# Wireless-Channel Key Distribution Based on Laser Synchronization
## 4 Numerical Results  
We carried out numerical simulations to verify our experimental observations and investigate the impact of transmission distance. For simplicity, we exclusively consider the scenario in free space. The path loss of free space is given by Friis transmission formula [35]  
\[L_{f}=20\lg(4\pi d/\lambda)+L_{M}-G_{t}-G_{r}, \tag{8}\]  
where \(d\) is transmission distance, \(\lambda\) is the wavelength of the drive signal, \(L_{M}\) is the atmospheric attenuation, and \(G_{t}\) and \(G_{r}\) are the antenna gains of the transmitting and receiving antennas, respectively. The atmospheric attenuation is given by \(L_{M}=\gamma_{a}\cdot d\), where \(\gamma_{a}\) is the attenuation coefficient. Assuming perfect polarization matching between the transmitting and receiving antennas, the transfer function of free space is given by [36] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Measure** & **Symbol** & **Value** \\ \hline Cross-correlation & CC & 0.950 \\ Retained ratio by dual-threshold quantization & \(R_{dt}\) & 0.623 \\ Mutual information between [PERSON] and [PERSON] & \(I(Y_{At}Y_{E})\) & 0.489 \\ BER between [PERSON] and [PERSON] & \(R_{fail}\) & \(3.12\times 10^{-3}\) \\ Secret bit rate & \(R_{b}\) & 0.240 \\ Key generation rate & \(r\) & 150 Mbit/s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Statistical evaluation on the generated bits. The threshold coefficients for the dual-threshold quantization are set to \(C_{+}=0.400\) and \(C_{-}=0.408\). Figure 6: NIST statistical tests results for \(p\)-value. Using 1 Mbit data and a significant level of \(\alpha=0.01\), the \(p\)-value should be greater than 0.01 to pass the test. \[H_{f}(f,d)=10^{-L_{f}/20}\text{exp}(-j2\pi d/\lambda), \tag{9}\]  
The impulse response of the channel is defined as  
\[h_{f}(t,d)=F^{-1}\ \Big{\{}H_{f}(f,d)\Big{\}}, \tag{10}\]  
where \(F^{-1}[\cdot]\) is the inverse Fourier transform. The received signal \(S_{r}(t,d)\) can be obtained by using  
\[S_{r}(t,d)=S_{t}(t)\otimes h_{f}(t,d)+n(t), \tag{11}\]  
where \(S_{t}(t)\) is the drive signal, \(\otimes\) is the convolution operator, and \(n(t)\) is the noise from the receiving antenna output, given by  
\[n(t)=\sqrt{kT_{a}B}\cdot\xi(t), \tag{12}\]  
where \(\xi(t)\) is the normalized Gaussian noise, \(k=1.38\times 10^{-23}\) J/K is the Boltzmann's constant, \(T_{a}\) is the equivalent noise temperature of the receiving antenna, and \(B\) is the bandwidth of the receiving antenna. The drive light injected into the DFB laser can be represented by  
\[E_{inj}(t,d)=E_{0}(t)\text{exp}[j\pi g_{a}\cdot S_{r}(t,d)], \tag{13}\]  
where \(E_{0}(t)\) is the CW light input to the phase modulator, and \(g_{a}\) is the voltage gain of the amplifier. In our simulations, it's used the transmission line laser model (TLLM) in VPItransmissionMaker software (Version 9.1) to simulate the dynamic response of DFB lasers to light injection [37; 38]. The parameter values used in our simulations are as follows: \(G_{t}=10\) dBi, \(G_{r}=10\) dBi, \(\gamma_{a}=0.01\) dB/km, \(T_{a}=303.15\) K, \(B=2.8\) GHz. The center wavelength of the drive signal is 6.7 GHz, and the bandwidth is 2.8 GHz. The transmission power \(P_{t}\) of the drive signal is 13 dBm, and the modulation depth of the phase modulator is set to 0.7. The laser inner parameters are listed in Table 2. The laser current is set to \(I=2.4\ I_{th}\), where the current threshold \(I_{th}\) is 20 mA. With this configuration, the relaxation oscillation frequency of the laser is 5.5 GHz. The injection strength is set to \(-8.5\) dB, and the optical frequency detuning of the center frequencies of the DFB lasers and drive light is set to 1.6 GHz. We also use a raw sample rate of \(f_{s}=1\) GHz and a dual-threshold quantization method to generate bits. First, let us consider the case where the distance from the drive to Alice (\(d_{DA}\)) is equal to the distance from the drive to Bob (\(d_{DB}\)). The simulation results are shown in Figure 7. With the increase in transmission distance \(d\), the cross-correlation between the