# Vulnerability of Satellite Quantum Key Distribution to Disruption from Ground-Based Lasers
## 2 Methods
### Free-Space Link Loss  
The rate of excess noise photons, \(n_{s}\), detected by the QKD ground station after reflecting off the QKD satellite can be calculated from the following equation adapted from [10]  
\[n_{s}=\frac{P_{0}\lambda}{\hbar c}\frac{2}{\pi(\theta_{d}R_{1})^{2}}\exp[-2( \frac{\theta_{p}}{\theta_{d}})^{2}][\frac{1}{1+(\theta_{j}/\theta_{d})^{2}}][ \frac{\sigma_{sat}A_{\tau}}{4\pi R_{2}^{2}})\eta_{r}T_{a}T_{c}, \tag{1}\]  
where \(P_{0}\) is the launch power of the laser terminal, \(\lambda\) is the wavelength of the laser, \(hc\) is [PERSON]'s constant multiplied by the speed of light, \(\theta_{d}\) is the divergence of the laser beam, \(R_{1}\) is the distance from the laser terminal to the satellite, \(\theta_{p}\) is the pointing error of the laser terminal, \(\theta_{j}\) is the RMS jitter of the terminal pointing, \(\sigma_{sat}\) is the optical scattering cross section of the satellite, \(A_{\tau}\) is the area of the QKD ground station receiving telescope, \(R_{2}\) is the distance from the satellite to the ground station, \(\eta_{r}\) is the optical throughout efficiency of the ground station, \(T_{a}\) is the transmissivity of the atmosphere at the laser frequency, and \(T_{c}\) is the transmissivity of high cirrus cloud. In general, Equation (1) takes form: received photon rate equals the launched photon rate, times the launch optic gain, times the pointing error losses, times the receiving optic gain, times the propagation path losses. Figure 1: Simplified schematic of the setup to determine the optical scattering profile of a wrinkled gold foil target. Setup (**a**) simulates the scenario where light from the laser terminal scatters from the satellite and is detected by the QKD ground station, while setup (**b**) simulates the scenario where light from the laser terminal is detected by a satellite-based QKD receiver due to photons entering the receiving optic highly off-axis and scattering internally. SPD, single photon detector; SMF, single-mode fiber; MMF, multi-mode fiber. For the case where the mobile laser terminal is directly targeting a QKD receiver on board a satellite, Equation (1) becomes  
\[n_{s}=\frac{P_{0}\lambda}{hc}\frac{2}{\pi(\theta_{d}R_{1})^{2}}\exp[-2(\frac{ \theta_{p}}{\theta_{d}})^{2}][\frac{1}{1+(\theta_{j}/\theta_{d})}]\sigma_{rec}T_ {a}T_{c}, \tag{2}\]  
where \(\sigma_{rec}\) represents the cross section for scattering of laser photons into the receiving optic on board the satellite. All of the parameters in Equations (1) and (2) are well characterized thanks to conventional satellite laser ranging operations, except for \(\sigma_{sat}\) and \(\sigma_{rec}\). The experimental determination of the optical scattering profiles is outlined in the previous section. In this analysis, we will assume the laser terminal launches a single-mode Gaussian beam. This is a reasonable assumption because modern commercial off-the-shelf lasers are able to launch on the order of several kilowatts of power with beam quality factors near unity. The divergence of the laser beam, \(\theta_{d}\), and pointing jitter, \(\theta_{j}\), are heavily influenced by turbulence in the lower atmosphere. For the purposes of this analysis, we assume that the laser terminal has very good tracking and the loss due to pointing error, \(\theta_{p}\), is much smaller than those due to divergence and pointing jitter, so we set \(\theta_{p}=0\). The divergence is given by,  
\[\theta_{d}=\sqrt{(\frac{4\lambda}{\pi D_{L}})^{2}+(\frac{2\lambda}{\pi r_{0}} )^{2}}, \tag{3}\]  
where \(D_{L}\) is the diameter of the laser terminal aperture, and \(r_{0}\) is the Fried parameter (coherence scale) of the atmosphere, while the turbulence-induced pointing jitter is given by,  
\[\theta_{j}=\frac{2.92}{D_{L}^{1/3}}(\frac{1}{\sin\theta_{a}})\int_{h}C_{n}^{2} (h)\,dh, \tag{4}\]  
where \(\theta_{a}\) is the elevation angle of the satellite from the laser terminal, and \(C_{n}^{2}(h)\) is the turbulence profile of the atmosphere with height \(h\).