## 3 Results and Discussion
### Modulation Efficiency  
It is also important to consider the magnitude of the electric field across the LC-filled region. As indicated in [33], an electric field of \(\sim\)0.2 \(\times\) 10\({}^{4}\) V/cm would be required to obtain a nematic LC 4-n-pentyl-4\({}^{\prime}\)-cyanobiphenyl (5 CB) molecular orientation from parallel (0\({}^{\circ}\)) to perpendicular (90\({}^{\circ}\)). Moreover, to take into account the additional electric field needed to overcome the mechanical anchoring strength of the alignment layer [9], a total electric field of \(\sim\)0.4 \(\times\) 10\({}^{4}\) V/cm would be necessary to have a perpendicular LC molecule orientation. Figure 6b shows the minimum electric field obtained in the LC-filled trench region with different reverse bias voltage values between the two aluminum (Al) contacts placed between the LC-filled trench at various distances between the Al contact and the trench, \(S\); as in Figure 6a, \(W_{SiNx}\)-900 nm, \(H_{SiNx}\)-200 nm, \(W_{LC}\)\(\sim\)1500 nm, \(H_{LC}\)-600 nm, \(D\)\(\sim\)25 nm, \(n\)-2, and \(\lambda\)-1.55 \(\upmu\)m, according to the results shown in Figures 3-5. As shown in Figure 6b, lower values of \(S\) lead to higher values of the obtainable minimum electric field in the LC-filled trench region. Significantly, a reverse bias as low as \(\sim\)1 V (resp. 2 V) can be used to obtain a perpendicular LC molecule orientation using a moderate \(S\) distance value of 0.5 \(\upmu\)m (resp. 1 \(\upmu\)m), which is compatible with the available drivable voltage (e.g., \(\leq\)2 V [34]) from Si-based complementary metal-oxide semiconductor (CMOS) circuits. Figure 6c shows that the electric field can be effectively and uniformly applied across the LC-filled trench region with 1 V. It is important to note that the Al contact needs to be placed slightly lower than the LC-filled trench region by \(\sim\)0.2 \(\upmu\)m, as indicated in Figure 6a, in order to have a relatively well distributed electric field across the LC-filled trench. Last but not least, to determine the modulation performance of an optical modulator based on an LC phase shifter on a SiNx waveguide, a common Figures of Merit (FoM) when using  
Figure 5: The optical power transmission efficiency from the SiNx waveguide to the LC-based phase shifter structure at different LC-filled trench taper length (\(L_{Laper,LC}\)) values and a taper tip width (\(t_{LC}\)) of \(t_{LC}\) = (**a**) 50 nm, (**b**) 100 nm, (**c**) 200 nm, and (**d**) 300 nm. According to the inset of Figure 5b, an additional optical loss of \(<\)0.5 dB can be maintained as long as the length of the air void does not exceed 5 \(\upmu\)m.  
MZI is the modulation efficiency, \(V_{\pi}L_{\pi}\), which is the product between the voltage and the length needed to obtain a \(\pi\) phase shift [35]. \[V_{\pi}L_{\pi}=V_{\pi}\cdot\frac{\lambda}{2\Delta n_{eff}(V_{\pi})}\]  
where \(\Delta n_{eff}(V_{\pi})\) is the change in the effective refractive index of the phase shifter structure between no bias voltage (LC molecules oriented in parallel (0\({}^{\circ}\))) in one arm of the MZI and \(V_{\pi}\) (LC molecules oriented perpendicularly (90\({}^{\circ}\))) in the other arm. Figure 6d shows that a \(V_{\pi}L_{\pi}\) of -0.3 \(\times\) 10\({}^{-2}\)\(V\cdot cm\) can be expected; this is one order of magnitude less than that typically obtainable with forward-biased silicon Mach-Zehnder modulators [36; 37], and it is consistent with previous data reported in a recent demonstration of an LC-based optical modulator [9], indicating that our simulation is well calibrated.