## 4 Compact Rate Equations  
The SCH-QW-G-B bandgap structure in Section 2 is modeled using the set of rate equations expressing the carrier and photon dynamisms. In this section, the rate equation model describes the intrinsic behaviors of a MQW VCSEL with more complete carrier and photon dynamisms than previously reported [18; 19]. We first derive the rate equations for the two SCH layers (SCH\({}_{1}\) and SCH\({}_{2}\)) of the active regions. Carriers injected into SCH\({}_{1}\) make their way to SCH\({}_{2}\). The carrier movement between a SCH layer and an unconfined state in an adjacent QW layer determines the carrier quantity. In addition, electron-hole recombination in the SCH layer with high carrier density is included. The variations in carrier numbers with respect to time in the two SCH regions are expressed by the differential equations below,  
\[\frac{dN_{SCH(1)}}{dt}=\frac{\eta_{inj}I_{a}(t)}{q}-R_{SCH(1)}-\frac{N_{SCH(1 )}}{\tau_{D}}+\frac{N_{G(1)}}{\tau_{D}} \tag{5}\]  
\[\frac{dN_{SCH(2)}}{dt}=-\frac{I_{lank}}{q}-R_{SCH(2)}-\frac{N_{SCH(2)}}{\tau_ {D}}+\frac{N_{G(n_{w})}}{\tau_{D}} \tag{6}\]  
where \(N_{SCH(i)}\) and \(R_{SCH(i)}\) represent the carrier number and the combination rate in SCH\({}_{1}\) (\(i=1\)) and SCH\({}_{2}\) (\(i=2\)). \(\eta_{inj}\) is the current injection efficiency (dimensionless) and \(q\) is the electron charge. The injection current \(I_{a}(t)\) is a time-varying high-speed signal, and \(I_{lank}\) is the leakage current. \(N_{G(j)}\) is the unconfined carrier number in the gateway state of the QW layer, where the subscript \(j\) denotes the gateway state in the \(j\)th QW layer counting from the bottom of the active region (in Figure 1b). The number of gateway states in the active region is the same as the number of QWs (denoted as \(n_{w}\)), therefore \(1\leq j\leq n_{w}\). \(\tau_{D}\) is the effective carrier transport time between a SCH or barrier layer and its adjacent gateway state. Next, the gateway rate equations for the carriers are derived. In the first gateway state (at the bottom of the active region in Figure 1b) and the last gateway state (at the top), the following effects are considered: the carrier exchange with the adjacent SCH or barrier layer, the electron-hole recombination, and the carrier exchange with the confined states within the QW layer. Equation (7) represents the rate equation for the unconfined carriers in the bottom QW layer (\(j=1\)),  
\[\frac{dN_{G(1)}}{dt}=\frac{N_{SCH(1)}}{\tau_{D}}-\frac{2N_{G(1)}}{\tau_{D}}+ \frac{N_{B(1)}}{\tau_{D}}+\frac{N_{W(1)}}{\tau_{esc}}-\frac{N_{G(1)}}{\tau_{ cap}}-R_{G(1)} \tag{7}\]where \(N_{B(1)}\) is the carrier number in the bottom barrier layer, \(N_{W(1)}\) and \(R_{G(1)}\) are the confined carrier number and the recombination rate of the unconfined carriers in the bottom QW layer, respectively. \(\tau_{esc}\) and \(\tau_{cap}\) are the QW carrier escape, and capture lifetimes, respectively. The rate equation for unconfined carriers in the top QW layer is expressed in (8),  
\[\frac{dN_{G(n_{w})}}{dt}=\frac{N_{SCH(2)}-2N_{G(n_{w})}+N_{B(n_{w}-1)}}{\tau_{D} }+\frac{N_{W(n_{w})}}{\tau_{esc}}-\frac{N_{G(n_{w})}}{\tau_{cap}}-R_{G(n_{w})} \tag{8}\]  
where the subscript \(n_{w}\) represents the top layer. For gateway states in the other QW layers (i.e., \(2\leq j\leq n_{w}\)-1) which are adjacent to two barrier layers, the rate equation is expressed by Equation (9):  
\[\frac{dN_{G(j)}}{dt}=\frac{N_{B(j-1)}-2N_{G(j)}+N_{B(j)}}{\tau_{D}}+\frac{N_{W (j)}}{\tau_{esc}}-\frac{N_{G(j)}}{\tau_{cap}}-R_{G(j)},\;2\leq j\leq n_{w}-1. \tag{9}\]  
Similarly, in the barrier layer, we consider the carrier exchange between the barrier layer and the two adjacent gateway states as well as the electron-hole recombination using Equation (10),  
\[\frac{dN_{B(k)}}{dt}=\frac{N_{G(k)}}{\tau_{D}}-\frac{2N_{B(k)}}{\tau_{D}}+ \frac{N_{G(k+1)}}{\tau_{D}}-R_{B(k)},\;1\leq k\leq n_{w}-1 \tag{10}\]  
where \(R_{B(k)}\) is the recombination rate in the \(k\)th barrier layer.