## 4 Numerical Simulation Results and Discussions
### Laser Dynamic States under Double External Cavity
#### 4.2.2 Region of Long Double OFB (\(L_{\rm ex2}=10^{-1}\) m)  
The impacts of double OFB on the dynamic states and the corresponding average Low-frequency RIN when f < 10 MHz of laser in a long external cavity will be investigated. We set the distance between the first and second reflectors at \(7\times 10^{-2}\) m, making the double OFB cavity approach the region of the long external cavity (\(L_{\rm ex2}=10^{-1}\) m). Figures 8-10 plot the bifurcation diagrams and the corresponding average Low-frequency RIN (\(f<10\) MHz) when the single OFB induces the same dynamic states as Figures 4-6. The figures also indicate that the dynamics of the LD begin with the same dynamic of the LD with a single OFB in the region of weak OFB strength (at small \(K_{ex2}\))--in both cases, when \(\psi_{2}=0.0\) and \(\omega(\tau_{1}+\tau_{2})\). However, the increase in the double OFB intensity of \(K_{ex2}\) does not take the laser to a route-to-CW. It takes it to a route-to-chaos, as occurred in the case when \(\psi_{2}=0.0\) in Figures 4b, 5b and 6b. The chaos dynamics range of \(K_{ex2}\) is narrowest at \(K_{ex1}=2\times 10^{-2}\) and \(\psi_{2}=0.0\), which agree with PO subject to single OFB, and widest at \(K_{ex1}=4\times 10^{-2}\), and \(\psi_{2}=0.0\) that agree with chaos dynamic state under single OFB. On the contrary, Figure 10a shows similar behavior as shown in Figure 6a: the rise in the double OFB intensity \(K_{ex2}\) takes the LD to a route-to-CW state when \(\psi_{2}=\omega(\tau_{1}+\tau_{2})\) and attracts the laser to a route-to-chaos state at \(\psi_{2}=0.0\), which is cleared in Figure 10b. The figures show that these routes to the chaos state do not mirror the congruent path from CW to the relevant dynamic at single OFB strength \(K_{ex1}\) in the bifurcation diagrams of Figure 2 under single OFB. The figures show that variations of PO frequency \(f_{PO}\) with \(K_{ex2}\) increase with increasing double OFB strength \(K_{ex2}\) through each dynamic state (PO, PD, and chaos) when \(\psi_{2}=0.0\) and \(\omega(\tau_{1}+\tau_{2})\), as mentioned in Ref. [12]. In Figures 8-10 and on the right \(Y\)-axis, the average Low-frequency RIN (\(f<10\) MHz) variation with the double cavity OFB intensity \(K_{ex2}\) corresponds to BDs shown in the same figures. The RIN, as \(\psi_{2}=0.0\), enhances to higher noise levels when dynamic states transfer from PO and PD to the chaos regime and never approach the solitary laser noise level (dashed line). Figures show that when the optical phase \(\psi_{2}=0.0\), the RIN displays a lower level than that as \(\psi_{2}=\omega(\tau_{1}+\tau_{2})\). As shown in the figures, when the optical phases \(\psi_{1}=\omega\tau_{1}\), and \(\psi_{2}=\omega(\tau_{1}+\tau_{2})\), the dynamic states of the laser are more stable than when \(\psi_{1}=\omega\tau_{1}\), and \(\psi_{2}=0.0\), and the RIN decreases to levels near quantum noise. Figures 4-6 (short external cavity regime) in comparison with Figures 8-10 (long external cavity regime) show that the double external cavity leads to setting the irregular states produced by the single external cavity and keeping the stable CW. This reduces the RIN to lower levels near the solitary laser noise level. This leads to improvements in the properties of the LD, especially in applications such as optical communication systems and others. The simulated results show that the RIN, which corresponds to unstable states of the LD, may be stabilized by operating the laser in a short external cavity regime and increasing the double OFB intensity \(K_{ex2}\) and optical phase \(\psi_{2}\). This overall helps in designing lasers with high dynamics and low noise levels. The impacts of the optical phase and double OFB \(K_{ex2}\) on the RIN, with an adapting laser design to adjust the values of \(K_{ex2}\) and \(\psi_{2}\), are introduced for the first time. This can be useful in increasing/decreasing stability/RIN of lasers, respectively.