## 3 Parasitic Circuit  
A complete VCSEL model includes the extrinsic parasitic circuit description and the intrinsic rate-equation-based model. The parasitic circuit originates from the cavity structure explained in Section 2 and determines the small-signal response. The rate-equation-based model reflects the variation in the number of carriers and photons inside the active region.  
The complete parasitic circuit model is shown in Figure 2a. The contact pads introduce resistance, capacitance and inductance to the parasitic circuit. Cp is the capacitance between the contacts, the resistance Rp represents the contact pad loss, and Lp is the inductance induced by the contacts. The resistance Rt and Rb model the top, and bottom Bragg reflectors, respectively. The oxide layer capacitance, Cox, is in series with the junction capacitance, Cj. The resistance of the active region, Ra, is parallel with the series connection of Cox and Cj. The time-varying current flowing through Ra serves as the input to the rate equation model. In Figure 2b, we simplify the parasitic circuit for further circuit analysis and modeling. The series connection of Cox and Cj leads to Ca, and the combination of the top and bottom Bragg reflector resistances leads to Rm.  
Based on Figure 2b, the input impedance of the parasitic circuit is expressed as,  
\[Z_{in}(f)=\left(\left(\left(j\cdot 2\pi f\cdot C_{p}\right)^{-1}+R_{p}\right)^{ -1}+\left(j\cdot 2\pi f\cdot L_{p}+R_{m}+\left(j\cdot 2\pi f\cdot C_{a}+R_{a} {}^{-1}\right)^{-1}\right)^{-1}\right)^{-1} \tag{1}\]  
where \(f\) is the frequency of the modulation signal and \(j\) is the imaginary unit.  
The active region injection current with respect to time \(t\), _la(t)_, is expressed in Equation (2),  
\[I_{a}(t)=\frac{I_{d}e^{j\omega t}\cdot Z_{in}}{R_{a}+R_{m}-\omega^{2}L_{p}C_{a} R_{a}+j\omega\left(C_{a}R_{a}R_{m}+L_{p}\right)} \tag{2}\]  
Figure 2: (**a**) The complete parasitic circuit of the VCSEL; (**b**) the simplified parasitic circuit model.  
where \(I_{d}\) is the amplitude of the time-varying high-speed driving current.  
Due to Joule ohmic heating and free carrier absorption loss, the VCSEL self-heating effect leads to temperature elevation of the laser cavity [17]. The parasitic elements in the active region, \(Rm\), \(Ra\) and \(Ca\), are modelled as the function of the active region temperature, as shown in Equation (3),  
\[f(T)=a_{0}+a_{1}(T-T_{0})+a_{2}(T-T_{0})^{2}+a_{3}(T-T_{0})^{3}+a_{4}(T-T_{0}) ^{4} \tag{3}\]  
where \(a_{0}\), \(a_{1}\), \(a_{2}\), \(a_{3}\) and \(a_{4}\) are the polynomial coefficients, \(T\) is the VCSEL cavity temperature in Kelvin, and \(T_{0}\) is the relative temperature subtracted from \(T\) to reduce the requirement for coefficient extraction precision.  
The active region temperature is expressed as follows due to the self-heating effect,  
\[T=T_{amb}+(IV-P)R_{th}-\tau_{th}\frac{dT}{dt} \tag{4}\]  
where \(T_{amb}\) is the ambient temperature, \(V\) is the device voltage, \(I\) is the driving current, \(P\) is the output light power, \(R_{th}\) is the thermal impedance of the VCSEL, and \(\tau_{th}\) is the thermal time constant.  
Considering the excellent thermal conductivity of the contact pads and relatively large gap between pads and the active region, it is accurate enough to regard \(\mathrm{Rp}\), \(\mathrm{Cp}\), and \(\mathrm{Lp}\) as temperature-independent.