## 5 Results on Small-Signal Modulation Response of EAM-VCSELs  
Figure 4 presents the EAM-VCSEL small-signal modulation response for the case of strong coupling between the EAM and the VCSEL cavities. Figure 4a represents the VCSEL current modulation case, i.e., \(H_{i}(\
u)\) from Equation (21) and Figure 5b represents the loss modulation case, i.e., \(H_{out}(\
u)\) from Equation (18). Different colors correspond to different bias points for the EAM losses as denoted in the figure caption. From Figure 4a we see that the modulation cut-off frequency for current modulation strongly improves as the bias absorption of the EAM is increased: from 13.5 GHz for \(\alpha_{EAM}=500\,\mathrm{cm}^{-1}\) (black curve) to 24 GHz for \(\alpha_{EAM}=8500\,\mathrm{cm}^{-1}\) (red curve). This is due to the fact that as \(\alpha_{EAM}\) is increased, the optical power distribution along the EAM-VCSEL is modified: it is decreased (increased) in the EAM (VCSEL) cavity. Consequently, the confinement factor in the VCSEL cavity and photon density are increased. However, one remains limited by the relaxation oscillation frequency. For the case of small-signal modulation of EAM losses shown in Figure 4b, the relaxation frequency peak is still well pronounced, however its amplitude strongly decreases with \(\alpha_{EAM}\) and the modulation curve is flatter compared to the current modulation case. Consequently, the cut-off frequency is considerably increased, up to about 45 GHz for \(\alpha_{EAM}=500\,\mathrm{cm}^{-1}\) (black curves) and beyond 100 GHz for \(\alpha_{EAM}\geq 1900\,\mathrm{cm}^{-1}\) (all other curves). This, together with the conclusions drawn when comparing the optical power distribution for strong and moderate coupling with and without absorption in the EAM, leads to the expectation that such a high cut-off frequency should be easily achieved for the cases of moderate and weak coupling. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Notation** & **Value** & **Units** \\ \hline Differential gain & \(a\) & \(4\times 10^{-8}\) & \(\upmu\mathrm{m}^{2}\) \\ \hline Transparency carrier density & \(n_{0}\) & \(2.5\times 10^{6}\) & \(\upmu\mathrm{m}^{-3}\) \\ \hline Carrier lifetime & \(\tau_{e}\) & 1 & ns \\ \hline Differential carrier lifetime & \(\tau_{Am}\) & \(\tau_{e}/1.5\) & ns \\ \hline Light group velocity & \(v_{g}\) & \(7.5\times 10^{4}\) & \(\upmu\mathrm{m}/\mathrm{ns}\) \\ \hline VCSEL oxide aperture radius & \(\rho\) & 2.7 & \(\upmu\mathrm{m}\) \\ \hline Gain compression coefficient & \(\epsilon\) & \(2\times 10^{-5}\) & \(\upmu\mathrm{m}^{3}\) \\ \hline Carrier injection efficiency & \(\eta\) & 1 & \\ \hline Injection current & \(l\) & 2.5 & mA \\ \hline \hline \end{tabular}
\end{table} 
Table 2: EAM-VCSEL parameters. Figure 3: (color online) Dependence of **(a)** total absorption \(\alpha_{tot}\) of the EAM-VCSEL, **(b)** its derivative \(d\alpha_{tot}/d\alpha_{EAM}\), **(c)** VCSEL cavity confinement factor \(\Gamma\) and **(d)** its derivative \(d\Gamma/d\alpha_{EAM}\) on the EAM absorption \(\alpha_{EAM}\). Color coding for the three representative cases of different coupling strength between the EAM and VCSEL cavities are the same as for Figure 2.  
Figure 5 presents the EAM-VCSEL small-signal modulation response for the case of moderate coupling between the EAM and the VCSEL cavities. Figure 5a represents the VCSEL current modulation case and Figure 5b--the losses modulation case. Different colors correspond to different bias points for the EAM losses in the same way as for Figure 4. From Figure 5a we see that the modulation cut-off frequency for current modulation is much less influenced as compared to the case of strong coupling (increasing from about 25 GHz to 28 GHz) as the bias absorption of the EAM is increased. This is due to the fact that the changes of \(\alpha_{EAM}\) now impact much less the optical power distribution along the EAM-VCSEL and, consequently, the confinement factor in the VCSEL cavity and photon density.