## 3 Results and Discussion
### Liquid-Crystal-Based Phase Shifter Integrated on a SiN\({}_{x}\) Waveguide Structure
Additionally, as shown in the inset of Figure 3d, too-thick values of \(\sim\)700 nm for \(H_{LC}\) push the optical mode to be largely confined in the upper LC region; this could potentially make it highly difficult to integrate the optical structure with a lower input SiNx waveguide, which is later discussed. In Figure 3e, expectedly, a smaller gap between the liquid-crystal-filled trench and the SiNx waveguide (\(D\)) helps to increase the obtainable phase shift values, as a higher portion of the optical mode is in the LC region instead of the SiO2 gap. Taking into account the fabrication uniformity of the modern CMOS process for photonics of \(\sim\)10 nm [25; 26; 27], a \(D\) value of 25 nm is preferable, as it can render a more relaxed fabrication tolerance together with phase-shifting performance comparable to that when using a \(D\) of 10 nm. The phase-shifting performance presented in Figure 3a-e is based on a refractive index (\(n\)) of SiNx-2 (stoichiometric SiNx); however, the \(n\) values of SiNx can be conveniently increased if the SiH4 and N2 gas ratio becomes larger during the deposition process [13]. As shown in Figure 3f, using SiNx with higher \(n\) values lead to a significant decrease in the obtainable phase shift, as the optical confinement becomes stronger in the  
Figure 2: (**a**) A cross-sectional schematic view of the liquid-crystal-based phase shifter section comprising the liquid-crystal-filled trench and SiNx waveguide. To thoroughly investigate the structure, the SiNx waveguide width (\(W_{SiNx}\)), SiNx waveguide thickness (\(H_{SiNx}\)), liquid-crystal-filled trench width (\(W_{LC}\)), liquid-crystal-filled trench height (\(H_{LC}\)), vertical gap between the liquid-crystal-filled trench and SiNx (\(D\)), SiNx refractive index (\(n\)), and optical wavelength (\(\lambda\)) in the C-band region are studied, allowing for an understanding of the potential and limitations of the phase-shifting performance at a given shifter length to be obtained. The optical guided fundamental quasi-TE mode in the liquid-crystal-based phase shifter when the liquid crystal molecules are aligned (**b**) perpendicular (90\({}^{\circ}\)) and (**c**) parallel (0\({}^{\circ}\)) to the SiNx waveguide propagation direction. SiN\({}_{\rm x}\) waveguide, leading to a smaller optical overlap with the LC region, as shown in the right inset. Last but not least, Figure 3g shows that the phase-shifting performance over the entire C-band wavelength region of the LC-based phase shifter integrated on the SiN\({}_{\rm x}\) waveguide structure is better in the lower wavelength values according to the LC molecular properties [22]. To summarize, the values of \(W_{SiNx}\), \(H_{SiNx}\), \(W_{LC}\), \(H_{LC}\), \(D\), and \(n\) need to be carefully considered with a view to obtaining competitive phase-shifting performance while maintaining single-mode propagation in the LC-based phase shifter section (\(W_{SiNx}\), \(H_{SiNx}\), \(W_{LC}\), and \(n\)), the ability to be later integrated with input SiN\({}_{\rm x}\) waveguides (\(H_{LC}\)), and relaxed fabrication tolerance (\(D\)). Figure 3: Phase-shifting performance of the liquid-crystal-based phase shifter integrated on a SiN\({}_{\rm x}\) waveguide structure with respect to (**a**) SiN\({}_{\rm x}\) waveguide width, (**b**) SiN\({}_{\rm x}\) waveguide thickness, (**c**) liquid-crystal-filled trench width, (**d**) liquid-crystal-filled trench height, (**e**) SiO\({}_{2}\) vertical gap between the liquid-crystal-filled trench and SiN\({}_{\rm x}\), (**f**) SiN\({}_{\rm x}\) refractive index, and (**g**) optical wavelength in the C-band region.