# Wavelength Selection for Satellite Quantum Key Distribution
## 5 Results and Discussion
### Wavelength Selection Through Atmospheric Turbulence  
The authors in [58] show that the use of adaptive optics combined with tight filtering can improve QKD performance particularly in the visible region. As the beam spot size is proportional to the wavelength, using smaller wavelengths allows for tighter spatial filtering, which results in a higher signal-to-noise ratio. Filtering becomes even more important in conditions of low visibility, where transmittance decreases while spectral radiance increases due to additional atmospheric scattering. Here, we investigate how \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{7}{c}{**Scenario 1: Tucson**} \\ \hline
**Parameter** & **Visible** & **NIR** & **SWIR** & **MWIR** & **LWIR** & **NIR\({}_{2}\)** & **SWIR\({}_{2}\)** \\ \hline Wavelength \(\lambda\) (\(\mu\)m) & 0.69739 & 1.2857 & 2.1344 & 3.7026 & 8.3031 & 0.867 & 1.556 \\ \hline Secret key rate \(R\) (bits/use) & 0.0287 & 0.0339 & 0.036 & 0.0358 & 0.0246 & 0.0322 & 0.0352 \\ \hline Receiver diameter \(D_{r}\) (m) & 1.3784 & 2.1613 & 3.1423 & 4.5 & 5.1342 & 1.6757 & 2.5477 \\ \hline Initial beam waist \(w_{0}\) (m) & 0.3261 & 0.4465 & 0.5819 & 0.7625 & 1.1538 & 0.3712 & 0.5067 \\ \hline \multicolumn{7}{c}{**Scenario 2: Waterford**} \\ \hline
**Parameter** & **Visible** & **NIR** & **SWIR** & **MWIR** & **LWIR** & **NIR\({}_{2}\)** & **SWIR\({}_{2}\)** \\ \hline Wavelength \(\lambda\) (\(\mu\)m) & 0.69739 & 1.2988 & 2.1407 & 3.7026 & 8.303 & 0.867 & 1.556 \\ \hline Secret key rate \(R\) (bits/use) & 0.0223 & 0.0271 & 0.029 & 0.0291 & 0.0222 & 0.0249 & 0.0282 \\ \hline Receiver diameter \(D_{r}\) (m) & 1.5865 & 2.3694 & 3.3405 & 4.936 & 5.273 & 1.8541 & 2.7856 \\ \hline Initial beam waist \(w_{0}\) (m) & 0.3261 & 0.4615 & 0.5819 & 0.7625 & 1.1538 & 0.3712 & 0.5067 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Wavelength, receiver diameter, and initial beam waist that maximise secret key rate in Tucson and Waterford using the parameters listed in the text. These are compared with commonly used wavelengths in the NIR\({}_{2}\) and SWIR\({}_{2}\) regions. Figure 4: Three-dimensional contour plots representing the secret key rate in **(a)** Tucson and **(b)** Waterford, for a range of receiver diameters \(D_{r}\) and initial beam waists \(w_{0}\). Wavelengths were selected to represent each window, where the \(D_{r},w_{0}\) combination (marked by the circle) optimizes the secret key rate. such a system affects wavelength selection. In Figure 5, the secret key rate is plotted for a range of coherence lengths, \(r_{0}\), using the optimal \(\lambda\), \(D_{r},w_{0}\) for (a) Tucson and (b) Waterford. The vertical lines at 5 cm and 50 cm are adopted from [58] and are a description of the wavefront resulting from uncompensated turbulence and post 200 Hz AO correction, respectively. There is a significant increase in performance gained by using AO correction, resulting in slightly higher secret key rates in Tucson than in Waterford. When the turbulence-distorted wavefront is compensated by AO, the SWIR and MWIR yield the best performance, followed by the NIR, visible and LWIR wavelengths, respectively. When the coherence length is small, indicating poorly compensated turbulence-induced wavefront aberrations, the MWIR performs significantly better than other wavelengths and could be used as an alternative to adaptive optics. We also compare the performance of the best NIR and SWIR wavelengths with wavelengths typically used in these regions. When the turbulent effects are compensated by a suitable AO system, there is not a large difference in secret key rates. However, when the turbulence-induced wavefront distortions are uncompensated, as indicated by the 5 cm coherence length, the secret key rate is much larger for the best wavelengths in each region. In Figure 6, the maximum secret key rates across the entire spectrum are plotted against receiver diameter for (a) Tucson and (b) Waterford.