## 3 Results and Discussion
### Linear Tapering of the LC Section for Optical Integration  
As also shown in Figure 1, to couple light from the 900 nm wide SiN\({}_{\rm x}\) waveguide (\(W_{SiNx}\)\(\sim\)900 nm) to the LC-based phase shifter on the same SiN\({}_{\rm x}\) waveguide layer, a linear tapering of the LC section (\(L_{taper,LC}\)) is required to accommodate optical integration between the two sections due to the mode size and effective index mismatch. Figure 5a-d show the optical power transmission efficiency from the SiN\({}_{\rm x}\) waveguide to the LC-based phase shifter structure at different LC-filled trench taper length (\(L_{taper,LC}\)) and taper tip width (\(t_{LC}\)) values. According to the knowledge obtained from the results shown in Figures 3 and 4, the values of other important parameters used in this investigation are as follows: \(H_{SiNx}\)\(\sim\)200 nm, \(W_{LC}\)\(\sim\)1500 nm, \(H_{LC}\)\(\sim\)600 nm, \(D\)\(\sim\)25 nm, \(n\)\(\sim\)2, and \(\lambda\)\(\sim\)1.55 \(\upmu\)m. Significantly, the investigation of the optical integration performance has to be considered for LC molecule alignments both parallel (0\({}^{\circ}\)) and perpendicular (90\({}^{\circ}\)) to the SiN\({}_{\rm x}\) waveguide propagation direction, because the intensity modulation capability of the optical signal is obtained from the difference in the phase-shifting effect (0\({}^{\circ}\) and 90\({}^{\circ}\)) between the two arms of the Mach-Zehnder interferometer (MZI). In Figure 5, as expected, larger \(L_{taper,LC}\) values lead to higher optical power transmission efficiency values, and an \(L_{taper,LC}\) of \(\sim\)50 \(\upmu\)m is required to obtain the maximum optical transmission efficiency from the SiN\({}_{\rm x}\) waveguideto the LC-based phase shifter structure. Interestingly, we find that using a lower taper tip width value (\(t_{LC}\)-50 or 100 nm) can help make the optical transmission performance of the integration scheme become more comparable between the parallel (0\({}^{\circ}\)) and perpendicular (90\({}^{\circ}\)) LC molecular alignments despite the difference in the effective index values between the two cases, e.g., as shown in Figure 2b,c. This is crucial, as too much variation in the optical intensity levels between the two arms of an MZI could lead to a spurious intensity modulation that can degrade the optical modulator performance [31]. Additionally, it should be noted that a trench width of 50 or 100 nm at the taper tip (\(t_{LC}\)) with a trench height \(H_{LC}\) of 600 nm would result in a trench aspect ratio of 6-12, which is still compatible with available advanced high-aspect-ratio SiO\({}_{2}\) etching processes [32].  
Figure 4: (**a**) Intensity profile of the optical propagation inside the 200 nm thick SiN\({}_{x}\) MMI (the same thickness value obtained in Figure 3b) at an optical wavelength of 1.55 \(\upmu\)m. The MMI can maintain a high optical power transmission of around 98% (-0.09 dB optical loss) over the entire C-band wavelength range. (**b**) Optical power transmission from a 3 \(\upmu\)m wide SiN\({}_{x}\) waveguide at the MMI outputs to a narrower SiN\({}_{x}\) waveguide necessary at the phase shifter section at different linear SiN\({}_{x}\) taper length (\(L_{Laper,\ SiNx}\)) values. To obtain \(\sim\)100% optical power transmission together with a relatively compact taper (\(L_{Laper,\ SiNx}\leq\) 50 \(\upmu\)m), the value of \(W_{SiNx}\) cannot be lower than 900 nm (light blue upward-pointing triangle).