## 4 Numerical Simulation Results and Discussions
### Laser Dynamic States under Double External Cavity  
#### 4.2.1 Region of Short Double OFB (\(L_{\text{ex2}}\) = 3.5 \(\times\) 10\({}^{-2}\) m)  
To investigate the impact of the double external cavity on the dynamic states and RIN of the laser, bifurcation diagrams are used. We will start by assuming that the distance between the first and second reflectors is \(5\times 10^{-3}\) m, making the cavity of the double OFB in the short cavity region (\(L_{\text{ex2}}\) = 3.5 \(\times\) 10\({}^{-2}\) m). Dynamic states of lasers with short external cavities are highly influenced by the number of round-trips in short external cavities [32]. In this paper, we consider the number of round-trips in our simulations. Figures 4a, 5a and 6a plot the bifurcation diagrams and the corresponding average Low-frequency RIN (\(f\) < 10 MHz) spectrum versus OFB intensity \(K_{\text{ex2}}\) with optical phase \(\psi_{2}=\omega(\tau_{1}+\tau_{2})\) when the single OFB strength is set to dynamic states corresponding to PO, PD, and chaos states at single OFB strength \(K_{\text{ex1}}\) = 0.02, 0.025, and 0.04, respectively. The shown figures are characterized by three features at each dynamic state: (1) the laser dynamic state in the weak OFB region begins with the dynamic state of a single OFB and (2) increases in the strength of the double OFB \(K_{ex2}\), leading the LD to a route-to-CW (more stable dynamics). Finally, (3) this route-to-CW is a reflector to the identical route-from-CW to a pertinent dynamic at this level of single OFB strength \(K_{ex1}\) in the bifurcation diagrams of Figure 2 under a single external cavity. When the optical phase \(\psi_{2}=0.0\), the LD dynamic states in the presence of double OFB are shown in Figures 4b, 5b and 6b. Figures show that reducing the optical phase \(\psi_{2}\) to 0.0 moves the laser dynamic state to unstable states and changes the route-to-CW when \(\psi_{2}=\omega(\tau_{1}+\tau_{2})\) to the route-to-chaos state at \(\psi_{2}=0.0\). The width of the CW dynamic is wider as \(\psi_{2}=\omega(\tau_{1}+\tau_{2})\) than that as \(\psi_{2}=0.0\), as shown in Figure 4. Figure 5 shows the same scenario but in the PD state, where, by increasing the double OFB strength \(K_{ex2}\), the dynamic states transfer from the PD state to tour and chaotic states, which is the route to the chaos state. Figure 6 shows the dynamic states of the double external cavity at (\(K_{ex1}=0.04\), chaos), the chaos state is directly the route to the CW dynamic state, and the CW state width is wider as \(\psi_{2}=0.0\) than that as \(\psi_{2}=\omega(\tau_{1}+\tau_{2})\). In the same Figures 4-6 and on the right \(Y\)-axis, the average RIN at Low-frequency (\(f<10\) MHz) versus the double cavity OFB intensity \(K_{ex2}\) corresponds to the BDs shown. Figure 4a shows the Low-frequency RIN in the PO regime; as shown in the figure, it is almost in the range of \(10^{-15}\) to \(10^{-14}\) and decreases to approach a solitary laser noise level (dashed line) with increasing the double OFB strength \(K_{ex2}\). As \(\psi_{2}=0.0\), Figure 4b displays RIN enhancement when dynamic states transfer from PO, PD, and chaos regimes till reaching the CW state. The level of the RIN is reduced near the level of solitary laser noise. Figures 5b and 6a,b show the highest level of the RIN, which occurs almost in the chaos regime. We notice that by reducing the optical phase \(\psi_{2}\) to 0.0, the RIN displays a lower level than that as of \(\psi_{2}=\omega(\tau_{1}+\tau_{2})\) in the CW, PO, and PD regimes. As shown in the figures, when the optical phases \(\psi_{1}=\omega\tau_{1}\), and \(\psi_{2}=\omega(\tau_{1}+\tau_{2})\), the dynamic of the LD is more stable than that as \(\psi_{1}=\omega\tau_{1}\), and \(\psi_{2}=0.0\), and the RIN suppresses to a level near the solitary laser noise, especially when the laser operates in the CW or PO regime.