## 5 Modulators
### Slow Light Modulators  
To increase the modulation efficiency, the use of slow light structures combined with any given refractive index change mechanism allows a higher level of interaction between light and matter. This enhancement is directly proportional to the increase in group index (or decrease in group velocity) with respect to conventional waveguides. Slow light modulators are far more complex than their conventional (rib-waveguide-based) counterparts. Indeed, to minimize unwanted reflections and prohibitive insertion loss, slow light modulators require additional efforts to achieve smooth optical transitions between \"fast-light\" (rib/strip-waveguides) and \"slow-light\" sections (corrugated waveguide). Figure 17 illustrates the improvement of using adiabatic tapers generated by using a Blackman apodization function. This simultaneously reduces the Fabry-Perot resonances produced by the reflecting input/output of the slow light waveguide and losses at moderate group index values (up to 20) with respect to the unapodized version of the waveguide.  
The modulation efficiency enhancement is directly proportional to the slow down factor, i.e., the ratio of the group index in the slow light region and that in the fast light region (s = n\({}_{\text{g,slow light}}\)/n\({}_{\text{g,fast light}}\)). The maximum experimentally measured group index (n\({}_{\text{g}}\) = 20) here leads to a slow down factor of 5, which enables a size reduction of the modulator by the same factor, while the losses follow a linear or even sub-linear [46] trend versus group index. Based on the efficiency of the conventional modulator, the slow version would be 5 times shorter (186 \(\upmu\)m) with similar losses, provided that a suitable tapering section such as the one proposed in Figure 17 is used.  
Figure 16: (**a**) An optical micrograph of the push-pull modulator and (**b**) DC characterization results. The extinction ratio is \(>\)10 dB across 1270-1350 nm range and NRZ OOK eyes diagrams with (**c**) 1.6 V\({}_{\text{PP}}\), (**d**) 2.3 V\({}_{\text{PP}}\) and (**e**) 3.2 V\({}_{\text{PP}}\) in a differential driving scheme (the drive voltage applied on each arm).  
Moreover, it is of paramount importance that the optical group velocity matches the electrical group velocity. This mismatch may significantly alter the E/O bandwidth in certain conditions (note Section 6.1). While the slowdown of the group velocity of the propagating light is achievable with properly designed slow light structures, the slowdown of the group velocity of the electrical wave is not as straightforward. To solve this issue, the electrical group index must match the optical group index, i.e., both electrical and optical waves must travel at the same speed in a travelling wave configuration. Through the combination of inductive and capacitive elements, a microwave index up to 11.6 has been demonstrated at 40 GHz and beyond while keeping the impedance value close to 50 \(\Omega\). This value is the largest to date obtained in planar transmission lines [47].  
Overall, we have shown various aspects of slow light modulators and found solutions for achieving low losses, low Fabry-Perot resonances via adiabatic slow-light tapers, as well as slow-wave electrodes to keep benefiting from the high slow light modulator efficiency at high frequencies. The fabrication of fully integrated slow light modulators is currently in the final stage.