# Wavelength Selection for Satellite Quantum Key Distribution
## 3 Channel Model
### Turbulence  
As the beam propagates through the atmosphere, it may encounter atmospheric turbulence caused by random fluctuations in temperature and pressure. This alters the spatial and temporal refractive index of air, which can cause the beam to spread and become distorted [52; 53]. The refractive index structure of the atmosphere, \(c_{n}^{2}(x)\), is given by Equation (4) as follows:  
\[\begin{split} c_{n}^{2}(x)=0.00594(v/27)^{2}(10^{-5}x)^{10}\exp \left(-x/1000\right)\\ +2.7\times 10^{-16}\exp\left(-x/1500\right)+A\exp\left(-x/100 \right)\end{split} \tag{4}\]  
where \(x\) is the altitude, the wind speed \(v\) is 21 m/s and \(A\) is the value of \(C_{n}^{2}(0)\), set to \(1.7\times 10^{-14}\) for the Hufnagel-Valley model \(HV_{5/7}\) variant. Most turbulent effects occur within the lower 20 km of the atmosphere. In the downlink, the beam travels hundreds of kilometres before it interacts with significant turbulence, at which point the beam waist is large and more resilient to turbulence. To quantify the amount of beam spread and beam wander in the downlink scenario, we use the phase screen model [52; 54; 55; 56], where the propagation path is subdivided into smaller regions and the random phase changes of each region are accounted for by a thin phase screen placed at the beginning of each region (details of the simulation are provided in Appendix A). Figure 2 shows the evolution of two beams as they propagate from a 500 km satellite down through the turbulent atmosphere. The left beam shows the intensity profile of the beam at the point of initialization, while the centre and right images show the intensity profile that result from a 1.55 um beam and a 10 um beam, respectively, after each has propagated through a statistically independent realization of the satellite downlink channel.  
The wavefronts at the receiver differ in two meaningful ways. Firstly, the 15 cm beam waist has grown to 1.65 m in the case of the SWIR beam and to 10.6 m for the LWIR beam. Secondly, the turbulence-induced phase changes have distorted both wavefronts but significantly less for the 10 um beam. This retains much of its original Gaussian distribution unlike the intensity of the 1.55 um beam which is now distributed in small pockets. The scintillation index (\(\sigma_{l}^{2}\)) is a useful measure to quantify the strength of atmospheric turbulence and is defined as the normalized variance of irradiance fluctuations. Calculations confirm that for zenith angles less than 60\({}^{\circ}\), the downlink channel operates under the condition of weak turbulence \((\sigma_{I}^{2})<1\)[57]. Thus, we can consider turbulence-induced beam spread negligible. Beam spread is almost entirely caused by diffraction and can be accurately approximated by a Gaussian spatial profile using Equation (3). The phase screen algorithm is computationally intensive to run, so the Strehl ratio, \(S\), is adopted [58] to account for the reduced spatial coherence at the receiver entrance pupil, which leads to a broadened spot size in the focal plane.  
\[S=\left[1+\left(\frac{D_{\text{R}}}{r(\lambda)}\right)^{5/3}\right]^{-6/5} \tag{5}\]  
where \(r(\lambda)\) is the coherence length given by  
\[r(\lambda)=r_{0}\Big{(}\frac{\lambda}{\lambda_{0}}\Big{)}^{6/5} \tag{6}\]  
where \(r_{0}\) is the coherence length measured at \(\lambda_{0}\) = 500 nm and is inversely related to the strength of turbulence.