## 6 Results and Discussion  
The model validation and parameter extraction are divided into two parts: One is for the parasitic circuit and the other is for the circuits of carrier and photon dynamisms. For parameter extraction of the parasitic circuit, the S\({}_{11}\) data are utilized. For the circuits of carrier and photon dynamisms, the measurement results of output optical powers under different bias currents are utilized.  
The experimental setup for S\({}_{11}\) measurement is shown in Figure 4. A bias-tee is employed to combine the bias current and the small signal from a high-precision time-domain reflectometer (TDR). After the bias-tee, an 850 nm VCSEL chip is driven. S\({}_{11}\) data are measured by the TDR module. The self-wire-bonded VCSEL chip under test is manufactured by II-VI Incorporated.  
The measurement for I-V-P data (I for injection current; V for VCSEL voltage; P for optical power) is performed automatically using the computer program, and the experimental schematic diagram for I-V-P measurement is shown in Figure 5. The DC power supply is controlled by a personal computer (PC) to change the output voltage. A resistor of 462 \(\Omega\) is used to protect the VCSEL, which voltage is recorded by a multi-meter. Therefore, the current passing through the resistor and the VCSEL can be obtained using [PERSON]'s law. The fiber couples the light of the VCSEL and is connected to an optical power meter. The optical power meter data is recorded by a PC. Based on the I-V-P data, the VCSEL temperatures under different bias currents are obtained using Equation (4) with \(R_{th}=2\times 10^{3}\) K/W. It needs to be pointed out that the differential term in the right-hand side of Equation (4) is equal to zero for the steady state.  
The expression for the magnitude of S\({}_{11}\) is shown below based on Equation (1),  
\[H(f)=20\cdot\text{log}_{10}\left|\frac{Z_{in}(f)-Z_{0}}{Z_{in}(f)+Z_{0}}\right| \tag{36}\]  
where \(Z_{0}=50\)\(\Omega\) is the measurement system impedance.  
Figure 4: The experimental setup for S\({}_{11}\) measurement.  
Figure 3: The compact circuit model for carrier and photon dynamisms of the MQW VCSEL.  
Curve fitting of large-signal DC responses is utilized to extract the parameters for the rate equations which describe the VCSEL electro-optical dynamisms. With the obtained VCSEL temperatures, the coefficients b0-b4 are extracted by minimizing the sum of the squares of the residuals between the simulation and measurement results of the leakage current. \(T_{0}\) in Equation (23) is chosen as 300 K. Under steady states, rate equations are simplified into the non-differential nonlinear equations. The temperature-independent parameters of nonlinear equations are extracted by fitting the I-P curve. Table 3 presents parameters of the rate equations, which are extracted by curve fitting. The simulated I-P curve and the experimental result are depicted in Figure 6b, and they match with each other very well. Based on the extracted parameters in Tables 1-3, the equivalent circuit model is realized in Cadence Tool Suits (Virtuoso Environment 5.10.41). The 25 Gbps PRBS-7 transient responses under 6 mA bias current for both simulation and measurement are shown in Figure 7. The peak-to-peak jitter in the experiment is 11.3 ps, while the simulation result is 11.4 ps. The measured vertical and horizontal eye-opening ratios are 0.55 and 0.71, while the simulated ratios are 0.58 and 0.70, respectively. The measurement and simulation show satisfactory match, proving that the proposed model have strong ability to simulate the VCSEL performance.