# Parasitic Parameters Extraction of High-Speed Vertical-Cavity Surface-Emitting Lasers
## 3 Results and Discussion  
### Parameter Extraction and Analysis  
Figure 5a shows the \(S_{11}\) data fit of our fabricated device. As seen in the figure, the data fitting curve has a relatively excellent fit to the measurement curves, and the fitting error is still kept to three decimal places despite the noise in the measurement data at high frequency.  
It is interesting to know how the \(-3\) dB modulation bandwidth \(f_{-3\text{ dB}}\) and the values of the small-signal equivalent circuit elements change with the VCSEL oxide aperture diameter. Thus, we fabricated and utilized 3, 5, 7, and 9 \(\upmu\)m oxide aperture diameter 850 nm VCSELs and extracted their parasitic data. Figure 5b shows the \(-3\) dB bandwidth \(f_{-3\text{ dB}}\) of different oxide aperture VCSELs at different bias currents. As shown in the figure, the \(f_{-3\text{ dB}}\) of the same oxidation aperture device increases with the current in a logarithmic curve and decreases as the diameter of the oxide aperture increases.  
Figure 6a-f show the polynomial fitting curve of extracted values and 95% fit confidence bounds of \(R_{p}\), \(C_{p}\), \(C_{a}\), \(R_{a}\), \(R_{m}\), and \(L\), respectively, as a function of different bias currents for the 3, 5, 7, and 9 \(\upmu\)m 850 nm oxide aperture diameter VCSELs at 25 \({}^{\circ}\)C. From Figure 6a,b, we can observe that the values of \(R_{p}\) and \(C_{p}\) decrease with increasing cur  
Figure 5: (**a**) Real and imaginary parts of the measured \(S_{11}\) spectra of a 3 \(\upmu\)m oxide aperture VCSEL. The measurement is performed at room temperature over a frequency range of 40 GHz at 3 mA, 8 mA, and 12 mA. Both parts are fitted using the calculated analytical function from the equivalent circuit model (dashed curve), and (**b**) the \(-3\) dB frequency \(f_{-3\text{ dB}}\) of the 3, 5, 7, and 9 \(\upmu\)m 850 nm oxide aperture VCSELs at different bias currents.  
rent. Additionally, the larger the mesa diameter, the smaller \(R_{p}\) and \(C_{p}\). According to Equation (4), \(C_{p}\propto 1/f_{-3\,\mathrm{dB}}\). The different amounts of variation in the \(C_{p}\) value of the device at the same current are related to the different filling areas due to the different mesa diameters of the device and the different filling thicknesses during the device fabrication process. As shown in Figure 6c, the value of \(C_{a}\) increases with increasing current and oxide aperture diameter. The oxide layer capacitance can be calculated by the plate capacitor formula [25], and the value of \(C_{a}\) is directly proportional to the oxidized area.  
As shown in Figure 6d,e, the resistance \(R_{a}\), \(R_{m}\) decreases with increasing current and mesa diameter. Figure 6f shows that the \(L\) of the device decreases with increasing current and decreasing oxide aperture diameter. According to equation \(L=\left(2\pi f\sigma\delta\right)^{-1}\), we know that \(L\propto 1/f\). \(\sigma\) is the electrical conductivity, \(\delta\) is the skin depth, and \(f\) is the frequency. From Equation (4), it is known that the device bandwidth increases as \(L\) decreases. From Figure 6 can be seen that 95% fit confidence bounds are quite narrow, and the polynomial fitting curves are obeying the rules of physics, apart from the abnormal data of 9 \(\upmu\)m oxidation aperture for \(R_{p}\) and \(C_{p}\) at a 3 mA current. The abnormal in individual data may be caused by test errors.