## 3 Chaos Synchronization
Figure 2: Chaotic attractors in the phase space of the laser’s intensity output and the carrier density. (**a**) Transmitter laser and (**b**) Receiver laser. but has a definite functional dependence on a parameter such as injection rate [50]. Despite the intra-cavity propagation delay magnitude is in the order of pico-seconds, cannot be discounted as marginal addition of knowledge and hence in this work we have systematically carried out its effect on secure optical communication using diode lasers. To further characterize the intra-cavity propagation delay and synchronization properties, we have analyzed the robustness of synchronization and the associated \(\tau_{PD}\) against variations of the coupling parameters. The maximum correlation coefficient \(C_{m}\) and \(\tau_{PD}\) are computed in the plane of the injection parameters (frequency detuning (\(\Delta f\)) versus injection rate (\(\eta\))). Thus we obtained two maps for maximal correlation \(C_{m}\) variation and intra-cavity propagation delay \(\tau_{PD}\) which are displayed in Figure 4 and 5, respectively. In Figure 4, we observe that a stronger injection is needed to obtain synchronization for larger detuning and shows asymmetry with respect to the zero detuning. The maximal correlation of \(C_{m}\geq 0.93\) (dark red) is expected for higher injection. For lower injection with larger detuning, lower degrees of correlation (blue region) is observed. The shape of the synchronization region is similar to the one reported for different schemes using edge emitting semiconductor lasers [15]. Figure 4: Maximum cross-correlation coefficient \(C_{m}\) map in the injection parameters (frequency detuning \(\Delta f\), injection rate \(\eta\)) plane for keeping \(\kappa=9\) ns\({}^{-1}\), \(\tau_{ext}=10\) ns, and \(\tau_{f}=0\). Figure 3: Cross-correlation coefficient between transmitter and receiver laser for \(\kappa=9\) ns\({}^{-1}\), \(\eta=15\) ns\({}^{-1}\), \(\Delta f=0\), \(\tau_{ext}=10\) ns, and \(\tau_{f}=0\). The inset figure shows the expanded version of CC plot near zero in time scale. The intra-cavity propagation delay \(\tau_{PD}\) is found to be 32.8 ps. The correlation time-shift (intra-cavity propagation delay \(\tau_{PD}\)) required to obtain the maximum correlation value \(C_{m}\) in the plane of injection parameters is shown in Figure 5. For a lower injection near zero detuning, high positive magnitude of \(\tau_{PD}\) is observed (orange color region). The \(\tau_{PD}\) takes the negative value (yellow region) for positive detuning above +20 GHz with variations of injection rates. Whereas in the negative detuning region, \(\tau_{PD}\) is observed positive as well as negative with higher magnitudes as negative detuning increased and found to decrease for higher injection rates. In Figure 5, at low injection rates and moderately away from the zero detuning region, the intra-cavity propagation delay value found high in magnitude (in the order of \(10^{4}\) ps) since the quality of synchronization is worse in those regions. The asymmetry in the results with respect to the detuning is due to the influence of asymmetry of synchronization performance \(C_{m}\) (Figure 4), which is determined by the nonzero linewidth enhancement factor and external injection [51; 52]. In Figure 6, we show the trend of maximum correlation coefficient \(C_{m}\) (Figure 4) and the associated \(\tau_{PD}\) (Figure 5) for keeping \(\Delta f=0\) with \(\eta\) variation, which is indicated vertical dashed line in Figures 4 and 5. The \(C_{m}\) and \(\tau_{PD}\) for a range of injection rates are shown in Figure 6. The magnitude of \(\tau_{PD}\) (blue trace) decreases from 56.4 ps to 0.4 ps with increasing injection rate, and the respective \(C_{m}\) value (red trace) found to increase from 0.75 to 0.98. The decreasing nature of \(\tau_{PD}\) for higher injection indicates that the injected signal dominates the dynamics and has greatly controlled the receiver laser's independent emission [53].