## 3 Results and Discussion
### Multi-Mode Interferometer (MMI) Splitter for Compact LC-Based Optical Modulator  
As shown in Figure 1, a multi-mode interferometer (MMI) splitter is needed to transfer light from one SiN\({}_{\rm x}\) waveguide into two separate LC-based phase shifters integrated on the same SiN\({}_{\rm x}\) layer. As the SiN\({}_{\rm x}\) waveguide thickness (\(H_{SiNx}\)) value should be consistent over the entire structure to allow for low fabrication complexity, and as larger \(H_{SiNx}\) values significantly decrease the obtainable phase shift (Figure 3b), we focus on an \(H_{SiNx}\) of 200 nm. The inset of Figure 4a shows the intensity profile of the optical propagation inside the 200 nm thick SiN\({}_{\rm x}\) MMI at an optical wavelength of 1.55 \(\upmu\)m. After optimization, we arrive at \(W_{MMI}\) = 11 \(\upmu\)m, \(L_{MMI}\) = 66 \(\upmu\)m, \(G\) = 2.6 \(\upmu\)m, and \(W_{SiNx,\ MMI}\) = 3 \(\upmu\)m. It should be noted that the tolerance of an MMI device to wavelength variation depends essentially on the width of the access waveguide, \(W_{SiNx,MMI}\); a wider value of \(W_{SiNx,MMI}\) will relax the optical wavelength and dimension variation constraints [16]. Indeed, SiN\({}_{\rm x}\) MMIs have been previously studied because of their benefits of a lower refractive difference with SiO\({}_{2}\) compared to Si and reduced sensitivity to temperature changes [28; 29; 30]. Our designs are based on our previous investigation indicating that a relatively large access waveguide width reduces the refraction caused by a narrower entrance section and allow them to properly cover the entire wavelength of interest [16]. Therefore, with a relatively large value of \(W_{SiNx,\ MMI}\) = 3 \(\upmu\)m, the 200 nm thick SiN\({}_{\rm x}\) MMI can maintain a high optical power transmission of around 98% (\(\sim\)0.09 dB optical loss) over the entire C-band wavelength range, as shown in Figure 4a. On the contrary, as previously discussed (Figure 3a), a larger value of the SiN\({}_{\rm x}\) waveguide width (\(W_{SiNx}\)) is detrimental to the obtainable phase-shifting performance of the LC-integrated SiN\({}_{\rm x}\) waveguide structure. Therefore, a suitable tapering structure is required to transfer the optical mode from the 3 \(\upmu\)m wide SiN\({}_{\rm x}\) waveguide at the MMI outputs to a narrower one necessary in the LC-based phase shifter section. Figure 4b shows the linear SiN\({}_{\rm x}\) taper length (\(L_{taper,\ SiNx}\)) values needed to achieve 100% optical power transmission of the optical mode from the 3 \(\upmu\)m wide SiN\({}_{\rm x}\) waveguide at the MMI outputs to a SiN\({}_{\rm x}\) waveguide with different values of \(W_{SiNx}\) used at the phase shifter. To obtain \(\sim\)100% optical power transmission together with a relatively compact taper (\(L_{taper,\ SiNx}\)\(\leq\) 50 \(\upmu\)m), the value of \(W_{SiNx}\) should not be lower than 900 nm (light blue upward-pointing triangle.) Interestingly, a 900 nm wide SiN\({}_{\rm x}\) waveguide results in only slightly lower phase-shifting performance than a 500 nm wide one (Figure 3a). As a result of the studies in this section, we conclude that a wideband and low-loss MMI is possible using a relatively thin SiN\({}_{\rm x}\) waveguide (\(H_{SiNx}\)\(\sim\)200 nm), which is required to have good phase-shifting performance in the LC-based section; however, a compromise needs to be made in terms of the SiN\({}_{\rm x}\) waveguide width value (\(W_{SiNx}\)\(\sim\)900 nm) in order to obtain not only competitive phase-shifting performance but also overall device compactness.