## 4 EAM-VCSEL: Structure and Eigenmodes  
Details of the EAM-VCSEL structure designed to operate at a wavelength of 850 nm are listed in Table 1. We based our choice of the structure on the recently reported experimental GaAs/AlGaAs EAM-VCSEL [13] with enhanced modulation bandwidth [14]. In order to make the comparison with more elaborated numerical approach [23], some details as grading layers in the DBRs and the cavity cladding layers are replaced by average refractive index uniform layers and EAM and VCSEL quantum well (QW) and barrier regions are lumped in single uniform layers. We calculate the EAM-VCSEL eigenmodes following the transfer matrix method and imposing the condition that there is no in-coming field to the whole multilayer structure as outlined in [21,24]. However, when considering EAM-VCSEL it is important to take into account the refractive index changes with carrier density in the QW region of the VCSEL cavity and the one of the EAM QW region with \(\alpha_{EAM}\), i.e., the linewidth enhancement factors \(\alpha_{H}^{VCL}\) and \(\alpha_{H}^{EAM}\)[23]. The refractive index change \(\Delta n_{VCL}^{re}\) in the QW regions of the VCSEL when changing carrier density by \(\Delta n\) is  
\[\Delta n_{VCL}^{re}=\frac{dn_{VCL}^{re}}{dn}\Delta n=-\frac{\lambda a}{4\pi} \alpha_{H}^{VCL}\Delta n. \tag{22}\]  
We consider that \(n_{VCL}^{re}\) starts changing as soon as \(n\
eq n_{0}\). The eigenmode and steady-state carrier density calculations are done repeatedly updating QW refractive index accordingly to Equation (22) until \(\Delta n_{VCL}^{re}<1\times 10^{-5}\). The change of the real part of the refractive index of the EAM QW region is given by  
\[\Delta n_{EAM}^{re}=\frac{\lambda}{4\pi}\alpha_{H}^{EAM}\Delta\alpha_{EAM}. \tag{23}\]  
Here, the EAM cavity thickness \(d_{EAM}=0.2125+2\times t_{EAM}\) is taking three values in order to treat the representative cases of strong (\(d_{EAM}=0.4972\,\upmu\)m), moderate (\(d_{EAM}=0.5\,\upmu\)m) and weak (\(d_{EAM}=0.5023\,\upmu\)m) coupling between the EAM and the VCSEL cavities. We illustrate the importance of considering the linewidth enhancement factors in Figure 1 where we present the optical power distributions along the EAM-VCSEL obtained in a similar way as in [21; 24] for strong coupling between the VCSEL and the EAM cavity and \(\alpha_{EAM}=1000\,\mathrm{cm}^{-1}\). The blue curves correspond to the case when the refractive indices of both VCSEL and EAM QW regions are constant, i.e., \(\alpha_{H}^{VCL}=\alpha_{H}^{EAM}=0\). When only the VCSEL active region antiguiding effect is taken into account, i.e., \(\alpha_{H}^{VCL}=2,\quad\alpha_{H}^{EAM}=0\) (green curve), the coupling between the two cavities improves. It also further improves when both linewidth enhancement factors are accounted for--red curve for \(\alpha_{H}^{VCL}=2,\quad\alpha_{H}^{EAM}=1\). In order to self-consistently calculate the impact of antiguiding effect in the VCSEL active region, we consider typical VCSEL parameters for the steady-state solution of EAM-VCSEL rate in Equations (10) and (11), as listed in Table 2.  
In Figure 2, we present the optical power distributions along the EAM-VCSEL for the composed cavity mode situated primary in the VCSEL cavity for these three cases and \(\alpha_{H}^{VCL}=2\), \(\alpha_{H}^{EAM}=1\). The coupling between the EAM and VCSEL cavities is determined by their relative optical thicknesses as demonstrated in [21; 24]. It is the strongest at the designed wavelength when both cavities optical thicknesses are integer numbers of \(\pi\). Increasing \(d_{EAM}\), the EAM cavity is tuned out of resonance with the VCSEL cavity, the optical field distribution becoming more asymmetric and concentrated into the VCSEL cavity--c.f. the blue, green and red curves in Figure 2b. The result is an increased