## Appendix A Amplitude and Phase Conditions  
To obtain the revised lasing conditions resulting from the additional feedback component, extracting the amplitude and phase of the effective reflection coefficients is needed. In polar notation:  
\[\rho_{j}^{\text{eff}}=\left|\rho_{j}^{\text{eff}}\right|e^{i\varphi_{j}}, \tag{11}\]  
where the magnitude is computed as:  
\[\left|\rho_{j}^{\text{eff}}\right|^{2}\bigg{/}\rho_{j}^{2} \overset{\eqref{eq:2}}{=}\left[1+\kappa_{j}t_{\text{cav}}\cos( \phi_{j})\right]^{2}+\left[\kappa_{j}t_{\text{cav}}\sin(\phi_{j})\right]^{2}\] \[=1+2\kappa_{j}t_{\text{cav}}\cos(\phi_{j})+\kappa_{j}^{2}t_{\text {cav}}^{2}. \tag{12}\]Assuming condition (1) of weak external feedback, the last term on the right-hand side of Equation (A2) is neglected, resulting in:  
\[\left|\rho_{j}^{\text{eff}}(\omega_{\text{\tiny FB}})\right|\Big{/}\rho_{i}= \sqrt{1+2\kappa_{j}t_{\text{cav}}\cos(\phi_{j})}\overset{\text{(1)}}{\approx}1+ \kappa_{j}t_{\text{cav}}\cos(\phi_{j}).\] (A3)  
The phase of the effective reflection coefficient is extracted from:  
\[\varphi_{j}=\arctan\left[\left(\rho_{j}^{\text{eff}}\right)^{\prime\prime}/ \left(\rho_{j}^{\text{eff}}\right)^{\prime}\right]\overset{\text{(2)}}{ \approx}\arctan\left[\frac{\pm\kappa_{j}t_{\text{cav}}\sin(\phi_{j})}{1+\kappa _{j}t_{\text{cav}}\cos(\phi_{j})}\right]\]  
\[\overset{\text{(1)}}{\approx}\arctan[\pm\kappa_{j}t_{\text{cav}}\sin(\phi_{j} )]\overset{\text{(1)}}{\approx}\pm\kappa_{j}t_{\text{cav}}\sin(\phi_{j}).\] (A4)  
The fields traveling forward and backward in the laser cavity, \(\mathcal{E}_{\text{f}}\) and \(\mathcal{E}_{\text{b}}\) shown in Figure A1, can now be related by the effective reflection coefficients:  
\[\mathcal{E}_{\text{f}}(z=0)=\rho_{1}^{\text{eff}}\,\mathcal{E}_{\text{b}}(z=0)\] (A5a) \[\mathcal{E}_{\text{b}}(z=L)=\rho_{2}^{\text{eff}}\,\mathcal{E}_{\text{f}}(z=L).\] (A5b)  
Using the propagation constant:  
\[\beta\equiv n\omega/c\,\] (A6)  
with \(n\) the effective refractive index of the lasing mode and \(c\) the speed of light in vacuum, the fields can be written as:  
\[\mathcal{E}_{\text{f}}=A_{f}e^{-i\beta z+\frac{1}{2}(\Gamma g-\alpha)z}\] (A7a) \[\left|\rho_{j}^{\text{eff}}\right|^{2}\bigg{/}\rho_{j}^{2} \overset{\eqref{eq:2}}{=}\left[1+\kappa_{j}t_{\text{cav}}\cos( \phi_{j})\right]^{2}+\left[\kappa_{j}t_{\text{cav}}\sin(\phi_{j})\right]^{2}\]0 (A7b)  
where \(g\) is the gain coefficient and \(\alpha\) is the attenuation coefficient. Replacing Equation (A5) into Equation (A7):  
\[\left|\rho_{j}^{\text{eff}}\right|^{2}\bigg{/}\rho_{j}^{2} \overset{\eqref{eq:2}}{=}\left[1+\kappa_{j}t_{\text{cav}}\cos( \phi_{j})\right]^{2}+\left[\kappa_{j}t_{\text{cav}}\sin(\phi_{j})\right]^{2}\]1 (A8a) \[\left|\rho_{j}^{\text{eff}}\right|^{2}\bigg{/}\rho_{j}^{2} \overset{\eqref{eq:2}}{=}\left[1+\kappa_{j}t_{\text{cav}}\cos( \phi_{j})\right]^{2}+\left[\kappa_{j}t_{\text{cav}}\sin(\phi_{j})\right]^{2}\]2 (A8b)  
and inserting Equation (A8a) into Equation (A8b) results in:  
\[\left|\rho_{j}^{\text{eff}}\right|^{2}\bigg{/}\rho_{j}^{2} \overset{\eqref{eq:2}}{=}\left[1+\kappa_{j}t_{\text{cav}}\cos( \phi_{j})\right]^{2}+\left[\kappa_{j}t_{\text{cav}}\sin(\phi_{j})\right]^{2}\]3 (A9)  
Once lasing has been established, the gain assumes its threshold value:  
\[\left|\rho_{j}^{\text{eff}}\right|^{2}\bigg{/}\rho_{j}^{2} \overset{\eqref{eq:2}}{=}\left[1+\kappa_{j}t_{\text{cav}}\cos( \phi_{j})\right]^{2}+\left[\kappa_{j}t_{\text{cav}}\sin(\phi_{j})\right]^{2}\]4  
Figure A1: Schematic of the effective cavity of the laser, resulting from calculating effective reflection coeffitients. where \(g_{\rm FB}\) is the threshold gain with feedback.