  
Atmospheric turbulence has the characteristics with irregularity and randomness. In the turbulence, the temperature, velocity, pressure, refractive index, and other physical properties fluctuate randomly along with time. Atmospheric turbulence is the main factor that restricts the performance of free-space optical communication. It will lead to beam wandering and beam broadening. Ultimately, it will result in the reduction of the reliability and stability for the whole communication system.  
When the beam propagates in turbulent atmosphere, the diameter or area of the spot received by the aperture will increase and the center of the spot will deflect at the receiving plane, which is the phenomenon of beam wandering and broadening. The geometry for this model is shown in Figure 2. Where, \(a\) is the radius of receiver, \(W\) is the radius of beam, and \(r_{w}\) is the distance of wandering [9].  
Assuming that the beam fluctuates in the central plane of the receiver aperture, the probability density distribution can be expressed by Weibull distribution [9]:  
\[P(T_{1})=\frac{2R^{2}}{\sigma^{2}QT_{1}}(2 ln\frac{T_{0}}{T_{1}})\hat{\delta}^{-1 }exp(-\frac{1}{2\sigma^{2}}R^{2}(2 ln\frac{T_{0}}{T_{1}})^{\frac{2}{\delta}}). \tag{16}\]  
\(\sigma^{2}\) is variance of wandering:  
\[\sigma^{2}=1.919C_{n}^{2}L^{3}(2W_{0})^{-1/3}, \tag{17}\]  
where \(L\) is the distance of transmission and \(W_{0}\) is the beam spot radius at the radiation source. \(C_{n}^{2}\) is the index of refraction structure function. In this paper, we use the Hufnagel-Valley model for \(C_{n}^{2}\). It is a height distribution model of \(C_{n}^{2}\) proposed based on experimental measurement that is closely related to height. This model includes stratosphere and troposphere and is widely used [18],  
\[C_{n}^{2}=0.00594(V_{w}/27)^{2}(h/10^{-15})^{10}e^{-h/1000}+2.7\times 10^{-16}e^{ -h/1500}+Ae^{-h/100}, \tag{18}\]  
where \(V_{w}\) is the wind speeds, \(h\) is the altitudes, and \(A\) is constant. Usually, we set it as \(1.7\times 10^{-14}\).  
\(R\) is scale parameter, \(Q\) is shape parameter [9],  
\[R=a[ln(\frac{2T_{0}^{2}}{1-exp(-4\frac{a^{2}}{W^{2}})I_{0}(4\frac{a^{2}}{W^{2} })})]^{-\frac{1}{\delta}}, \tag{19}\]  
\[Q=8\frac{a^{2}}{W^{2}}\frac{exp(-4\frac{a^{2}}{W^{2}})I_{1}(4\frac{a^{2}}{W^{2} })}{1-exp(-4\frac{a^{2}}{W^{2}})I_{0}(4\frac{a^{2}}{W^{2}})}\times[\ln(\frac{2 T_{0}^{2}}{1-exp(-4\frac{a^{2}}{W^{2}})I_{0}(4\frac{a^{2}}{W^{2}})})]^{-1}. \tag{20}\]  
\(I_{i}\) is modified Bessel function and \(T_{0}\) is the maximum transmission efficiency for a given beam radius \(W\), which can be written as [9] :  
\[T_{0}^{2}=1-exp(-2\frac{a^{2}}{W^{2}}). \tag{21}\]  
Figure 2: Beam wandering and broadening.  
The transmission efficiency of the beam with a beam radius \(W\) on the receiving plane can be expressed as follows [9]:  
\[T_{1}^{2}=T_{0}^{2}\textit{exp}(-(\frac{r}{R})^{Q}). \tag{22}\]  
Figure 3 shows the relationship between refractive index structure function and wind speeds. It can be seen clearly from the figure that the higher the altitude, the greater the influence of refractive index structure function on wind speed. At altitude of 0 and 1 km, the refractive index structure function is almost unchanged with wind speed. In addition, the lower the altitude, the larger the refractive index structure function, which indicates that severe atmospheric fluctuations could cause drastic turbulence effects. When the altitude is above 10 km, the refractive index structure function is very small, and the atmospheric turbulence effect can be negligible.