## 3 Simulation Results and Discussions
### RIN at Low Frequency with LEF and SEF  
Figure 3, which corresponds to the BDs shown in Figure 1, shows the variation in the average value of the low-frequency RIN (\(f\leq 10\) MHz) with OFB intensity \(K_{ex}\) when the LEF \(\alpha=1,3\), and 5 and the SEF C = \(10^{-4}\), \(10^{-5}\), and \(10^{-6}\). One can see that the average RIN almost coincides with the solitary laser noise level (the LD without OFB) in the weak OFB regime. Through the route to chaos, the RIN displays higher levels and reaches maximum values when the laser state becomes chaotic in medium and some strong OFB regimes. At the Hopf bifurcation point, the average value of the RIN attains minimum levels, which happens to be lower than the solitary RIN level. Under strong OFB, the average value of the RIN exhibits lower levels than the corresponding solitary laser RIN level. As shown in Figure 3(a1), the average value of the RIN level almost coincides with the solitary laser noise level under the CW state and when the LEF \(\alpha=1\). By increasing the LEF, the RIN becomes higher than that of the LD without OFB depending on the operating state of the LD (CW, PO, or chaos) and OFB intensity \(K_{ex}\), which is shown in Figure 3(a1-a3,b1-b3,c1-c3). Figure 3(a1-c1,a2-c2,a3-c3) exhibits the impact of the SEF on the average value of the low-frequency RIN. As shown in Figure 3(a1), the average value of the RIN level almost coincides with the solitary laser noise level under the CW state and when the SEF C = \(10^{-4}\). By decreasing the SEF, the RIN attains higher values compared with the solitary laser noise level, which depends on the operating state of the LD (CW, PO, or chaos) and OFB intensity \(K_{ex}\), as shown in Figure 3(a1-c1,a2-c2,a3-c3). Figures 1 and 3 show clearly the significant roles played by the SEF in the behavior of the dynamic state. As shown in Figure 3(a1,b1), the dynamic states have been changed dramatically, while in Figure 3(b1,c1), such a change is not noticeable when decreasing the SEF, which specifies the ratio of the spontaneous field entered to the lasing mode and affects the width of the laser linewidth. The SEF and stimulated emission coefficient are identical in principle. Decreasing the SEF and increasing the threshold current level gives a high noise level and a wide linewidth when operating a laser at low power levels.  
As shown in Figure 3, decreasing/increasing the LEF/SEF causes a reduction in the value of the average value of the RIN at low frequency, and this leads to an improvement in the properties of the LD with short EC, especially in applications such as optical communication systems and others. The simulated results in Figure 3 show that the average value of the RIN, which corresponds to the operation state instability of the LD with short EC shown in Figure 1, can be improved by decreasing/increasing the LEF/SEF, enhancing the OFB, which helps for drawing LDs with high dynamics and a low noise level. The impacts of the LEF and SEF with OFB on the average value of the RIN with selecting the LD design to adjust the values of the LEF and SEF are introduced for the first time, and this could be effective at increasing/decreasing the stability/RIN of LDs with short EC.  
Figure 2: OFB \(K_{ex}\) variation with (**a**) LEF and (**b**) SEF, at HB point and \(I/I_{th0}=1.5\).