## 2 Time-Delay Model of Two-Reflector OFB  
A model of the rate equation, which includes multiple round-trips in the external cavity and optical phase due to OFB, has been established [5]. The model is extended to study the theoretically dynamic states of the LDs subject to double external OFB [12]. An improved rate equation of LD with double OFB from two external cavities for photon numbers, phase, and carrier numbers is formulated as follows:  
\[\frac{dS}{dt}=\frac{1}{2}[G-G_{th}]\ S+\frac{a\xi}{V}N+F_{S} \tag{1}\]  
\[\frac{d\theta}{dt}=\frac{1}{2}a\frac{a\xi}{V}\big{(}N-\overline{N}\big{)}- \frac{c}{n_{D}L_{D}}\phi+F_{\theta} \tag{2}\]  
\[\frac{dN}{dt}=\frac{I}{e}-\frac{N}{\tau_{s}}-\frac{a\xi}{V}\big{(}N-N_{g}\big{)} S+F_{N} \tag{3}\]  
\[G=\frac{a\xi}{V}\big{(}N-N_{g}\big{)}S-BS, \tag{4}\]where \(G\) is the optical gain, \(a\) is the slope of the optical gain with the changes in carrier number \(N\), \(\xi\) is the confinement factor, \(V\) is the active region volume, \(\overline{N}\) is the \(N\) time-averaged value, \(\alpha\) is the LEF, \(I\) is the injection current, \(\tau_{s}\) is the electron lifetime due to spontaneous emission, and \(N_{g}\) is the carrier number at transparency. The universal constant \(e\) defines the electron charge, and \(B\) is the nonlinear gain coefficient, respectively [19; 20; 21]. The threshold gain is given using  
\[G_{th}=G_{thD}-\frac{c}{n_{D}L_{D}}\ln\lvert U\rvert, \tag{5}\]  
where \(G_{thD}\) is the threshold gain [21]:  
\[G_{thD}=\frac{c}{n_{D}}\left[\kappa+\frac{1}{2L_{D}}\ln\frac{1}{R_{f}R_{b}} \right], \tag{6}\]  
where \(k\), \(L_{D}\), \(R_{f}\), and \(R_{b}\) are the internal loss, laser cavity length, and front and back facet reflectivity, respectively. The function \(U\) expressing the double OFB in the rate equations is given using [12]:  
\[ \begin{array}{ll}U=&1-K_{ex1}\sqrt{\frac{S(I-\tau_{1})}{S(I)}}e^{\lvert \theta(t-\tau_{1})-\theta(t)-\omega\tau_{1}\rvert}\\ &-K_{ex2}\sqrt{\frac{S(I-\tau_{2})}{S(I)}}e^{\lvert\theta(t-\tau_{2})-\theta( t)-\omega\tau_{2}\rvert}\end{array}  \tag{7}\]  
where \(\omega\) is the angular frequency of the laser emission, \([\theta(t-\tau_{1})-\theta(t)]\) and \([\theta(t-\tau_{2})-\theta(t)]\) are the optical phase deviations due to chirping induced by delay in external cavity 1 and external cavity 2, respectively. \(K_{ex1}\) and \(K_{ex2}\) are the OFB strengths from the single external cavity and double external cavity, respectively, and are determined with the front facet reflectivity, first reflector reflectivity, and second reflector reflectivity, respectively [12]:  
\[K_{ex1}=\left(1-R_{f}\right)\frac{\sqrt{R_{ex1}}}{\sqrt{R_{f}}} \tag{8}\]  
\[K_{ex2}=\left(1-R_{f}\right)\left(1-R_{ex1}\right)\frac{\sqrt{R_{ex2}}}{\sqrt {R_{f}}} \tag{9}\]  
Figure 1 illustrates a model scheme of semiconductor lasers subjected to double OFB. The laser emitted from the front travels a round-trip in the cavity between the front and first reflector, \(R_{ex1}\). Then, a round-trip is made between the front facet and double reflector \(R_{ex2}\), which then re-enters the laser cavity. That is, there are time delays \(\tau_{1}\) and \(\tau_{2}\) between the laser reflected by the front of the laser facet into the cavity of the laser and the laser entered back from single and double reflectors, respectively [12]. By supposing \(L_{ex1}\) and \(L_{ex2}\) to be the lengths of the single and double external cavities, respectively, and the external cavity refractive index \(n_{ex}\), the feedback laser has time delays of  
\[\tau_{1}=\frac{2n_{ex1}L_{ex1}}{c},\text{ and }\tau_{2}=\frac{2n_{ex}L_{ex2}}{c} \tag{10}\]  
The Langevin noise sources \(F_{S}(t)\), \(F_{N}(t)\), and \(F_{\theta}(t)\), are Gaussian probability distributions with their correlation functions, \(\left\langle F_{x}(t)\ F_{y}(t^{\prime})\right\rangle=V_{xy}\ \delta(t-t^{\prime})\), \(x,y=S,N\ or\ \theta\), and have zero mean values, \(\left\langle F_{S}(t)\right\rangle=\left\langle F_{N}(t)\right\rangle=\left\langle F _{\theta}(t)\right\rangle=0\).