## 3 Chaos Synchronization  
In this section, we establish the chaos synchronization between the transmitter and receiver laser and present the influence of intra-cavity propagation delay \(\tau_{PD}\) in the chaos synchronization. The transmitter and receiver laser rate equations, Equations (1)-(4), are numerically solved using the Runge-Kutta algorithm. We consider the chaos shift keying (CSK) scheme for encoding of message signal [37]. The transmitter laser's pump magnitude is modulated with a message and transmitted to the receiver laser. A digital message signal with 1.2 Gbit/s bit rate is encoded at the transmitter laser with modulation depth \(b=0.2\). In the simulation, we keep the parameter values as, \(\kappa=9\) ns\({}^{-1}\), \(\tau_{ext}=10\) ns, \(J_{m}=1.10I_{th}\), \(J_{R}=1.08I_{th}\), \(\eta=15\) ns\({}^{-1}\), \(\Delta f=0\), and \(\tau_{f}=0\). Although a higher bias currents of laser provides less distortion in the output intensities by the relaxation oscillation as well as provide higher modulation bandwidth, the influence of intra-cavity propagation delay \(\tau_{PD}\) is found to be more significant in the chaos synchronization of semiconductor lasers when the lasers are operated at low bias currents. Hence, we kept low bias currents for the TL and RL in our investigations. The time evolution output of the transmitter (black trace) and receiver laser (red trace) are obtained and shown in Figure 1. Here, the TL's time evolution output (black trace) contains both chaotic component arising due to the optical feedback and the message components due to the modulation of bias current. The difference between the laser's intensity is shown by the grey trace. The red trace (receiver laser) and grey trace (TL-RL) in Figure 1 are shifted vertically for clarity. Figure 2a shows the chaotic attractor of the transmitter laser in the phase space of the intensity and the carrier density. And, Figure 2b corresponds to the chaotic attractor of the receiver laser. The attractors in the transmitter and receiver lasers show the moderately different orbit since generalized synchronization and also due to the encoded message signal at the transmitter laser. The synchronization quality and the time delay between the coupled lasers are estimated by performing cross-correlation (CC) analysis between TL and RL. From the CC analysis, the prominent peak value (\(Cm\)) evaluates the synchronization quality, and location of the peak in time corresponds to the time delay between TL and RL output. A large value of \(C_{m}\) indicates that good synchronization has been achieved. The obtained normalized correlation coefficients is illustrated in Figure 3. The inset figure shows the expanded cross-correlation plot near zero and it can be seen clearly that, although the time of flight (\(\tau_{f}\)) between transmitter and receiver laser is set to zero, the prominent correlation peak is not occurring exactly at zero. This peak shift in time, named as intra-cavity propagation delay (\(\tau_{PD}\)), is eventually due to the propagation of transmitter laser's output within the receiver's cavity [48], and it is found to be 32.8 ps in this case. The corresponding maximum correlation (\(C_{m}\)) value is found to be 0.85. Essentially, this intra-cavity propagation delay time correction should be incorporated in order to obtain a better synchronization between transmitter and receiver laser. Experimental observation of this additional-time delay has been recently demonstrated in chaos synchronization of coupled semiconductor lasers. It is shown that such additional time delay is not arbitrary in character  
Figure 1: Time traces of the transmitter and receiver lasers: The black (red) traces correspond to TL (RL) output (chaotic intensity including message signal) respectively for \(J_{m}=1.10I_{th}\), \(J_{R}=1.08I_{th}\), \(\kappa=9\) ns\({}^{-1}\), \(\eta=15\) ns\({}^{-1}\), \(\Delta f=0\), \(\tau_{ext}=10\) ns, and \(\tau_{f}=0\). The grey trace corresponds to the difference between the intensities of the transmitter and receiver lasers.