## 2 Theory and System Model  
The system under investigation consists of an external cavity semiconductor laser serving as the transmitter laser (TL), and another solitary semiconductor laser playing a role of receiver laser (RL). The TL is rendered chaotic by the external cavity optical feedback, and the chaotic output of TL is uni-directionally injected to RL. For our numerical analysis both TL and RL are modeled by suitably adapting the Lang-Kobayashi (L-K) rate Equations [13]. L-K model comprises of the laser rate Equations for slowly varying complex electric field \(E(t)\) and carrier density \(N(t)\) for both the lasers. The rate Equations for TL and RL are;  
\[\dot{E}_{T}(t)=\frac{1+i\alpha}{2}\left[G_{T}(t)-\frac{1}{\tau_{p}}\right]E_{T }(t)+\kappa E_{T}(t-\tau_{ext})e^{-i\omega_{T}\tau_{ext}}+\sqrt{2\beta_{T}N_{T }(t)}\xi_{T}(t) \tag{1}\]  
\[\dot{N}_{T}(t)=\frac{I_{T}}{e}-\frac{N_{T}(t)}{\tau_{n}}-G_{T}(t)|E_{T}(t)|^{2} \tag{2}\]  
\[\dot{E}_{R}(t)=\frac{1+i\alpha}{2}\left[G_{R}(t)-\frac{1}{\tau_{p}}\right]E_{R }(t)+\eta E_{T}(t-\tau_{f})e^{-i(\omega_{T}\tau_{f}+\Delta\omega t)}+\sqrt{2 \beta_{R}N_{R}(t)}\xi_{R}(t) \tag{3}\]  
\[\dot{N}_{R}(t)=\frac{I_{R}}{e}-\frac{N_{R}(t)}{\tau_{n}}-G_{R}(t)|E_{R}(t)|^{2} \tag{4}\]In these equations, the indices \(T\) and \(R\) refer to the transmitter and receiver lasers respectively. \(J\) is the lasers' injection current, \(e=1.602\times 10^{-19}\) C is the electron charge. \(G(t)=g_{0}(N(t)-N_{th})/(1+\epsilon|E(t)|^{2})\) is the optical gain, where carrier number at threshold \(N_{th}=1.5\times 10^{8}\), linear gain coefficient \(g_{0}=12\times 10^{3}\), gain saturation coefficient \(\epsilon=5\times 10^{-7}\), linewidth enhancement factor \(\alpha=3.8\), carrier life time \(\tau_{n}=2\) ns, photon life time \(\tau_{p}=2\) ps. The lasers are operated at the wavelength \(\lambda=830\) nm. The frequency detuning between the laser is \(\Delta\omega=\omega_{T}-\omega_{R}=2\pi\Delta f\), and the external-cavity round trip time is \(\tau_{ext}=10\) ns. The Gaussian noise sources \(\xi\) is with zero mean and unity variance [14]. Spontaneous emission rate \(\beta=10^{-6}\) ns\({}^{-1}\). \(\kappa\) is the transmitter laser's feedback rate and \(\eta\) is the injection rate between TL and RL. Throughout our investigations, the time of flight (\(\tau_{f}=0\)) between TL and RL is kept zero. The time step is 0.2 ps in our simulations.  
The message signal is modulated (encoding message) with bias current of transmitter laser by CSK method [42]. In the first term of Equation (2), for encoding purpose the bias current of TL \(J_{T}\) can be modified to \(J_{T}=J_{m}(1+bM(t))\), here \(b\) is modulation depth, \(M(t)\) is pseudo random sequence, i.e., \(M(t)=1/2(-1/2)\) for a \"1\" (\"0\") bit and \(J_{m}\) is bias current of transmitter laser.  
The synchronization quality and the associated time delay between the transmitter and receiver laser are estimated by performing cross-correlation (CC) analysis between the outputs of transmitter and receiver laser. The CC in terms of time-shift (\(\Delta t\)) is given by  
\[C(\Delta t)=\frac{\langle[I_{T}(t-\Delta t)-\langle I_{T}\rangle][I_{R}(t)- \langle I_{R}\rangle]\rangle}{\sqrt{\langle[I_{T}(t-\Delta t)-\langle I_{T} \rangle]^{2}\rangle\langle[I_{R}(t)-\langle I_{R}\rangle]^{2}\rangle}} \tag{5}\]  
where the expectation denoted by \(\langle...\rangle\) is calculated via time average and \(I_{T,R}=\)\(|E_{T,R}|^{2}\) is the total intensity output of the laser. In CC analysis plot, the prominent peak value evaluates the synchronization quality, and location of the peak corresponds to the time delay between TL and RL output. For the ideal case, \(C\) would be 1, which relates to perfect synchronization condition.