## 4 Results and Discussions  
Figure 3 shows plots of the experimentally measured transmittances of the three types of phase shifters. The cumulative transmittances for the THz phase shifters of type A, B, and C, reduced due to absorption losses of fused silica, LCs, and ITO NWh's were 20%, 25%, and 31%, respectively. The difference is mainly due to the numbers of substrates and ITO NWh's layers that were employed in the three configurations. The transmittance of \(\pi\)/2 phase shifters using ITO NWh's as electrodes (\(\geq\)75%) is much more superior than that using sputtered ITO film (\(\leq\)1%) [5]. Therefore, in this case of 2\(\pi\) waveplate with multi-cell structure using ITO NWh's as electrodes, transmittance of 20-30% is reasonable and much superior to the previous state-of-art result. Using complex permittivities or refractive indices of ITO NWh's, fused Silica, and LCs, reported in our previous works [18; 19; 20; 24] in a Jones matrix formulation with Fresnel equations, we can estimate the loss caused from different interfaces of the device. The analysis shows that the most serious sources of loss for our device come from the interfaces of fused Silica/ITO NWh's, and LCs/ITO NWh's. On the other hand, losses incurred at air/ fused Silica and LCs/ fused Silica interfaces are slight. Figure 3: Tunable terahertz (THz) transmittance of the phase shifters, type A, B, and C, respectively. In Figure 4, we have plotted phase shift as a function of frequency from 0.2 to 1.0 THz by varying the voltages for the three types of devices. As expected, the measured phase shift changed linearly with frequency for a given bias voltage. For the type-A phase shifter, the slope of linear fits varied from 0.40\(\pi\)/THz to 2.67\(\pi\)/THz as the applied voltage was ramped from 2.12 to 10.61 V (rms). For the type-B and type-C structures, they varied from 0.51\(\pi\)/THz to 2.49\(\pi\)/THz, and 0.60\(\pi\)/THz to 2.59\(\pi\)/THz, as the applied voltages was ramped from 2.40 to 10.61 V (rms) and 5.66 to 21.21 V (rms), respectively. Figure 4: THz Phase shift as a function of THz frequency for five driving voltages, respectively, for devices of (**a**) type A, (**b**) type B, and (**c**) type C configurations. In Figure 5, we have plotted experimentally measured phase shifts as a function of the driving voltage. The theoretical curve according to the model proposed in this study and those use earlier ones that are based on simpler models with approximations. References [3; 8] are also shown. In all cases, more than 360\({}^{\circ}\) degrees of phase shifts was achieved for all three types of devices. Further, Bias voltages for achieving 2\(\pi\) phase shift at 1.0 THz were V (rms) \(\approx\) 2.6, 5.3, 10.3 for types A, B, and C devices, respectively. Thus, the effect of ITO NWhs for assisting the alignment and reorientation of LC molecules is significant. Further, this characteristic for type A device is compatible with CMOS technology. We measured the response time properties of the present ITO NWhs phase shifter. The 20 to 80% rise time and fall time are around 5  
Figure 5: Phase shift as a function of applied voltage for devices of (**a**) type A, (**b**) type B, and (**c**) type C configurations. Overall, the agreement of experimental data with the present model is significantly better than previous ones. To check quantitatively the accuracy of the theoretical models, we define an error function, \(\Delta=\Sigma\left|\,\varLambda_{\mathrm{j,theo}}-\varLambda_{\mathrm{j,exp}}\,\right|\), where \(j\) equals 1, 2, 3,.... The parameters, \(\varLambda_{\mathrm{j,exp}}\) and \(\varLambda_{\mathrm{j,theo}}\), represent the experimental and theoretical phase shift when a bias voltage, \(V_{\mathrm{j}}\), was applied. For the case of data shown in Figure 5a for type A and employing the theoretical model that is described in Refs. [3; 8], and this work, \(\Delta\) were found to be 8.19, 8.38, and 6.51, respectively.