## 4 Compact Rate Equations
For confined carriers by the quantum wells, the dynamisms, such as carrier capture/escape, quantum tunneling, spontaneous and stimulated emissions, and recombination, are accounted. The following equation represents the rate equation for confined carriers in the first QW layer,  
\[\frac{dN_{W(1)}}{dt}=\frac{N_{G(1)}}{\tau_{cap}}-\frac{N_{W(1)}}{\tau_{esc}}- R_{W(1)}-\frac{N_{W(1)}-N_{W(2)}}{\tau_{tun}}-v_{g}\Gamma_{W(1)}G_{W(1)}S_{W(1)} \tag{11}\]  
where \(S_{W(1)}\) represents the photon number in this QW layer. \(\tau_{tun}\) is the tunneling time, and \(v_{g}\) is the group velocity of the lasing medium. \(R_{W(1)}\), \(\Gamma_{W(1)}\) and \(G_{W(1)}\) represent the recombination rate, the optical confinement factor, and the gain coefficient for the confined carriers. The rate equation for the confined carriers in the last QW layer is expressed as the following,  
\[\frac{dN_{W(n_{w})}}{dt}=\frac{N_{G(n_{w})}}{\tau_{cap}}-\frac{N_{W(n_{w})}}{ \tau_{esc}}-R_{W(n_{w})}-\frac{N_{W(n_{w})}-N_{W(n_{w}-1)}}{\tau_{tun}}-v_{g} \Gamma_{W(n_{w})}G_{W(n_{w})}S_{W(n_{w})} \tag{12}\]  
where the subscript is replaced to represent the top layer (\(n_{w}\)). The rate equation for confined carriers in all other QW layers is expressed as follows, where the subscript is generalized to \(j\) to represent the \(j\)th QW layer:  
\[\frac{dN_{W(j)}}{dt}=\frac{N_{G(j)}}{\tau_{cap}}-\frac{N_{W(j)}}{\tau_{esc}} +\frac{N_{W(j-1)}+N_{W(j+1)}-2N_{W(j)}}{\tau_{tun}}-R_{W(j)}-v_{g}\Gamma_{W(j )}G_{W(j)}S_{W(j)},\;2\leq j\leq n_{w}-1. \tag{13}\]  
Finally, Equation (14) accounts for the photon number generated in the \(j\)th QW,  
\[\frac{dS_{W(j)}}{dt}=-\frac{S_{W(j)}}{\tau_{p}}+R_{\omega\theta(j)}+v_{g} \Gamma_{W(j)}G_{W(j)}S_{W(j)} \tag{14}\]  
where \(\tau_{p}\) is the photon lifetime. \(R_{\omega\theta(j)}\) represents the rate of electron-hole recombination of the lasing mode. The rate equations describing these carrier and photon dynamisms are detailed above. The number of differential rate equations reaches 4-\(n_{w}\)+1 for a VCSEL where \(n_{w}\) corresponds to the number of QWs, leading to a challenging computation complexity. In addition, parameter extraction from the experimental results is required to simulate and evaluate the performance of the manufacturedVCSEL. The number of parameters included in the aforementioned equations is relatively large, which challenges parameter extraction. Furthermore, the proposed rate equations are strongly dependent on the number of QWs, such that the model cannot be generalized to all VCSEL designs as those in [12; 13], where the numbers of QWs in these designs are quite different. It is, therefore, necessary to manipulate the rate Equations (5)-(14) to make them independent from the number of QWs. The detailed analysis of carrier and photon behaviors builds a set of rate equations for their comprehensive dynamics, and the inclusion of the gateway states bridges SCH layers and QW layers. Consequently, the issues mentioned above are overcome by a series of transformations of rate equations. Combining Equations (5) to (10), Equation (15) below describes the variation in the total number of unconfined carriers, \(N_{S}\),  
\[\frac{dN_{SCH(2)}}{dt}=-\frac{I_{lank}}{q}-R_{SCH(2)}-\frac{N_{SCH(2)}}{\tau_ {D}}+\frac{N_{G(n_{w})}}{\tau_{D}} \tag{6}\]0  
where \(N\) is the total number of confined carriers inside each a QW.