## 4 Conclusions  
In conclusion, we thoroughly investigated the potential of employing a SiN\({}_{\rm x}\) waveguide platform to enable the use of liquid-crystal-based phase shifters for on-chip optical modulators using optical 3D-FDTD and electrical simulations. Contrary to previous demon  
Figure 7: FDTD simulation of the entire structure from the SiN\({}_{\rm x}\) input to the SiN\({}_{\rm x}\) output waveguide, including two MMIs, the linear tapering of the LC section, and the two arms of the LC-based phase shifter for **(a)** ON-mode and **(b)** OFF-mode operations at an optical wavelength of 1.55 \(\upmu\)m. (c) ER and IL of the LC-based MZI optical modulators with LC section lengths (\(L_{LC}\)) of 30 and 50 \(\upmu\)m at different wavelength values over the C-band region. Significantly, a longer phase shifter length (50 \(\upmu\)m) can contradictorily worsen the ER value, as the phase shifter length required to obtain a \(\pi\) phase shift could be as low as a few tens of microns; therefore, it is necessary to carefully select the phase shifter length of the LC-based MZI optical modulator. strations, we chose to focus on using only one SiN\({}_{\rm x}\) waveguide layer for the entire device in both the passive and active regions in order to achieve a potentially simpler fabrication process. The entire structure of the liquid-crystal-based MZI optical modulator, comprising mainly MMI splitters, linear tapers to transfer the optical mode between different structures, and LC-based phase shifters integrated on a SiN\({}_{\rm x}\) waveguide, was systematically investigated. The trade-offs between several important parameters in each part were discussed in order to holistically consider the obtainable phase-shifting performance, single-mode propagation, compact integration, and fabrication tolerance. Based on the investigations, out of eleven studied parameters, we project that a wideband, relatively compact, high-extinction ratio, and low-voltage SiN\({}_{\rm x}\) waveguide-integrated MZI optical modulator can be obtained while using only one SiN\({}_{\rm x}\) waveguide layer for both the passive and active regions by carrying out the following: (1) employing a relatively thin SiN\({}_{\rm x}\) layer (\(H_{SiNx}\)) with a moderately high refractive index value (\(n\)) of stoichiometric Si\({}_{3}\)N\({}_{4}\) to enhance optical overlapping with the LC region; (2) using a relatively large MMI access waveguide width (\(W_{SiNx,MMI}\)) to obtain wideband and low-loss operation; (3) utilizing a narrow width for the tip of the LC-filled trench taper (\(t_{LC}\)) to obtain a low-loss and comparable optical coupling in both arms of the MZI with LC molecules aligned parallelly (0\({}^{\circ}\)) and perpendicularly (90\({}^{\circ}\)) to avoid a spurious intensity modulation that can degrade the optical modulation; (4) maintaining compromising values of the SiN\({}_{\rm x}\) waveguide width (\(W_{SiNx}\)), LC-filled trench width (\(W_{LC}\)), and height (\(H_{LC}\)) to retain device compactness and single-mode operation; (5) carefully positioning the metal contacts with respect to the LC-filled trench (\(S\)) to ensure a homogeneous and high field in the LC trench with a bias voltage potentially as low as 1 V; and (6) attentively determining the phase shifter length of the LC-based MZI optical modulators to obtain proper destructive and constructive interference at the modulator outputs, as a too-long phase shifter length can contradictorily worsen the optical modulation. Via well-calibrated finite-difference time-domain techniques, together with electrical simulation, this work contributes toward obtaining a deeper and holistic understanding of the potential, limitations, and design rules to achieve a competitive LC-based Mach-Zehnder interferometer (MZI) optical modulator for compact, wideband, and low-voltage on-chip optical modulation. Conceptualization, P.C. ; formal analysis, P.J., N.C., W.T. and P.C. ; investigation, P.J. ; data curation, P.J. ; writing--original draft preparation, P.J. and P.C.