# Bias Current of Semiconductor Laser: An Unsafe Key for Secure Chaos Communication
## 2 Theoretical Model  
Figure 1 shows the schematic diagram of the optical chaos communication system, with a pair of mutually-coupled, authorized external-cavity lasers (\(\mathrm{SL_{1}}\) and \(\mathrm{SL_{2}}\)). Two kinds of eavesdroppers were considered. Eavesdropper \(\mathrm{Eve_{A}}\) was disguised as an authorized transceiver which was bidirectionally coupled with the transmitter \(\mathrm{SL_{1}}\) (in this way, \(\mathrm{Eve_{A}}\) could not only eavesdrop the message but could also send false information to \(\mathrm{SL_{1}}\)). Eavesdropper \(\mathrm{Eve_{B}}\) simply tapped the transmitted signal from \(\mathrm{SL_{1}}\) and unidirectionally injected into the eavesdropping laser \(\mathrm{SL_{EB}}\). Note that the ECLs of the communication users and eavesdroppers had the same structure and the same semiconductor lasers. In addition, we simulated the spectra of \(\mathrm{SL_{1}}\) with and without considering \(\mathrm{SL_{2}}\). It was found that the relaxation oscillation frequency did not show any obvious change. For brevity, we omitted the equations of \(\mathrm{SL_{2}}\) in this manuscript. The ECLs were modeled by the following Lang-Kobayashi equations [19]. \[\frac{dE_{1}}{dt}=\frac{1+i\alpha}{2}\Bigg{[}\frac{g(N_{1}-N_{0})}{1+\varepsilon |E_{1}|^{2}}-\frac{1}{\tau_{P}}\Bigg{]}E_{1}+\frac{k_{1}}{\tau_{in}}E_{1}(t- \tau)e^{-i\omega_{1}\tau}+\frac{k_{A}}{\tau_{in}}E_{A}(t-\tau_{c})e^{-i(\omega _{A}\tau_{c}-\Delta\omega t)}, \tag{1}\]  
\[\frac{dE_{A,B}}{dt}=\frac{1+i\alpha}{2}\Bigg{[}\frac{g(N_{A,B}-N_{0})}{1+ \varepsilon|E_{A,B}|^{2}}-\frac{1}{\tau_{P}}\Bigg{]}E_{A,B}+\frac{k_{1}}{\tau _{in}}E_{A,B}(t-\tau)e^{-i\omega_{A,B}\tau}+\frac{k_{A,B}}{\tau_{in}}E_{1}(t- \tau_{c})e^{-i(\omega_{A,B}\tau-\Delta\omega t)}, \tag{2}\]  
\[\frac{dN_{1,A,B}}{dt}=\frac{I_{1,A,B}}{qV}-\frac{N_{1,A,B}}{\tau_{N}}-\frac{g( N_{1,A,B}-N_{0})}{1+\varepsilon|E_{1,A,B}|^{2}}\Big{|}E_{1,A,B}\Big{|}^{2}, \tag{3}\]  
Figure 1: Schematic diagram of two kinds of eavesdroppers: \(\mathrm{Eve_{A}}\) acted a disguiser that was bidirectionally coupled to the transmitter, and \(\mathrm{Eve_{B}}\) tapped and unidirectionally injected the transmitted light to its laser. \(\mathrm{SL}\)—semiconductor laser; OC—optical coupler; OI—optical isolator; EDFA—erbium-doped optical fiber amplifier; \(\mathrm{I}\)—bias current. \(\mathrm{SL_{1}}\) and \(\mathrm{SL_{2}}\) are lasers of legal users. where \(E(t)\) is the complex amplitude of optical field and \(N(t)\) represents the corresponding carrier density. The subscripts '1', 'A', and 'B', represent the legal user, \(\text{Eve}_{\text{A}}\), and \(\text{Eve}_{\text{B}}\), respectively. \(I\) is the bias current. \(k\) is the amplitude feedback strength. \(\tau=5\) ns is the feedback delay time. \(I_{\text{th}}=12\) mA is the laser threshold current. \(k_{\text{A,B}}=0.447\) is the amplitude coupling strength. Note that, we set \(k_{\text{A}}=0\) in the \(\text{Eve}_{\text{B}}\) simulation. \(\tau_{\text{c}}=19\) ns represents the coupling delay time. \(\Delta\omega=\omega_{1}-\omega_{\text{A,B}}=0\) denote the detuning angular frequency of the legal user's laser and the eavesdropper's laser. The other intrinsic parameters are listed as follows--transparency carrier density \(N_{0}=0.5\times 10^{5}\)\(\upmu\)m\({}^{-3}\), differential gain \(g=2.125\times 10^{-3}\)\(\upmu\)m\({}^{3}\)ns\({}^{-1}\), gain saturation parameter \(\varepsilon=1\times 10^{-5}\)\(\upmu\)m\({}^{3}\), carrier lifetime \(\tau_{\text{N}}=2.2\) ns, photon lifetime \(\tau_{\text{p}}=1.6\) ps, linewidth enhancement factor \(\alpha=6.0\), round-trip time in laser cavity \(\tau_{\text{in}}=7.3\) ps, active layer volume \(V=100\)\(\upmu\)m\({}^{3}\), and the elementary charge \(q=1.602\times 10^{-19}\) C. The fourth-order Runge-Kutta method, with a step of 2.5 ps was used to solve these equations in the simulation.