## 3 Results and Discussion  
### Liquid-Crystal-Based Phase Shifter Integrated on a SiN\({}_{x}\) Waveguide Structure  
Firstly, in order to gain an understanding of the potential and limitations of an LC-based phase shifter integrated on a SiN\({}_{\mathrm{x}}\) waveguide, as shown in Figure 2a, the effects of seven important parameters, namely, the SiN\({}_{\mathrm{x}}\) waveguide width (\(W_{SiNx}\)), SiN\({}_{\mathrm{x}}\) waveguide thickness (\(H_{SiNx}\)), liquid-crystal-filled trench width (\(W_{LC}\)), liquid-crystal-filled trench height (\(H_{LC}\)), SiO\({}_{2}\) vertical gap between the liquid-crystal-filled trench and SiN\({}_{\mathrm{x}}\) (\(D\)), SiN\({}_{\mathrm{x}}\) refractive index (\(n\)), and optical wavelength in the C-band region (\(\lambda\)), are holistically investigated. As shown in Figure 2b,c, the change in the alignment of LC molecules from parallel (\(0^{\circ}\)) to perpendicular (\(90^{\circ}\)) with respect to the SiN\({}_{\mathrm{x}}\) waveguide propagation direction results in variations in the optical mode effective index values. This will introduce the phase-shifting effect as discussed above; hence, knowledge on the obtainable phase shift dependence on each parameter between \(0^{\circ}\) and other LC orientations from \(15^{\circ}\) to \(90^{\circ}\) is valuable. Figure 1: First half of a Mach–Zehnder interferometer (MZI) optical modulator consisting of a liquid-crystal-based phase shifter integrated on a SiN\({}_{\mathrm{x}}\) waveguide, a multi-mode interferometer (MMI) for a compact optical modulator, and a linear tapering of the liquid crystal (LC) section. In total, 11 parameters are holistically investigated to achieve a competitive LC-based MZI optical modulator with good modulation efficiency over the C-band optical wavelength region. The output part of the MZI is a mirror copy of the first half in order to combine the two arms of the MZI back into a single output waveguide. Contrary to Ref. [9], our investigation focuses on using the same SiN\({}_{\mathrm{x}}\) waveguide layer for the entire device. Figure 3a-g systematically shows the variation in phase-shifting performance with respect to each of the aforementioned parameters at a chosen shifter length of 50 \(\upmu\)m and different LC molecule alignment values; the latter can be controlled via external electric fields and is later discussed. For consistency in every case reported in Figure 3, when the parameters are not varied, the values are as follows: \(W_{SiNx}\)-500 nm, \(H_{SiNx}\)-200 nm, \(W_{LC}\)-1500 nm, \(H_{LC}\)-600 nm, \(D\)-25 nm, \(n\)-2, and \(\lambda\)\(\sim\)1.55 \(\upmu\)m. The data points include only the cases in which the structures support single-mode propagation. As shown in Figure 3a, larger values of \(W_{SiNx}\) tend to decrease the obtainable phase shift values, as the optical mode is increasingly confined in the SiNx waveguide. Interestingly, a too-narrow value of \(\sim\)400 nm for \(W_{SiNx}\) also leads to a reduction in the obtainable phase shift; this is because the optical mode becomes too loosely confined in the SiNx waveguide, and a significant part of the mode interacts with the LC sidewall, as shown in the top inset of Figure 3a [9]. In Figure 3b, as expected, larger values of \(H_{SiNx}\) significantly decrease the obtainable phase shift values because the optical mode is increasingly confined in the SiNx waveguide, as shown in the right insets. Nevertheless, \(H_{SiNx}\) values lower than 200 nm are not possible when attempting to achieve a well-supported optical mode in the SiNx waveguide. In Figure 3c, a larger LC-filled trench width (\(W_{LC}\)) and height (\(H_{LC}\)) both positively increase the obtainable values of the phase shift. Nevertheless, as shown in the inset of Figure 3c, too-wide values of \(\sim\)2500 nm for \(W_{LC}\) lead the structure to support a higher-order optical mode.