## 2 Rate Equations Model
### Power Spectral Density and Laser Intrinsic Linewidth  
The next step is to calculate the FN PSD, and from it the laser intrinsic linewidth. By solving the system in Equation (21) it is possible to find an expression for the PSD [45]:  
\[S_{f}^{(1)}(\Omega)=\frac{\Omega^{2}}{2\pi^{2}}\langle|\phi_{\text{0p}}(\Omega)|^ {2}\rangle, \tag{22}\]  
which, using the following definitions:  
\[\text{F}_{0} \equiv\zeta_{\text{s}}^{2}+\text{K}_{\text{c}}^{2}\tau_{\text{e}} ^{-2}-2\text{K}_{\text{c}}a_{g}G_{0}S_{0} \tag{23a}\] \[\text{F}_{1} \equiv\text{K}_{\text{s}}^{2}+\text{K}_{\text{c}}^{2}\] (23b) \[\Lambda_{4} \equiv 4\text{F}_{1}D_{\phi\phi}\] (23c) \[\text{F}_{2} \equiv\text{K}_{\text{c}}\alpha_{\text{H}}+\text{K}_{\text{s}}\] (23d) \[\text{F}_{3} \equiv\text{K}_{\text{c}}-\alpha_{\text{H}}\text{K}_{\text{s}}\] (23e) \[\Delta_{4} \equiv\text{F}_{1}^{2}, \tag{23f}\]  
and:  
\[\Lambda_{2} \equiv a_{g}^{2}\text{F}_{2}^{2}D_{NN}+4D_{\phi\phi}\Big{(}\text{ K}_{\text{s}}^{2}\tau_{\text{e}}^{-2}+\text{F}_{0}\Big{)}-2\frac{\text{K}_{ \text{s}}}{S_{0}}D_{SS}a_{g}\Big{(}\tau_{\text{e}}^{-1}\text{F}_{2}-\zeta_{ \text{s}}\alpha_{\text{H}}-\alpha_{\text{H}}G_{0}\Big{)} \tag{23g}\] \[\Lambda_{0} \equiv\alpha_{\text{H}}^{2}a_{g}^{2}\Big{[}D_{SS}\Big{(}\zeta_{ \text{s}}^{2}+G_{0}^{2}+2\zeta_{\text{s}}G_{0}\Big{)}+\zeta_{\text{s}}N\tau_{ \text{sp}}^{-1}\Big{]}+\Big{(}\tau_{\text{e}}^{-1}\zeta_{\text{s}}+a_{g}G_{0}S_ {0}\Big{)}^{2}4D_{\phi\phi}\] (23h) \[\Delta_{2} \equiv\text{K}_{\text{c}}^{2}\zeta_{\text{s}}^{2}+\tau_{\text{e} }^{-2}\text{F}_{1}^{2}-2\text{F}_{1}a_{g}G_{0}S_{0}\text{F}_{3}\] (23i) \[\text{F}_{0} \equiv\zeta_{\text{s}}^{2}+\text{K}_{\text{c}}^{2}\tau_{\text{e}} ^{-2}-2\text{K}_{\text{c}}a_{g}G_{0}S_{0} \tag{23a}\]0  
can be written as:  
\[4\pi^{2}S_{f}^{(1)}\overset{\eqref{eq:f_f_f_f_f_f_f_f_f_f_f_f_f_f_f_f_f_f_f_f_f _f_f_f_f_f_f_f_f_f_f_f_f_f_f_f_f_f_f_f  
where \(C=1\) is the critical value that separates both behaviours. Numerical solutions of the proposed system under conditions (8) and (11) are thus compared to the single feedback case for three cases:  
Case 1: \(C=0.5\). This represents the single-feedback case with a single solution, and results for various values of \(\Delta\phi_{\mathrm{m}}\) are shown in \[\text{F}_{0} \equiv\zeta_{\text{s}}^{2}+\text{K}_{\text{c}}^{2}\tau_{\text{e}} ^{-2}-2\text{K}_{\text{c}}a_{g}G_{0}S_{0} \tag{23a}\]1 Column A, B and C show the numerical solutions for the variations in lasing frequency, threshold gain and intrinsic linewidth, given by Equations (5) and (26), respectively. By plotting the logarithm of the latter, negative values correspond to linewidth narrowing. The blue and orange plots represent the double-feedback and the single-feedback case, respectively, and these labels are maintained throughout the document. The upper row shows the case where \(\Delta\phi_{\mathrm{m}}=0\), i.e., additional phase shifts at, for example, the external mirrors (\(\phi_{\mathrm{m}_{j}}\)) are kept at zero. Linewidth narrowing for a wide range of frequencies can be observed, evidenced, as mentioned, by \(ln(F)<1\). As an example, when \(f_{\mathrm{ref}}t_{2}=0.75\), a 74% and a 42% reduction in linewidth is seen with respect to the free running laser and the single feedback case, respectively. Additionally, the signal is singled valued in the full domain, yet it is close to the critical point where multi-valued solutions arise. As \(\Delta\phi_{\mathrm{m}}\) increases, the amplitude of the lasing frequency shift is reduced until becoming zero for all values when condition (12) is met, as expected from previous analysis.