# Parasitic Parameters Extraction of High-Speed Vertical-Cavity Surface-Emitting Lasers
## 2 Materials and Methods
### Equivalent Circuit Model  
The modulation bandwidth can be improved by both changing the internal factors and minimizing the external parasitic components [20]. The parasitic parameters are structure dependent and can be extracted using an equivalent circuit model [19]. Figure 4 shows a schematic diagram of the oxide-confined VCSEL marked with the equivalent circuit, which includes all the major extrinsic parasitic effects. \(L\) is the inductance, \(C_{p}\) represents the frequency-dependent pad capacitance between the metal contact on the BCB resin and the bottom mirror stack at the ground, \(R_{p}\) represents the frequency-dependent dielectric losses in the BCB, \(R_{mirror}\) represents the series equivalent resistance of both the p-type and n-type DBR mirrors, \(R_{sheet}\) is the sheet resistance along with the bottom contact, and \(R_{contact}\) is the resistance of both p-type and n-type contact pads. The total resistance \(R_{m}\) can be written as \(R_{m}=R_{mirror}+R_{sheet}+R_{contact}\). \(R_{n}\) is the resistance of the active region and is associated with the oxide aperture. \(C_{oxide}\), \(C_{j}\), and \(C_{intrinsic}\) represent the capacitance of the oxide layer, junction, and intrinsic layer, respectively. The junction capacitance \(C_{j}\) is the sum of the depletion capacitance and diffusion capacitance; however, under normal forward bias conditions, \(C_{j}\) is dominated by the diffusion capacitance. \(C_{intrinsic}\), \(C_{oxide}\), and \(C_{j}\) are grouped together as [21]: \(C_{a}\)=\(C_{j}\)+(1/\(C_{oxide}\)+1/\(C_{intrinsic}\))\({}^{-1}\).  
Figure 3: (**a**) Measured small-signal modulation response at different bias currents for the 3 \(\upmu\)m oxide aperture VCSEL, and (**b**) the resonance frequency plotted against the square root of the bias current above the threshold with the D-factor and MCEF fits.  
Figure 2: (**a**) L-I-V curves of the VCSEL with a 3 \(\upmu\)m oxide aperture, and (**b**) lasing spectra at different bias currents.  
The values of the electrical components in the equivalent circuit model were obtained by fitting the measured reflection coefficient \(S_{11}(f)\), which can be expressed as [15; 22]:  
\[S_{11}(f)=\frac{Z_{T}(f)-Z_{0}}{Z_{T}(f)+Z_{0}}, \tag{1}\]  
where \(Z_{T}(f)\) is the total impedance and \(Z_{0}\) denotes the characteristic impedance of the RF-driving source, which typically holds a value of 50 \(\Omega\), and  
\[Z_{T}(f)=(\frac{1}{Z_{SM}+i2\pi fL}+(R_{P}+\frac{1}{i2\pi fC_{p}})^{-1})^{-1}, \tag{2}\]  
Here, \(Z_{SM}\) represents the sub-input impedance containing the three circuit elements \(R_{m}\), \(R_{a}\) and \(C_{a}\) and can be described as:  
\[Z_{SM}(f)=R_{m}+(\frac{1}{R_{a}}+i2\pi fC_{a})^{-1}, \tag{3}\]  
According to the extrinsic transfer function [14],  
\[H_{ext}(f)=\frac{Z_{0}i_{a}}{V_{\rm s}}=\frac{Z_{1}(f)}{Z_{2}(f)}\cdot\frac{Z _{4}(f)}{Z_{4}(f)+Z_{0}}\cdot\frac{Z_{0}}{R_{a}}, \tag{4}\]  
where the impedance \(Z_{1}(f)\) to \(Z_{4}(f)\) can be expressed as shown in (5)-(8):  
\[Z_{1}(f)=\frac{R_{a}}{1+i2\pi fR_{a}C_{a}}, \tag{5}\]  
\[Z_{2}=Z_{1}(f)+R_{m}+i2 fL, \tag{6}\]  
\[Z_{3}(f)=\frac{1}{i2\pi fC_{p}}+R_{p}, \tag{7}\]  
\[Z_{4}(f)=(\frac{1}{Z_{2}(f)}+\frac{1}{Z_{3}(f)})^{-1}, \tag{8}\]  
where \(i_{a}\) is the small-signal current across the active region, and \(\rm V_{s}\) represents the small-signal voltage generated by the source.