## 1 Introduction  
To meet the demands of applications in emerging technologies, such as sub-terahertz (THz) radio-over-fiber communication link [1], numerous quasi-optic components, such as phase shifters [2; 3; 4; 5; 6; 7; 8; 9; 10], polarization converter [11; 12; 13], filters [14], phase gratings [15], and polarizers [16], have been reported. In particular, a number of tunable THz devices using liquid crystals (LCs) have received attention [2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17]. For example, an LC spatial THz modulator that can tune THz wavefronts is pivotal in the demonstration of a THz vortex system that can execute orbital angular momentum conversions [17].  
Among LC THz devices, magnetically tunable 2\(\pi\) phase shifters have been shown to work effectively [2]. They are, however, somewhat bulky and are not so easy to use in some applications. Electrically tunable phase shifters are, therefore, attractive. However, early designs require a high driving voltage (larger than 100 V), because they use widely separated lateral electrodes for the lack of electrodes that are transparent in the THz band [3].  
Graphene exhibit outstanding electrical conductivity and high transparency in the THz frequency range [4,8,10]. Several groups have demonstrated THz phase controllers with graphene electrodes [4,8,10]. [PERSON] and et al. achieved the phase shift, which is 10.8 degree with a 50 \(\upmu\)m LC cell [8]. By applying the randomly aligned LCs cell with graphene electrodes T. [PERSON] and et al. demonstrated a phase shift of 0.11 rad at 1.5 THz [10]. A phase shift of more than 90 degrees, which is approximately ten times better than that reported in previous designs using LCs and graphene electrodes, was accomplished by our group [4].  
Besides, graphene, poly(3,4-ethylenedioxythiophene)/poly(styrenesulfonate) (PEDOT/PSS) films [11], and subwavelength metallic gratings [12] have been employed by [PERSON] et al. in electrically tunable THz polarization converters based on twisted nematic (TN) LC cells as transparent electrodes. The three main advantages are low operating voltages, high transmittance, and simple structures.  
Previously, we have shown that indium-tin-oxide (ITO) nanomaterials exhibit outstanding transmittance in the THz frequencies [18, 19, 20]. Their conductivity and DC mobility are also comparable to the sputtered ITO films [18, 19, 20]. Later, we also demonstrated the quarter-wave operation or a phase shift of around \(\pi/2\) at 1.0 THz with ITO nanomaterials as transparent electrodes [5]. Interestingly, ITO nanow whiskers have the additional functionality of aligning liquid crystal molecules [5].  
Due to the long wavelength of THz waves, thick LC cells are unavoidable. Lest the devices become too unwieldy, a sandwiched LC cell structure for THz application was proposed and demonstrated by us in magnetically tuned phase shifters [2] and patented [21]. With a sandwiched LC cell structure, it is possible to achieve a phase shift of 2\(\uppi\) at 1.0 THz with a relatively low voltage. Preliminary results were reported by us in a conference paper [22].  
In this work, we present in more detail the experimental and theoretical aspects of the device [22]. The frequency-and voltage-dependent characteristics of the THz phase shifters are described. In particular, a theoretical formulism that can handle thick (approximately 1 mm) LC cells is presented. Pretilt angles, which were measured by novel THz techniques, were taken into account. Experimental data are in good agreement with this theoretical formulism.