## 4 Numerical Simulation Results and Discussions
### Laser Dynamic States under Single External Cavity  
LD dynamic states induced via the single external cavity at two values of optical phase \(\psi_{1}\) = 0.0 (red color) and \(\omega\tau_{1}\) (black color) are shown in Figure 2. The dynamic states are calculated with BD of the photon number \(S(t)\), peak value, with the OFB strength \(K_{ex1}\). [PERSON] et al. investigated the influence of including the noise sources in the rate equations on the amplitude of pulsation and found that it is larger when counting the noise sources in the rate equations [6]. Since including the noise sources in the rate equations has a very little variation on the dynamic states induced via OFB, especially in the transition border to another state [6]. Langevin noise sources are omitted from the rate Equations (1)-(3) in the calculations of the BDs to get a clear vision of the dynamic states of the laser. The LD continues oscillating with the CW state for sufficiently small values of OFB strength \(K_{ex1}\), which is represented by no points in the figure. When \(K_{ex1}\) goes beyond the HB point (pointed by the first arrow) of \(K_{ex1}\) = 0.0129 and 0.0117 in the case of optical phase \(\psi_{1}\) = 0.0 and \(\omega\tau_{1}\), respectively, the LD dynamic state is identified with PO output for relaxation oscillations undamped. This PO is of the one-period kind with frequency increases with an increase in \(K_{ex1}\). The POs continue up to \(K_{ex1}\) = 0.0227 and 0.0231, with optical phases \(\psi_{1}\) = 0.0 and \(\omega\tau_{1}\), respectively. Beyond the PO regime, the bifurcation diagram shows two branches indicating the appearance of oscillations with period doubling (PD) (pointed by the second arrow). When \(K_{ex1}\) > 0.027 and 0.029 and optical phase \(\psi_{1}\) = 0.0 and \(\omega\tau_{1}\), respectively, the PD oscillation bifurcates more and shows many random points and tours, which represents a chaotic dynamic state (pointed by the third arrow). The shown PD route-to-chaos recognizes the LD with a short external cavity of \(f_{ex}\)/\(\tau_{r}\) > 1.0, as shown in Ref. [30]. By reducing the optical phase \(\psi_{1}\) to zero the strength of the LD is decreased, and the dynamic states are shifted toward increasing the OFB intensity \(K_{ex1}\) up to the PO region, which leads to more stability, as shown in Ref. [10]. However, after the PO region, the dynamical regimes such as PD and chaos start to appear at lower values of \(K_{ex1}\). As shown in the figure, reducing the optical phase \(\psi_{1}\) from \(\omega\tau_{1}\) to 0.0 has no effect on the route-to-chaos, which keeps the PD route-to-chaos. Figure 2: BD of the LD under single OFB at two values of optical phase \(\psi_{1}\) = 0.0 (red color) and \(\omega\tau_{1}\) (black color). In the regime when OFB \(K_{\text{ex1}}\) = 0.042-0.05, the laser starts to be attracted to a less chaotic or coherence collapse region, and this is due to entering the laser with a strong OFB, which induces stability in the dynamic states of the laser [5; 6]. Examples of the different dynamic states of the single external cavity laser shown in Figure 2 are characterized in Figure 3, when optical phase \(\psi_{1}\) = 0.0 and \(\psi_{1}\) = \(\omega\tau_{1}\); namely, the PO, PD, and chaos at OFB strength \(K_{\text{ex1}}\) = 0.02, 0.025, and 0.04, respectively. Figure 3 plots the photon numbers \(S(t)\)/\(S_{AV}\) time variations and the corresponding RIN power spectrum, which is calculated using the fast Fourier transform of \(S(t)\) using Equation (14). The intensity of \(S(t)\)/\(S_{\text{av}}\) is decreased, and the phase is shifted by decreasing the optical phase \(\psi_{1}\) from \(\omega\tau_{1}\) to zero in all laser states, as shown in Figure 3(a1-a3). The effect of varying optical phase \(\psi_{1}\) due to OFB on the laser fluctuations has been noticed in Refs. [9; 11; 31] and confirmed experimentally in Ref [11; 31]. The reduction in the intensity of \(S(t)\)/\(S_{\text{av}}\) is shown in Figure 3b as a little reduction of RIN.