## 1 Introduction  
Extensive research on liquid crystal (LC) materials has revealed their profound electro-optical properties, including birefringence behaviors [1], making them promising for various photonic applications [2; 3; 4]. The electro-optical properties of LC materials are involved in refractive index tuning between ordinary and extraordinary index values by external electric fields, causing the rotational motions of the LC molecules. Thanks to this capability, LC materials have been applied in photonics to achieve, for example, optical switching [5; 6], tunable filters [7], and phase shifters [8; 9; 10; 11; 12]. As pointed out in [11], optical phase shifters are critical elements in various devices in which the required response times can be varied from a few microseconds to a few milliseconds. In the latter case, phase shifters prioritize device compactness and a low driving voltage over speed, and LC-based phase shifters are more favorable than thermo-optic-based and pin junction-based devices in terms of device size and/or power dissipation. However, while several previous works on LC-based waveguide phase shifters have focused on the silicon (Si) waveguide platform [7; 8; 11; 12], SiN\({}_{\rm x}\)'s benefits of a relatively large transparency window, compatibility with silicon CMOS processes, and a moderate refractive index contrast with SiO\({}_{2}\) can be advantageous in terms of fabrication tolerance and a high optical power handling capability for wideband operation [13; 14; 15; 16; 17]. Moreover, although state-of-the-art work used a high-aspect-ratio SiN\({}_{\rm x}\)structure to achieve a very low propagation loss [18], relatively thick SiN\({}_{\rm x}\) waveguides can also lead to a low-loss propagation loss for an on-chip application [19]. Therefore, recently, the integration of LC materials into amorphous materials such as silicon nitride (SiN\({}_{\rm x}\)) has received great attention, although it could still be considered as being in its initial stages [9; 10; 20; 21]. [PERSON] et al. [10] and [PERSON] et al. [20] recently reported the integration of LC materials as a top cladding over a SiN\({}_{\rm x}\) waveguide for a phase shifter based on the electro-refraction effect. Nevertheless, to further improve performance, it was stated that the optimization of the LC around the SiN\({}_{\rm x}\) waveguide, to improve the optical mode overlap, as well as the optimization of the LC configuration and electrode distances, would be required [20]. Moreover, the increased insertion loss due to the abrupt change in the refractive index between the waveguide and LC interface needed to be addressed [10]. [PERSON] et al. [9] subsequently reported the integration of LC materials with SiN\({}_{\rm x}\), employing two different layers of SiN\({}_{\rm x}\) waveguides for passive optical components (waveguide and multi-mode interferometer (MMI)) and the active phase shifter region with a liquid-crystal-filled trench; therefore, an adiabatic escalator was required to vertically transfer the optical mode between the two SiN\({}_{\rm x}\) waveguide layers and ensure low insertion loss. In particular, Ref. [9] presented one of the first prototypes of a low-loss SiN\({}_{\rm x}\) waveguide-integrated MZI optical modulator. Additionally, [PERSON] et al. [21] recently reported an LC-based thermo-optics effect phase shifter using joule heating with an electrical current injection; a racetrack structure was employed to achieve highly temperature-sensitive operation. In this paper, contrary to [9], to investigate the potential of employing a SiN\({}_{\rm x}\) waveguide platform with a potentially simpler fabrication process for low-loss LC-based phase shifters and MZI optical modulators, we focus on achieving such an optical device using the same SiN\({}_{\rm x}\) waveguide layer for the entire device, including both the passive and active regions.