## 2 Rate Equations Model
### Power Spectral Density and Laser Intrinsic Linewidth
Under this condition the intrinsic linewidth does not experience fluctuations either, and the amplitude of the threshold gain fluctuations is lower than in the single feedback case, indicating better stability across the three analyzed parameters with respect to the  
\[\text{F}_{0} \equiv\zeta_{\text{s}}^{2}+\text{K}_{\text{c}}^{2}\tau_{\text{e}} ^{-2}-2\text{K}_{\text{c}}a_{g}G_{0}S_{0} \tag{23a}\]2 Numerical solutions for \(C=0.5\) under conditions (8) and (11) for different \(\Delta\phi_{\mathrm{m}}\). Double-feedback case shown in blue, single feedback case shown in orange. Column (**A**) shows the lasing frequency shift results. Column (**B**) shows the threshold gain shift. Column (**C**) shows the intrinsic linewidth variations. single feedback case. For the case of \(\Delta\phi_{\rm m}=\pi\), the threshold gain shows no fluctuations as predicted by Equation (14), and while the lasing frequency and intrinsic linewidth fluctuations are no longer zero, they are less pronounced than in the single feedback case. Further increases in \(\Delta\phi_{\rm m}\) show an increase in the fluctuations across all functions, and for \(\phi_{\rm m}>6\) multi-valued solutions arise. Case 2: \(C=1\). This represents the limiting case between single and multivalued solutions in the single-feedback case. Results for various values of \(\Delta\phi_{\rm m}\) are shown in \[\text{F}_{0} \equiv\zeta_{\text{s}}^{2}+\text{K}_{\text{c}}^{2}\tau_{\text{e}} ^{-2}-2\text{K}_{\text{c}}a_{g}G_{0}S_{0} \tag{23a}\]3 As expected, the threshold gain is stable for \(\Delta\phi_{\rm m}=\pi\), and meeting condition (12) results in a stable lasing frequency and intrinsic linewidth. As \(\Delta\phi_{\rm m}\) deviates from these optimal points in either direction, the amplitude of fluctuations increase until reaching multi-valued solutions for \(\Delta\phi_{\rm m}<1.5\) and \(\Delta\phi_{\rm m}>3.5\). Comparing these results with the previous case shows that as feedback increases, the single valued solutions become more sensitive to the value of \(\Delta\phi_{\rm m}\). Case 3: \(C=1.3\). This represents the single-feedback case with multi-valued solutions. Results for various values of \(\Delta\phi_{\rm m}\) are shown in \[\text{F}_{0} \equiv\zeta_{\text{s}}^{2}+\text{K}_{\text{c}}^{2}\tau_{\text{e}} ^{-2}-2\text{K}_{\text{c}}a_{g}G_{0}S_{0} \tag{23a}\]4 In the single feedback case, multi-valued solutions are present in a given frequency range, and this span increases with increasing feedback strength. The multi-valued characteristics are evidenced experimentally with unstable regimes characterized by mode hopping and eventually coherence collapse for sufficiently high feedback. In contrast, the system proposed in this work shows that by tuning the value of \(\Delta\phi_{\rm m}\) to meet condition (12), even with increasing feedback it is possible to achieve stable performance regardless of frequency. In the case shown in  
\[\text{F}_{0} \equiv\zeta_{\text{s}}^{2}+\text{K}_{\text{c}}^{2}\tau_{\text{e}} ^{-2}-2\text{K}_{\text{c}}a_{g}G_{0}S_{0} \tag{23a}\]5 Numerical solutions for \(C=1\) under conditions (8) and (11) for different \(\Delta\phi_{\rm m}\). Double-feedback case shown in blue, single feedback case shown in orange. Column (**A**) shows the lasing frequency shift results. Column (**B**) shows the threshold gain shift. Column (**C**) shows the intrinsic linewidth variations. Figure 4 for \(C=1.3\), single valued solutions can be found for \(\Delta\phi_{\rm m}\in(1.5,3.3)\) which is equivalent to a phase variation of more than \(90^{\circ}\). Still, comparing with previous cases it is possible to see that as feedback increases, the single valued solutions tolerance with respect to \(\Delta\phi_{\rm m}\) decreases. Nevertheless, it is an improvement with respect to the single feedback case which shows no single value solutions across all frequencies for \(C>1\). Taking all cases into account, it can be seen that linewidth narrowing regions are present for all cases of \(\Delta\phi_{\rm m}\) analyzed.