# Digitally Chirped Multilayer Quantum Dot Lasers with Dual-Wavelength Lasing Emissions
## 3 Theoretical Simulations
The excitons (electron-hole pairs) in the QW, ES1, GS1, ES2 and GS2 states of \(j\)-th QD group (\(N_{w}\), \(N_{e_{1},j}\), \(N_{g_{1},j}\), \(N_{e_{2},j}\) and \(N_{g_{2},j}\), respectively) as well as \(k\)-th spectral photon energy originated from GS1 and GS2 (\(S_{1,k}\) and \(S_{2,k}\), respectively) in DC-MQD lasers are coupled by the following system of nonlinear differential equations:  
\[\frac{dN_{w}}{dt}= \frac{I}{e}+\sum_{j}\left\{\frac{N_{e_{1},j}}{\tau_{e_{1}w}}-\frac {N_{w}}{\tau_{ww_{1}}}G_{1j}\left(1\!-\!\frac{N_{e_{1},j}}{M_{e_{1},j}}\right) +\frac{N_{e_{2},j}}{\tau_{e_{2}w}}-\frac{N_{w}}{\tau_{ww_{2}}}G_{2j}\left(1\! -\!\frac{N_{e_{2},j}}{M_{e_{2},j}}\right)\right\}-\frac{N_{w}}{\tau_{w}}, \tag{5}\] \[\frac{dN_{e_{1},j}}{dt}= \frac{N_{w}}{\tau_{ww_{1}}}G_{1j}\left(1\!-\!\frac{N_{e_{1},j}}{M _{e_{1},j}}\right)-\frac{N_{e_{1},j}}{\tau_{e_{1}w}}+\frac{N_{g_{1},j}}{\tau_{ g_{1}e_{1}}}\left(1-\frac{N_{e_{1},j}}{M_{e_{1},j}}\right)-\frac{N_{e_{1},j}}{ \tau_{e_{1}g_{1}}}\left(1\!-\!\frac{N_{g_{1},j}}{M_{g_{1},j}}\right)-\frac{N_{ e_{1},j}}{\tau_{e_{1}}},\] (6) \[\frac{dN_{g_{1},j}}{dt}= \frac{N_{e_{1},j}}{\tau_{e_{1}g_{1}}}\left(1\!-\!\frac{N_{g_{1},j }}{M_{g_{1},j}}\right)-\frac{N_{g_{1},j}}{\tau_{g_{1}e_{1}}}\left(1\!-\!\frac{ N_{e_{1},j}}{M_{e_{1},j}}\right)-\frac{C_{0}\Gamma_{a1}}{n_{r}}\sum_{k}g_{kj}^{ GS1}\left(S_{1,k}+S_{2,k}\right)-\frac{N_{g_{1},j}}{\tau_{g_{1}}},\] (7) \[\frac{dS_{1,k}}{dt}= \frac{C_{0}\Gamma_{a1}}{n_{r}}\sum_{j}g_{kj}^{GS1}S_{1,k}-\frac{S _{1,k}}{\tau_{p}}+\beta\frac{N_{g_{1},k}}{\tau_{g_{1}}},\] (8) \[\frac{dN_{e_{2},j}}{dt}= \frac{N_{w}}{\tau_{ww_{2}}}G_{2j}\left(1\!-\!\frac{N_{e_{2},j}}{M _{e_{2},j}}\right)-\frac{N_{e_{2},j}}{\tau_{e_{2}w}}+\frac{N_{g_{2},j}}{\tau_{ g_{2}e_{2}}}\left(1\!-\!\frac{N_{e_{2},j}}{M_{e_{2},j}}\right)-\frac{N_{e_{2},j}}{ \tau_{e_{2}g_{2}}}\left(1\!-\!\frac{N_{g_{2},j}}{M_{g_{2},j}}\right)-\frac{N_{ e_{2},j}}{\tau_{e_{2}}},\] (9) \[\frac{dN_{g_{2},j}}{dt}= \frac{N_{e_{2},j}}{\tau_{e_{2}g_{2},j}}\left(1\!-\!\frac{N_{g_{2}, j}}{M_{g_{2},j}}\right)-\frac{N_{g_{2},j}}{\tau_{g_{2}e_{2}}}\left(1\!-\!\frac{N_{e_{2},j}}{M _{e_{2},j}}\right)-\frac{C_{0}\Gamma_{a2}}{n_{r}}\sum_{k}g_{kj}^{GS2}\left(S_{1, k}+S_{2,k}\right)-\frac{N_{g_{2},j}}{\tau_{g_{2}}},\] (10) \[\frac{dS_{2,k}}{dt}= \frac{C_{0}\Gamma_{a2}}{n_{r}}\sum_{j}g_{kj}^{GS2}S_{2,k}-\frac{S _{2,k}}{\tau_{p}}+\beta\frac{N_{g_{2},k}}{\tau_{g_{2}}}, \tag{11}\]  
where \(I\) is pumping current, \(e\) is elementary charge, \(C_{0}\) is vacuum speed of light, \(n_{r}\) is refractive index, \(\beta\) is spontaneous emission factor, \(\Gamma_{a1}\) and \(\Gamma_{a2}\) are optical confinement factors for \(2\cdot QD_{1}\) and \(3\cdot QD_{2}\) stacks, respectively. The photon lifetime (\(\tau_{p}\)) is determined by the internal and mirror losses of about 7.7 cm\({}^{-1}\). The spontaneous lifetime for each energy state and transition time between adjacent energy states are also defined in Table 1. The time constants associated with GS1/ES1 states are referenced from [17], while those associated with GS2/ES2 states are obtained by fitting the second threshold current. Note that the carrier and photon rate equations associated with the second chirped wavelength are treated independently in Equations (9)-(11). More chirped wavelengths can be included accordingly. In addition, nonlinear coupling between photons and carriers in Equations (7) and (10) deals with total spectral photons (\(S_{1,k}+S_{2,k}\)). In comparison to these equations with individual spectral photons, the calculated light intensities are overestimated. Figure 2a shows simulated photon number spectra (\(S_{1,k}+S_{2,k}\)) at four injection currents. The photon numbers are also scaled in logarithmic scale and shifted in vertical axis. The flattened lasing peaks are due to insufficient photon spectral resolution (\(\Delta E=1\) meV) in the simulation. The spectra integrated photon number versus injection current for GS1, GS2 and total emitted photons are shown in Figure 2b.