## 5 Results on Small-Signal Modulation Response of EAM-VCSELs
For VCSEL current modulation, the cut-off frequency remains limited by the relaxation oscillation frequency. For the case of small-signal modulation of EAM losses shown in Figure 5b, the cut-off frequency remains beyond 100 GHz for all the cases of different \(\alpha_{EAM}\) considered, except for the magenta curve for \(\alpha_{EAM}=6700\,\mathrm{cm}^{-1}\). The modulation curve is quite flat with features only around the relaxation oscillation frequency. Moreover, the shape of these features considerably  
Figure 4: (color online) Small-signal modulation response of EAM-VCSEL for the case of strong coupling (\(d_{EAM}=0.4972\,\mathrm{\SIUnitSymbolMicro m}\)) and when modulating (**a**) injection current, i.e., \(H_{i}(\
u)\) from Equation (21) and (**b**) EAM losses \(\alpha_{EAM}\), i.e., \(H_{out}(\
u)\) from Equation (18). Different colors correspond to different bias points for the EAM losses \(\alpha_{EAM}\): \(8500\,\mathrm{cm}^{-1}\)(red), \(6700\,\mathrm{cm}^{-1}\) (magenta), \(5100\,\mathrm{cm}^{-1}\) (green), \(3500\,\mathrm{cm}^{-1}\) (blue), \(1900\,\mathrm{cm}^{-1}\) (cyan), \(500\,\mathrm{cm}^{-1}\) (black). The dotted straight lines denote the 3 dB limit for the modulation curves with the same color. Figure 5: (color online) Same as Figure 4 but for the case of moderate coupling between EAM and VCSEL cavities (\(d_{EAM}=0.5\,\mathrm{\SIUnitSymbolMicro m}\)). (**a**) and (**b**) correspond to modulation of injection current, i.e., \(H_{i}(\
u)\) from Equation (21) and EAM losses \(\alpha_{EAM}\), i.e., \(H_{out}(\
u)\) from Equation (18), respectively. depends on the bias \(\alpha_{EAM}\); while it has a well pronounced maximum, for small \(\alpha_{EAM}\) (black and cyan curves), it flattens, transits a region of both maximum and minimum (\(\alpha_{EAM}=3500\,\mathrm{cm}^{-1}\), blue curve) and then remains a sharp minimum only (green, magenta and red curves)). Such profound change of the shape of the modulation curve has been indeed found experimentally in [14] but remained unexplained. Our results therefore, not only provide an explanation for the change of the shape of the modulation curve but give the insight that the high frequency cut-off of about 30 GHz for the EAM-VCSEL structure investigated in [14] is due to electrical parasitics limitation. Finally, Figure 6 presents the EAM-VCSEL small-signal modulation response for the case of weak coupling between the EAM and the VCSEL cavities. The tendencies observed for the previous two cases remain the same. From Figure 6a we see that the modulation cut-off frequency for current modulation is now hardly influenced by the bias EAM absorption and modulation curves for loss modulation becomes quite flat and displays only a minimum at the relaxation oscillation frequency which becomes very shallow with increased EAM absorption. The quite flat response of the modulation curves for different biased \(\alpha_{EAM}\) indicates to the possibility to realize a large-signal modulation response beyond 100 GHz, which we investigate in the next section. It is worthy to note that in our way of calculations we consider that the optical field reaches its steady-state distribution on a time scale that is much shorter than the rest of the time scales (carrier dynamics and electro-optical modulation). This consideration is justified as the coupled-cavity VCSEL is very short (\(\,\mathrm{\SIUnitSymbolMicro m}\) length scale) and, therefore, the roundtrip time is also very short (\(f\)s time scale). Furthermore, there is no significant change of the optical field distribution during modulation and therefore, not many roundtrips are necessary for reaching optical field steady state.