# Digitally Chirped Multilayer Quantum Dot Lasers with Dual-Wavelength Lasing Emissions
## 3 Theoretical Simulations  
The theoretical model consists of a system of coupled rate equations that describe temporal evolution of carrier populations among energy states as well as their interaction with optical photons. Charged carriers of electrons and holes, populated in respective energy states and recombined in pairs, are treated as excitons. Three types of excitonic energy states are considered in QDs, i.e., QW state, excited state (ES) and ground state (GS). The InGaAs QW is a common reservoir for carrier relaxation to ES where transitions between ES and ES occur only in the same QDs. The rate equations for digitally chirped QD layers are modified from those for uniformly stacked multilayer QDs [15; 16]. Table 1 lists all device parameters used in the present simulation. Note that three chirped wavelengths were designed in the experiment but only two are modeled in the simulation for computational efficiency. The third wavelength did not lase because its steady-state GS carrier population is low and population inversion cannot be achieved in the measurement range. The inhomogeneous broadening of a QD ensemble is described by Gaussian distribution function [16] and discretized with an energy resolution (\(\Delta E\)) of 1 meV. The probability density functions for the \(j\)-th QD group of first and second chirped wavelengths are given by:  
\[ \begin{array}{ll}G_{1j}&=\frac{1}{\sqrt{2\pi}\sigma}exp\left(-\frac{E_{j}-E _{1}}{2\sigma^{2}}\right)\Delta E,\\ G_{2j}&=\frac{1}{\sqrt{2\pi}\sigma}exp\left(-\frac{E_{j}-E_{2}}{2\sigma^{2}} \right)\Delta E,\end{array}  \tag{1}\]where \(E_{1}\) and \(E_{2}\) are the mean energies as well as the lasing peaks of longer- and medium-wavelength QD stacks, respectively. The standard deviation (\(\sigma\)) is related to the inhomogeneous broadening factor (\(\Gamma_{inh}\)) or the full-width-of-half-maximum (FWHM) of Gaussian distribution by \(\Gamma_{inh}=(\sqrt{8\ln 2})\sigma\). The maximum numbers of GS and ES states for the \(j\)-th QD group, including stacking layer numbers (2 or 3), spin and energy degeneracies (\(D_{g}\) and \(D_{e}\)), are given by:  
\[ \begin{array}{ll}M_{g_{1}j}&=2D_{g}\cdot N_{s}\cdot W\cdot L\cdot G_{1j},\\ M_{e_{1}j}&=2D_{e}\cdot N_{s}\cdot W\cdot L\cdot G_{1j},\\ M_{g_{2}j}&=3D_{g}\cdot N_{s}\cdot W\cdot L\cdot G_{2j},\\ M_{e_{2}j}&=3D_{e}\cdot N_{s}\cdot W\cdot L\cdot G_{2j},\end{array}  \tag{2}\]  
where \(N_{s}\) is surface dot density per layer, and \(W\) and \(L\) are ridge width and cavity length, respectively. The optical gain (both GS and ES) can be simplified by multiplying a linear gain coefficient \((g_{0})\) by the carrier in excess of the transparency carrier and dividing by the total volume (\(V_{t}\)) [17]. Since longitudinal mode separation is less than 0.1 meV for 2.5-mm-long cavity; without loss of generality, we consider only photonic energies which are discretized exactly the same as excitonic energies. The constitutional optical gain of \(k\)-th photon energy that originated from \(j\)-th QD group of GS1 and GS2 is given by:  
\[ \begin{array}{ll}g_{kj}^{GS1}&=\frac{g_{0}}{V_{t}}\left(N_{g_{1}j}-\frac{M_{ g_{1}j}}{2}\right)L(E_{j},E_{k}),\\ \\ g_{kj}^{GS2}&=\frac{g_{0}}{V_{t}}\left(N_{g_{2}j}-\frac{M_{g_{2}j}}{2}\right) L(E_{j},E_{k}).\end{array}  \tag{3}\]  
The optical gain is homogeneously broadened by a Lorentz distribution function [16] given as  
\[L(E_{j},E_{k})=\frac{1}{\pi}\frac{(\Gamma_{hom}/2)}{(E_{j}-E_{k})^{2}+(\Gamma_ {hom}/2)^{2}}, \tag{4}\]