# Performance of Ship-Based QKD Under the Influence of Sea-Surface Atmospheric Turbulence
## 2 The Influence of Sea-Surface Turbulence on Laser Beams and Aiming Error  
The formation of sea-surface atmospheric turbulence is a complex process involving the interaction of multiple dynamic and thermodynamic conditions. Unlike inland atmospheric turbulence, oceanic atmospheric turbulence exhibits a clear pattern of daily variation, reaching extreme values in the late afternoon, and turbulence intensity tends to decrease with increasing altitude [15, 16, 17]. The atmospheric refractive index structure constant is usually used to characterize the magnitude of turbulence intensity, with larger values representing better atmospheric conditions. It varies from \(10^{-12}\) (strong turbulence) to \(10^{-17}\) (weak turbulence), with a mean value of \(10^{-15}\). Its relationship with altitude is shown in the following equation:  
\[C_{n}^{2}(h)=0.00594(W/27)^{2}\Big{(}10^{-5}h\Big{)}^{10}\exp(-h/1000)+2.7\times 1 0^{-16}\exp(-h/1500)+A\exp(-h/100) \tag{1}\]  
where \(h\) is the height in meters, \(A\) is the ground structure constant \(C_{n}^{2}(0)\), and \(W\) is the root mean square wind speed measured in meters per second. The change in oceanic atmospheric turbulence with seasons is not obvious, and the turbulence intensity in winter is slightly larger than that in summer; however, the spatial fluctuation is not significant. In general, the refractive index structure constants of offshore atmospheric turbulence at the surface range from \(10^{-17}\) to \(10^{-13}\)[15]. When a laser beam passes through a turbulent atmosphere, atmospheric turbulence causes the refractive index of light to change, resulting in beam intensity fluctuations, beam phases (up and down), beam drift and spread, and image point jitters. If the light intensity is measured in a turbulent atmosphere at some distance from the light source, it is found to fluctuate randomly around the mean value over time. This is due to the fact that the received laser is disturbed by atmospheric turbulence; alternating enhanced and weakened interference fringes appear in the transmission path, resulting in fluctuations in the detected radiation intensity. The scintillation index can describe the magnitude of this variation, and the model of atmospheric scintillation in the downlink can be expressed as:  
\[\sigma_{l}^{2}=2.25\mathrm{k}\frac{7}{6}\sec(\theta)\frac{11}{6}\int_{h_{0}}^{ H}C_{n}^{2}(h)(h-h_{0})^{\frac{5}{6}}dh \tag{2}\]  
A more precise Gaussian beam expression is needed for the uplink as follows:  
\[\sigma_{l}^{2}=2.25\mathrm{k}\frac{7}{6}\left(H-h_{0}\right)\frac{5}{6}\sec( \theta)\frac{11}{6}\int_{h_{0}}^{H}C_{n}^{2}(h)\bigg{(}1-\frac{h-h_{0}}{H-h_{0 }}\bigg{)}\frac{5}{6}\left(\frac{h-h_{0}}{H-h_{0}}\right)\frac{5}{6}dh \tag{3}\]  
where \(K=\frac{2\pi}{\lambda}\) is the wave numbers, and \(\theta\) is the zenith angle. The main difference in the scintillation model between the uplink and downlink is that the uplink needs to take into account variation in the atmospheric refractive structure constant; in other words, atmospheric scintillation has a greater impact on the uplink, as shown in Figure 1.  
The variance of aiming error angles from light intensity flicker can be expressed as:  
\[\sigma_{pe}^{2}=0.54(H-h_{0})^{2}\sec^{2}(\theta)\bigg{(}\frac{\lambda}{2 \omega_{0}}\bigg{)}^{2}\bigg{(}\frac{2\omega_{0}}{r_{0}}\bigg{)}^{\frac{5}{3}} \Bigg{[}1-\bigg{(}\frac{C_{r}^{2}\omega_{0}/r_{0}^{2}}{1+C_{r}^{2}\omega_{0}/r _{0}^{2}}\bigg{)}^{\frac{1}{6}}\Bigg{]} \tag{4}\]  
where \(C_{r}\) is the scintillation constant; \(\lambda\) and \(\omega_{0}\) are the wavelength and girdle radius of the laser beam, respectively; and \(r_{0}\) is the atmospheric coherence length.