## VCSEL Equivalent Circuit Model  
To realize the hybrid electronic-photonic modeling, the noise-inclusive rate Equations (25)-(27) are translated into a set of equations describing circuits. The equivalent circuit model is derived based on these circuit-described equations. To eliminate non-physical solutions of the carrier and photon dynamisms, Equation (31) is used to ensure that \(N_{S}\) and \(N\) are positive values,  
\[N_{S}=N_{SEQ}V_{act}e^{\frac{V_{\infty}}{2V_{T}}},\ N=N_{EQ}V_{QWe}e^{\frac{V_ {\infty}}{2V_{T}}} \tag{31}\]  
where \(N_{SEQ}\) and \(N_{EQ}\) are the densities of unconfined and confined carriers in thermal equilibrium at zero bias, and \(V_{act}\) and \(V_{QW}\) are the volumes of the active region and the single QW. \(V_{T}\), \(V\), and \(Vw\) are the thermal voltage, and the voltages across the SCH and QW layers, respectively. The relationship between the total number of photons \(S\) and the optical output power \(P_{o}\) is expressed as Equation (32),  
\[\frac{S}{P_{o}}=\frac{\lambda\tau_{p}}{2\pi\eta_{L}tshc} \tag{32}\]  
where \(\lambda\) is the free space wavelength, \(\tau_{p}\) is the photon lifetime, \(\eta_{c}\) is the output power coupling coefficient, \(\hbar\) is reduced Planck constant, and \(c\) is the light speed in free space. By substituting (31) into (25), Equation (33) is obtained to represent the equivalent circuit for the unconfined carriers in terms of current variables,  
\[I_{a}=I_{C}+I_{D}-I_{CW}-I_{Dw}+I_{1}+I_{RS}+I_{L}-I_{FNS} \tag{33}\]  
where  
\[I_{C}=\frac{qN_{SEQ}V_{act}}{\eta_{inj}\tau_{c}},\ I_{D}=\frac{qN_{SEQ}V_{act }}{\eta_{inj}\tau_{c}}(e^{\frac{V}{2V_{T}}}-1),\ I_{CW}=\frac{qN_{EQ}V_{QW}}{ \eta_{inj}\tau_{c}},\ I_{Dw}=\frac{qN_{EQ}V_{QW}}{\eta_{inj}\tau_{e}}(e^{\frac {V_{\infty}}{2V_{T}}}-1),\]  
\[I_{1}=\frac{qN_{SEQ}V_{act}}{2V_{T}\eta_{inj}}e^{\frac{V}{2V_{T}}}\frac{dV}{dt },\ I_{RS_{S}}=\frac{q}{\eta_{inj}}R_{N_{S}},\ I_{L}=\frac{I_{lank}}{\eta_{ inj}},\ I_{FNS}=\frac{q}{\eta_{inj}}F_{N_{S}}.\]  
In addition, Equation (34) is derived by substituting (31) and (32) into (26), which represents the equivalent circuit for the confined carriers in QWs,  
\[I_{C}+I_{D}=I_{CW}+I_{Dw}+I_{2}+I_{R1}+I_{S1}-I_{FN} \tag{34}\]  
where  
\[I_{2}=\frac{qN_{EQ}V_{QW}}{2V_{T}\eta_{inj}}e^{\frac{V_{\infty}}{2V_{T}}} \frac{dV_{w}}{dt},\ I_{R1}=\frac{q}{\eta_{inj}}R_{N},\ I_{S1}=\frac{q\lambda \tau_{p}v_{g}G_{N}P_{o}}{2\pi\eta_{inj}\eta_{c}tshc},\ I_{FN}=\frac{q}{\eta_{ inj}}F_{N}.\]  
Substituting (32) into (27), Equation (35) is obtained, describing the equivalent circuit for the photon dynamism,  
\[C_{p}\frac{dV_{o}}{dt}+\frac{V_{o}}{R_{p}}=I_{R2}+I_{S2}+I_{FS} \tag{35}\]  
where:  
\[R_{p}=1\,\Omega,\ C_{p}=\frac{2\tau_{p}}{R_{p}},\ V_{o}=\sqrt{P_{o}R_{p}},\ I_{R2}=\frac{2\pi\eta_{lc}tshc}{ \lambda V_{o}}R_{S},\ I_{S2}=v_{g}G_{N}\tau_{p}\frac{V_{o}}{R_{p}},\ I_{FS}= \frac{2\pi\eta_{lc}tshc}{\lambda V_{o}}F_{S}.\]  
The equivalent circuit schematic is shown in Figure 3, based on Equations (33)-(35). The equivalent circuit model is realized in the Cadence Virtuoso environment based on the Verilog-A language. As seen from the mathematical derivations above, \(I_{C}\), \(I_{Co}\) and \(I_{L}\) are considered as constant current sources,while \(I_{D}\) and \(I_{D\omega}\) are regarded as currents passing through a diode with the diode ideality factors equal to 2. Other current components, \(I_{RS}\), \(I_{R1}\), \(I_{R2}\), \(I_{S1}\), \(I_{S2}\), \(I_{1}\), \(I_{2}\), \(I_{FNS}\), \(I_{FS}\) and \(I_{FN}\), are generated by time-varying current-controlled current sources. The circuit in the bottom right corner of Figure 3 converts \(V_{O}\) to the practical optical power output, \(P_{O}\). \(R_{L}\) is an arbitrary resistor for completeness of the closed circuit, and its value does not influence the functionality of the circuit model.