# Free-Space QKD with Modulating Retroreflectors Based on the B92 Protocol
## 3 Results  
We now analyze the performance of our MRR-QKD system proposed in Section 2 and compare it with RM2018. With two decoy state technology, the secure key rate (SKR) of the QKD system is [28]. \[R=q\{-fQ_{\mu}H_{2}(E_{\mu})+Q_{1}^{L,\
u_{1},\
u_{2}}[1-H_{2}(e_{1}^{U,\
u_{1},\
u_{2}})]\} \tag{2}\]  
where \(q\) is the protocol efficiency, \(f\) is the bi-directional error correction efficiency, \(Q_{\mu}\) is the gain of the signal state, \(E_{\mu}\) is the overall QBER, \(Q_{1}^{L,\
u_{1},\
u_{2}}\) and \(e_{1}^{U,\
u_{1},\
u_{2}}\) are the lower bound of the gain and upper bound of the error rate for single photon state, respectively. \(H_{2}(\cdot)\) is the binary Shannon information function. In our system, the protocol efficiency of the B92 protocol is \(q=1/4\), whereas BB84 is used in RM2018, \(q=1/2\). The intensity of the signal state is \(\mu\). The maximum intensity range of the decoy state is \(\
u\in[\mu/r,\mu)\). In the asymptotic case, the smaller the value of \(\
u_{1}+\
u_{2}\), the higher the SKR [28]. Therefore, we chosen the signal state intensity \(\mu=0.48\), decoy state intensity \(\
u_{1}=\mu/r\) and vacuum state \(\
u_{2}=0\). Other simulation parameters are listed in Table 1.  
In Section 2, we showed that the extinction ratio of the MQW would cause leakage noise while the MQW is \"off\" in RM2018. This leakage noise would make the QBER increase. The smaller the extinction ratio, the higher the leakage noise. In our scheme, the extinction ratio does not generate leakage noise but only restricts the intensity of the decoy states. We \begin{table}
\begin{tabular}{c c c} \hline
**Symbol** & **Description** & **Value** \\ \hline \(\lambda\) & wavelength & 1550 nm \\ \(e_{detect}\) & misalignment-error probability 1  & 3.3\% \\ \(\
u_{0}\) & background rate & \(1.7\times 10^{-6}\) \\ \(\eta_{d}\) & detection efficiency 2  & 1 \\ \(f\) & error correction efficiency & 1.22 \\ \hline \end{tabular}
\end{table} 
Table 1: List of Simulation Parameters [29]. demonstrated the extinction ratio effect in our system, the simulation results are shown in Figure 2.  
The three curves with extinction ratios of 7.5, 15, and 22.5 dB overlap almost completely. This indicates that the extinction ratio does not have a significant impact on the SKR of our scheme. This is because the SKR is not sensitive to the intensity of the decoy state. Additionally, in contrast to RM2018, which stacks two MQWs to increase the extinction ratio, our system requires only one MQW in each path. Under different extinction ratios, we compare the SKR of our system and the RM2018. The results are shown in Figure 3. Considering the additional QBER resulting from noise leakage for the finite extinction ratio, the probability of a photon clicks the erroneous detector should be corrected to \(e^{\prime}_{detect}=\frac{r-1}{r+3}e_{detect}+\frac{2}{r+3}\)[23]. \(e_{detect}\) is the probability that a photon clicks the erroneous detector due to the optical misalignment and \(r\) indicates the extinction ratio. Figure 3: The relationship between the SKR and the transmission loss of our system and RM2018 under different extinction ratios. The dotted line represents the RM2018, and the solid line represents our system. The blue dotted line vanishes because the SKR is lower than zero. Figure 2: The relationship between the Secure key rate(SKR) and the transmission loss under different extinction ratios. As shown in Figure 3, the SKR of RM2018 is very sensitive to the extinction ratio. The SKR decreases as the extinction ratio reduces. When the extinction ratio was less than 15 dB, the SKR of RM2018 decreased below zero (vanishing blue dotted line). The performance of our system with only a 7.5 dB extinction ratio could reach at least 70% of RM2018 with a 22.5 dB extinction ratio, and even higher for the maximum transmission distance. This means that only one MQW is needed for each path in our system, whereas RM2018 requires three MQWs.