## 4 Compact Rate Equations
The total number of unconfined carriers, \(N_{S}\), and of confined carriers, \(N\), in all layers are expressed as the following:  
\[\frac{dN_{SCH(2)}}{dt}=-\frac{I_{lank}}{q}-R_{SCH(2)}-\frac{N_{SCH(2)}}{\tau_ {D}}+\frac{N_{G(n_{w})}}{\tau_{D}} \tag{6}\]1  
The total recombination rate for the unconfined carriers in all layers gives the equivalent recombination rate as, \(R_{NS}\), which shows the total number of recombined unconfined carriers per unit time:  
\[\frac{dN_{SCH(2)}}{dt}=-\frac{I_{lank}}{q}-R_{SCH(2)}-\frac{N_{SCH(2)}}{\tau_ {D}}+\frac{N_{G(n_{w})}}{\tau_{D}} \tag{6}\]2  
The equivalent escape lifetime \(\tau_{c}\) is equal to \(\tau_{esc}\), while the equivalent capture lifetime \(\tau_{c}\) is expressed as the following:  
\[\frac{dN_{SCH(2)}}{dt}=-\frac{I_{lank}}{q}-R_{SCH(2)}-\frac{N_{SCH(2)}}{\tau_ {D}}+\frac{N_{G(n_{w})}}{\tau_{D}} \tag{6}\]3  
The confined carrier dynamics are expressed below by combining Equations (11) to (13),  
\[\frac{dN_{SCH(2)}}{dt}=-\frac{I_{lank}}{q}-R_{SCH(2)}-\frac{N_{SCH(2)}}{\tau_ {D}}+\frac{N_{G(n_{w})}}{\tau_{D}} \tag{6}\]4  
where the photon number \(S\) is the total number of photons in QW layers, \(R_{N}\) is the total recombination rate for the confined carriers, and \(G_{N}\) is the gain coefficient weighted by optical confinement factors. These can be expressed as the following:  
\[\frac{dN_{SCH(2)}}{dt}=-\frac{I_{lank}}{q}-R_{SCH(2)}-\frac{N_{SCH(2)}}{\tau_ {D}}+\frac{N_{G(n_{w})}}{\tau_{D}} \tag{6}\]5  
For the photon dynamics, Equation (21) is derived from the summation of (14) over the QW position \(j\):  
\[\frac{dN_{SCH(2)}}{dt}=-\frac{I_{lank}}{q}-R_{SCH(2)}-\frac{N_{SCH(2)}}{\tau_ {D}}+\frac{N_{G(n_{w})}}{\tau_{D}} \tag{6}\]6  
With respect to Equations (15), (19) and (21), the equation set for the carrier and photon dynamics consists of three differential equations, an important reduction in computation for the modeling of a laser source with MQWs. Assuming each differential equation consumes the same computation time, the calculation efficiency, defined as the reciprocal of the computation time for the whole equation set, improves by 3.3-fold for a laser source with three QWs and by 6-fold for five QWs. The recombination rate terms \(R_{N_{S}}\), \(R_{N}\) and \(R_{S}\) consist of three main types of recombination: Shockley-Read-Hall (SRH) non-radiative recombination, radiative recombination and Auger non-radiative recombination. A three-order polynomial based on the ABC model [20] describes recombination terms, where \(A_{N_{S}}\), \(A_{N}\) and \(A_{S}\) are SRH recombination coefficients for unconfined carriers, confined carriers and photons, \(B_{N_{S}}\), \(B_{N}\) and \(B_{S}\) are corresponding radiative recombination coefficients, and \(C_{N_{S}}\), \(C_{N}\) and \(C_{S}\) are corresponding Auger recombination coefficients. The expressions for \(R_{N_{S}}\), \(R_{N}\) and \(R_{S}\) are expressed in Equation (22). Specifically, the radiative recombination dominates the photon generation, and non-radiative recombination effects are negligible. Therefore, values of \(A_{S}\) and \(C_{S}\) are set to 0. \[\frac{dN_{SCH(2)}}{dt}=-\frac{I_{lank}}{q}-R_{SCH(2)}-\frac{N_{SCH(2)}}{\tau_ {D}}+\frac{N_{G(n_{w})}}{\tau_{D}} \tag{6}\]7  
The strong thermal dependence of VCSEL I-P characteristics is mainly attributed to the temperature-dependent carrier leakage out of the active region [21]. The carrier leakage is represented by the leakage current \(I_{Ieak}\) in Equation (15), which is expressed by a fourth-order polynomial of temperature in Equation (23). It is worth mentioning that the fourth-order polynomial provides sufficient parameter fitting accuracy and acceptable computation complexity,  
\[\frac{dN_{SCH(2)}}{dt}=-\frac{I_{lank}}{q}-R_{SCH(2)}-\frac{N_{SCH(2)}}{\tau_ {D}}+\frac{N_{G(n_{w})}}{\tau_{D}} \tag{6}\]8  
where \(b_{0}\), \(b_{1}\), \(b_{2}\), \(b_{3}\), and \(b_{4}\) are the polynomial coefficients. In the same way as Equation (3), \(T_{0}\) is subtracted from \(T\) to reduce the requirement for coefficient fitting precision.