## 2 Rate Equations Model
### Rate Equations  
In order to obtain the frequency noise (FN) power spectral density (PSD), and from it the laser intrinsic linewidth, the laser rate equations for the intensity and phase, as well as one for the carrier number, need to be studied. The former two can be extracted from the dynamic equations for the field inside the laser cavity, following the Lang-Kobayashi [39] approach. Its full derivation is shown in Appendix C. Furthermore, Langevin noise terms are included to account for shot noise fluctuations. The following definitions are useful to simplify notation:  
\[\mathcal{A}(t)=\sqrt{S(t)}e^{-i\phi(t)} \tag{15a}\] \[\mathcal{S}_{j}^{\pm}\equiv\kappa_{j}\sqrt{S(t\pm t_{j})}\] (15b) \[\phi_{t_{j}}^{\pm}\equiv\phi(t\pm t_{j})\] (15c) \[\Delta\Phi_{1}^{+}\stackrel{{ (\text{15c})}}{{=}}\phi(t)-\phi_{t_{1}}^{+}-\phi_{1}\] (15d) \[\Delta\Phi_{2}^{-}\stackrel{{(\text{15c})}}{{=}}\phi(t)-\phi_{t_{ 2}}^{-}+\phi_{2}\] (15e) \[\Delta G\equiv G_{\text{FB}}-\tau_{\text{ph}}^{-1}, \tag{15f}\]  
with \(j=1,2\) relating to the EOF components from the left and right, respectively, and \(\tau_{\text{ph}}\) is the photon decay time, which accounts for cavity and mirror losses. The field amplitude \(\mathcal{A}\) is assumed to be slowly varying, where \(S\) is the photon number inside the laser cavity and \(\phi\) is the phase of the field. With these definitions, the rate equations of the system can be written as:  
\[\dot{S}\stackrel{{(\text{A27})}}{{=}}S\Delta G+2 \mathcal{S}_{2}^{-}\sqrt{S}\cos(\Delta\Phi_{2}^{-})+2\mathcal{S}_{1}^{+}\sqrt {S}\cos(\Delta\Phi_{1}^{+})+R_{\text{sp}}+F_{S} \tag{16a}\] \[\dot{\phi}\stackrel{{(\text{A28})}}{{=}}\frac{\alpha_{\text{H}} \Delta G}{2}-\delta\omega-\frac{\mathcal{S}_{2}^{-}}{\sqrt{S}}\sin(\Delta\Phi _{2}^{-})-\frac{\mathcal{S}_{1}^{+}}{\sqrt{S}}\sin(\Delta\Phi_{1}^{+})+F_{\phi}\] (16b) \[\ddot{N}\stackrel{{[\text{3}]}}{{=}}I-GS(t)-N\tau_{ \text{sp}}^{-1}+F_{N}, \tag{16c}\]  
where \(I\) is the effective rate of injected current (in electrons), \(\tau_{\text{sp}}\) is the carrier lifetime, and \(R_{\text{sp}}\) is the spontaneous recombination rate. This system is compatible with those presented in refs. [28; 31], by accounting for the sign change in the delayed feedback term corresponding to the left external mirror, due to it having the opposite propagation direction.  
The steady state solution of Equation (16) can be seen in Equation (A33). The Langevin noise sources \(F_{S}(t)\), \(F_{\phi}(t)\) and \(F_{N}(t)\) satisfy [44]:  
\[\langle F_{i}(t)\rangle =0 \tag{17a}\] \[\langle F_{i}(t_{1})F_{j}(t_{2})\rangle =2D_{ij}\delta(t_{1}-t_{2})\quad\text{with}\;\;i,j=S,\,\phi\, \text{or}\,N, \tag{17b}\]  
where:  
\[D_{\text{SS}}=R_{\text{sp}}S\quad;\quad D_{\phi\phi}=\frac{R_{\text{sp}}}{4S} \quad;\quad D_{\text{NN}}=R_{\text{sp}}S+N\tau_{\text{sp}}^{-1}\quad;\quad D_{ \text{SN}}=-R_{\text{sp}}S, \tag{18}\]  
are standard diffusion coefficients. A usual approach for solving the system from Equation (16) involves a small-signal analysis. Small deviations from a steady-state value are assumed:  
\[S \simeq S_{0}+S_{\Delta} =S_{0}+\int_{-\infty}^{\infty}e^{i\Omega^{\prime}t}S_{\text{op}}( \Omega^{\prime})\text{d}\Omega^{\prime}\qquad\text{with}\quad S_{0}\gg S_{\Delta}\] (19a) \[\phi \simeq\qquad\phi_{