Conclusion and Outlook

In this work, the performance characteristics of resonant graphene modulators were analyzed and compared to their rectlinear counterparts. It was shown that the resonant enhancement of MD only becomes significant if high IL is tolerated also. This relation was described semianalytically for a single resonator coupled to a waveguide. Devices resulting from inverse design, that are in principle not limited in the number of resonators follow the same limitation, suggesting that the found limitation holds for any resonant modulator with an arbitrary number of resonators and coupling between those resonators. The found limit is valid for a pure EA-modulation that does not affect the real part of the refractive index. Despite graphene exhibiting phase modulation, the effects are considered negligible for the following reasons: $\Im{\sigma}$ (and thus $\Re{n_{eff}}$) has an extremum at $\mu_c=h\nu/2$, located at the center of the switching slope (see ). A small modulation around this point will therefore not lead to significant phase modulation. Secondly, it lowers the system complexity to have broadband modulation, which is less dependent on the relative alignment between the laser line and the resonance frequency. Shifting a broad resonance requires a substantial phase modulation, which is more readily obtained in extended devices, as it accumulates over the propagation length. However, in this work highly compact resonators were considered. Comparing rectlinear and resonant modulators the relative difference in required laser power is found to be less than \qty{4.7}{%} for a thermal noise limited link and less than \qty{85}{%} for a shot noise limited link. For a link affected by thermal noise and strong RIN, the relative improvement is stronger compared to a link only limited by thermal noise. However, for most short-distance single-wavelength links thermal noise is likely the limiting noise term. It was found that resonant modulators have the potential to reduce the laser power and/or switching voltage beyond the aforementioned marginal improvements when local doping variations are considered. For a representative standard deviation in the chemical potential of \sigma_{dop}=\qty{0.1}{eV} these variations have a broadening effect similar in magnitude to the combined effects of a scattering rate of \gamma = \qty{3e13}{s^{-1}} and temperature T=\qty{300}{K}. It was shown that devices, which are small compared to the autocorrelation length $L_{corr}$ of the doping variations eliminate the effect of such variations. $L_{corr}$ was extracted from confocal Raman maps recorded by coworkers and was found to be approximately \qty{450}{nm}. Because this lies in the vicinity of the beam diameter it is controversial, whether these results are limited by the resolution of the measurement system. It is shown, that resolution-limited doping maps extracted with a gaussian beam exhibit a gaussian autocorrelation, while an exponential autocorrelation is observed in measurement. Furthermore, resolution limited measurements underestimate the doping standard deviation. Considering that the measured standard deviation is close to \qty{0.1}{eV} it seems unlikely that the actual variations are substantially higher. Further investigation into the magnitude, length scales, and causes of meso- and macroscopic doping variations is deemed necessary. Going forward a TERS measurement could be performed, which should eliminate doubts regarding the spatial resolution of the measurement system. Moreover, it should be evaluated how strongly the light can be confined by resonators in the planar integrated photonic platform at hand. \section{Implications and future Directions} The presented work has identified how resonant structures can serve to enhance the EA-modulation with graphene. Implications of the planned BEOL integration and the resulting changes in modeling that need to be incorporated are presented in appendix \ref{ap:beol}. Previously it was stated that the extinction in the critical coupling state is "significantly more sensitive to changes in loss than an electroabsorption modulator" Phare et al. (2015). This however does not lead to a significant performance improvement when considering the whole link as demonstrated in section Single Resonator Modulator. Furthermore, operating the modulator in the overcoupled regime (compared to the undercoupled regime) was shown to yield better IL at almost unchanged MD. Which is in contrast to Phare et al. (2015) stating that they overcome the bandwidth vs. modulation efficiency "tradeoff by exploiting the Zeno effect, in which an increase in loss in a coupled resonator increases the system transmission [...]" Phare et al., 2015. This scheme of operation is equivalent to biasing the modulator in the undercoupled state. The mentioned trade-off arises from the thickness of the insulator affecting the RC bandwidth and the modulation efficiency in an opposing way. For a fixed GIG cross-section and modulation voltage, the trade-off is represented by $f$ limiting the MD/IL ratio. This limitation has been investigated in great detail in this work. While no significant benefit of operating in the undercoupled regime was identified, it was found to have the drawback of higher loss when tuned away from critical coupling as is observed from the loss factor representation in figure (b). The primary benefit of resonant modulators has been identified as the reduction in footprint, especially of the active graphene area. As a result, the switching energy can be reduced, while potentially gaining bandwidth (see appendix \ref{ap:bw_experimental}). The higher extinction close to critical coupling yields only marginal gains with respect to the required laser power. However decreasing the graphene footprint can potentially yield further improvements in the on-off ratio, as it eliminates the influence of doping variations on large length scales compared to the device size. Eliminating doping variations becomes particularly relevant as other degradations like the finite scattering rate (mobility) improve with improved graphene handling and cleaning Tyagi et al., 2022.