## 3 Results and Discussion
### Modulation Efficiency
To reliably evaluate the extinction ratio (ER) and overall insertion loss (IL) of the LC-based MZI optical modulator, we perform a 3D-FDTD simulation of the entire structure from the SiN\({}_{x}\) input to the SiN\({}_{x}\) output waveguide, including two MMIs, the linear tapering of the LC section, and the two arms of the LC-based phase shifter integrated with the SiN\({}_{x}\) waveguide, as shown in Figure 7a,b for ON-mode (constructive interference at the output) and OFF-mode (destructive interference at the output) operations at \(\lambda\)-1.55 \(\upmu\)m with an LC section length (\(L_{LC}\)) of 30 \(\upmu\)m. It is important to note that another linear SiN\({}_{x}\) taper length (\(L_{taper,\ SiNx}\) = 50 \(\upmu\)m) is required in the output section to suppress the higher-order mode excited at the 3 \(\upmu\)m wide SiN\({}_{x}\) waveguide at the MMI output and to allow for only the fundamental mode to be transferred to the single-mode 0.9 \(\upmu\)m wide SiN\({}_{x}\) waveguide, at which the output of the entire structure is evaluated. In the 3D-FDTD simulation of the entire structure, a competitive ER of \(\sim\)15 dB and an IL lower than \(\sim\)1 dB are obtained, confirming the potential of a low-voltage LC-based MZI optical modulator for high extinction and low optical loss usage. It is worth mentioning that all optical losses, including material absorption together with the reflection and scattering losses at each optical component and interface, are taken into account in the 3D-FDTD simulation [38]. As each dB loss in optical power requires a dB increase in laser power [39], a low-loss optical modulator plays a crucial role in achieving a low-power optical interconnection with a good signal-to-noise level. Based on Table 1, we can project that a SiN\({}_{x}\) waveguide-integrated LC-based MZI optical modulator with a modulation efficiency (\(V_{\pi}L_{\pi}\)) comparable to that of the previous demonstration could be obtained while using only one SiN\({}_{x}\) waveguide layer for both the passive and active regions. It is important to note that, in the first demonstration work, information on the MMI and integration structure dimensions and the ER/IL values was not available. Significantly, Figure 7c reports the ER and IL values of the LC-based MZI optical modulators with LC section lengths (\(L_{LC}\)) of 30 and 50 \(\upmu\)m at different C-band wavelengths. We find that it is critical to carefully determine the phase shifter length of LC-based MZI optical modulators. A too-long phase shifter length can contradictorily worsen the optical modulation, as the phase shifter length required to obtain a \(\pi\) phase shift can be as low as \(\sim\)15-20 \(\upmu\)m (previous investigation shown in Figure 3g). To illustrate, our analysis shows that an LC-based MZI with a 30 \(\upmu\)m long phase shifter length will instead have a superior optical modulation performance to the one with a 50 \(\upmu\)m long phase shifter length. While the LC-based MZIs with 30 and 50 \(\upmu\)m long phase shifters have identical IL values (pink and purple data points), the ER values are significantly better when using the shorter phase shifter of 30 \(\upmu\)m (destructive interference; blue data points) than when using the longer phase shifter of 50 \(\upmu\)m (constructive interference; green data points). Additionally, regarding the potential device rapidity, it is important to mention that, although the typical response time of LC molecules can be expected to fall within the millisecond range, a response time within ten nanoseconds has been demonstrated to be possible using a twisted-nematic 5 CB LC [40]. It is worth noting that, to couple the investigated SiN\({}_{x}\) waveguide-integrated LC-based MZI optical modulator with an off-chip optical source, a SiN\({}_{x}\) spot-size converter could be employed [41; 42] to enable a low-loss optical interconnection.