# Airborne Quantum Key Distribution Performance Analysis under Supersonic Boundary Layer
## 3 Evaluation Result and Discussion
(**b**) The drifted offset of the beam, which reaches to the ground station. The speed of airborne is 0.7 Ma. Figure 5: (**a**) The deflection angle of transmitted photons. (**b**) The drifted offset of the beam, which via the supersonic boundary layer and reaches the ground station. Figure 6: (**a**) The photons distribution probability when the photons via the supersonic BL or not, which are denoted by a red triangle and blue point respectively. (**b**) The photons distribution probability when the photons via the supersonic BL. The azimuth angle is 10\({}^{\circ}\). Then, the receiving photons distribution probability ratio \(E_{boundary}\) in Equation (12) over the azimuth angle can be calculated, as shown in Figure 7, when the photon via the supersonic BL. The curve random jitter is more obvious, and \(E_{boundary}>1\) at some azimuth angle which shows that photons bunching has happened. Moreover, when the refractive index gradient is less than zero, the refraction angle is larger than incident angle. In this case, the photon spot dispersion caused by the small incident angle is more serious due to the limited number of grids. As shown in Figure 4, The small incidence angle will cause the beam to separate to the two sides, intensify the diffusion of light spots, and reduce the receiving photons distribution probability ratio \(E_{boundary}\). The case of the small incident angle with the negative refractive index gradient is indicated by the black box in Figure 3b. So, it is possible that significant dispersion phenomenon occurs around the zero azimuthal angle. When Figure 6 is converted into a top view, as shown in Figure 8, the distribution of photons can be well displayed with and without the effect of supersonic BL. By comparing the two cases where the azimuth is \(10^{\circ}\) and \(20^{\circ}\), the probability of photon appearing in the range of the receiving telescope after passing through the boundary layer is different. When the azimuth is \(10^{\circ}\), the photons passes through the boundary layer and are partially refracted out of the scope of the telescope. When the azimuth is \(20^{\circ}\), some photons that are not in the scope of the telescope are refracted into the telescope after passing through the boundary layer. By contrast, the case of \(E_{boundary}>1\) can be explained. Figure 8: **(a)** The receiving photons distribution when the azimuth angle is \(10^{\circ}\) with and without the effect of supersonic BL, which are denoted by a red triangle and blue point respectively. **(b)** The receiving photons distribution when the azimuth angle is \(20^{\circ}\) with and without the effect of supersonic BL. The green dotted line indicates the aperture of the receiving telescope. Figure 7: The receiving photons distribution probability ratio over the azimuth angle, when the photon via the supersonic BL. And the photons distribution probability with several different azimuth angles can be calculated, when the photons via the supersonic BL or not. As shown in Figure 9, different from the subsonic boundary layer, the supersonic boundary layer can diffuse the probability of photons distribution. It can cause the photon of receiving terminal to be so irregular that the transmission efficiency and secure key rate decrease. By taking the total transmission efficiency into the formula of the key rate, the QKD performance with the BL effects is evaluated and the result is shown in Table 1.0 When the photon via the BL, the key rate curve drops obviously. Compared with the subsonic surface layer, the effect of the supersonic surface layer is more obvious. The curve varies even more dramatically, and the effect of supersonic BL at individual points is very obvious. Therefore, the estimated average secure key rate is around 943 bit/s and 551 bit/s respectively when the photon via the subsonic and supersonic surface layer. If there's no boundary layer surrounding the aircraft, the estimated average secure key rate would be around 1468 bit/s.