## 2 Model of Analysis and Numerical Procedures  
### Modified Model of Analysis  
The model in [26] was applied here to explore the impact of the LEF and SEF on the dynamics and RIN of an AlGaAs/GaAs LD with OFB. In long EC counting, a single round trip in the rate equation is sufficient to investigate the LD dynamics, while in a short EC, we have to include the multiple round trips in the rate Equations (1)-(3) [26]. The rate equations, which depict the time-delayed LD states, are given by the rate of the photon number \(S(t)\), the optical phase \(\theta(t)\), and the electron number \(N(t)\) as  
\[\frac{dS}{dt}=\bigg{\{}\frac{a\xi}{V}(N-N_{g})-BS-G_{th0}+\frac{c}{n_{D}L_{D}} \ln|T|\bigg{\}}S+\frac{C}{\tau_{g}}N+F_{S}, \tag{1}\]  
\[\frac{d\theta}{dt}=\frac{aa\xi}{2V}(N-\overline{N})-\frac{c}{2n_{D}L_{D}}( \varphi-\overline{\varphi})+F_{\theta}, \tag{2}\]  
\[\frac{dN}{dt}=-\frac{a\xi}{V}(N-N_{g})S-\frac{N}{\tau_{S}}+\frac{I}{e}+F_{N}, \tag{3}\]  
where \(a\xi(N-N_{g})/V\) is the gain linearity, with \(a\) and \(N_{g}\) as material constants, and \(\xi\) is the confinement factor of the optical field inside the active region with volume \(V\), and \(C\) is the SEF. \(\alpha\) is the LEF, \(I\) is the injected current, \(\overline{N}\) is the mean electron number, \(c\) is the speed of light, and \(\tau_{s}\) is the lifetime. The threshold gain is given as \(G_{th0}\), which is given by:  
\[G_{th0}=\frac{c}{n_{r}}\left\{k+\frac{1}{2L}\ln\frac{1}{R_{f}R_{b}}\right\}, \tag{4}\]  
where \(k\) is the coefficient of loss of the LD. Many models have been introduced to introduce gain saturation [35; 36; 37]. The parameter \(B\) is the gain saturation, which is introduced by the gain third-order perturbation theory [35; 36]. \[B=\frac{9\hbar\omega_{o}}{4\epsilon_{o}n_{D}^{2}}\Big{(}\frac{\tau_{in}}{\hbar }\Big{)}^{2}\Big{(}\frac{\varsigma}{V}\Big{)}^{2}a|R_{cv}|^{2}(N-N_{s}),\ B=B_{c}(N-N_{s}), \tag{5}\]  
where \(R_{cv}\) is the dipole moment, \(\tau_{in}\) is the intra-band relaxation time, \(\hbar=h/2\pi\) is the plank constant, and \(N_{s}\) is an injection level. The coefficient \(T\), which represents the OFB term in the rate equation, is represented as [26]:  
\[ \begin{array}{ll}T&=1-\sum\limits_{m=1}(K_{ex})^{m}\Big{(}\frac{R_{f}}{1-R_ {f}}\Big{)}^{m-1}\exp\{-jm\psi\}\frac{S(t-m\tau)}{S(t)}\\ &=|T|\exp\{-j\phi\},\end{array} . \tag{6}\]  
where \(m\), \(R_{f}\), \(n_{ex}\), \(L_{ex}\), and \(\psi\) are the number of round trips, the front reflector reflectivity, the refractive index, the EC width, and the combined phase \(\psi=\phi_{ex}+\phi_{f}+\omega\tau\), respectively. Where \(\omega\), \(\phi_{ex}\), and \(\phi_{f}\) are the circular frequency emission, the phase retarded because of reflection from the external mirror, and the front facet, respectively, and \(\tau=2\pi n_{ex}L_{ex}/c\) is the time delay. The OFB strength coefficient \(K_{ex}\) is given by:  
\[K_{ex}=(1-R_{f})\sqrt{\frac{R_{ex}}{R_{f}}}, \tag{7}\]  
where \(R_{ex}\) is the external mirror reflectivity. The argument \(\phi\) of the OFB term \(T\) is given by:  
\[\phi=-\tan^{-1}\{\text{Im}[T]/\text{Re}[T]\}+\ddagger\pi \tag{8}\]  
where \(\ddagger\) is an integer. The \(\phi\) value is determined in the two-dimensional space (\(Re[T]\)-\(Im[T]\)). \(F_{S}(t)\), \(F_{\theta}(t)\), and \(F_{N}(t)\) are [PERSON] noise sources and are included in the rate Equations (1)-(3) to consider laser fluctuations due to intrinsic transitions [27; 38]. These noise sources have Gaussian probability distributions to be \(\delta\)-correlated processes and with zero means [27; 38]. The spectra of the \(RIN\) are then calculated by instantaneous fluctuation values \(\delta S(t)=S(t)-\overline{S}\) and using the fast Fourier transform as [39]:  
\[