# Electrically Tunable Hydrogen-Bonded Liquid Crystal Phase Control Device
## 2 Materials and Methods  
Figure 1 shows the molecular structure of the hydrogen-bonded LC. This material exhibited a thermotropic liquid crystalline phase due to the intermolecular hydrogen bonding between two identical molecules. Hydrogen bonding has an important role in supramolecular LCs [34]. In this study, a hydrogen-bonded liquid crystal (LC1) was introduced to a phase control device. LC1 is a mixture of molecules (as seen in Figure 1) that vary by alkyl chain length. We also studied a standard nematic LC (E44) for comparison. Figure 2 shows the structure of an electrically tunable phase shifter for terahertz waves. The LC material was injected into a normal sandwich cell. Both inner surfaces of the substrates were treated with antiparallel rubbing after coating the planar alignment layer with polyimide (SE2170, Nissan Chemical Industries, Tokyo, Japan) to obtain homogeneous alignment. The cell thickness was determined by using sheet spacers. The thickness of the LC layer was 800 \(\upmu\)m for terahertz operation. To keep the high transmittance of the terahertz wave, we used quartz substrates and Poly(3,4-ethylenedioxythiophene):Poly(styrenesulfonate) (PEDOT/PSS) electrodes. The direction of the LC molecules (director) could be controlled by an external electrical field. Figure 3 shows the experiment setup. In this study, the terahertz wave intensity profiles were measured by using an FIR CW laser as a signal source. This laser was the major source of a coherent CW with a powerful terahertz radiation above 0.3 THz. Here a CO\({}_{2}\) laser was used for pumping the CH\({}_{2}\)F\({}_{2}\) gas, and a frequency of 2.5 THz was used for our measurements. We placed the LC device between two wire-grid paralyzers, and the intensity of the terahertz wave was detected by a pyroelectric detector. To eliminate the influence of laser power variation, we checked the power of the THz wave by using  
Figure 1: Molecular structure of hydrogen-bonded liquid crystal. pyroelectric detector 1 at all times. An accurate transmittance was obtained by using the intensity of pyroelectric detector 2 normalized by the intensity of detector 1. The polarization state of the transmitted terahertz wave through the LC device was calculated by using the Jones matrix method [35]. Figure 4 shows the Jones matrix calculation model of a homogeneous-alignment LC device in the terahertz region. In this situation, the electric fields of the transmitted terahertz wave \(E_{x}\) and \(E_{y}\) are described as follows:  
\[\left[ \begin{array}{c}E_{x}\\ E_{y}\end{array} \right]=\mathbf{P_{A}}\mathbf{W}\left[ \begin{array}{c}\cos\Psi_{P} \\ \sin\Psi_{P}\end{array} \right], \tag{1}\]  
where \(\Psi_{p}\) is the angle of the polarizer as defined in Figure 4. The Jones matrix of the analyzer \(\mathbf{P_{A}}\) and that of the homogeneous LC device \(\mathbf{W}\) in Equation (1) are written as  
\[\mathbf{P_{A}}=\mathbf{R}(\Psi_{A})\left[ \begin{array}{cc}1&0\\ 0&0\end{array} \right]\mathbf{R}(-\Psi_{A}), \tag{2}\]  
\[\mathbf{W}=\mathbf{R}(\mathbf{\Psi_{i}})\left[ \begin{array}{c}exp\Big{(}-\frac{i\Gamma} {2}\Big{)}exp\Big{(}-\frac{2\pi n_{o}^{\prime\prime}d}{\lambda}\Big{)}&\mathbf{0 }\\ \mathbf{0}&exp\Big{(}\frac{i\Gamma}{2}\Big{)}exp\Big{(}-\frac{2\pi n_{o}^{\prime \prime}d}{\lambda}\Big{)}\end{array} \right]\mathbf{R}(-\mathbf{\Psi_{i}}), \tag{3}\]  
where \(\lambda\) is the wavelength of the incident terahertz wave, \(d\) is the thickness of the LC layer, and \(n^{\prime\prime}\) is the imaginary part of the complex refractive indices of the LC. Here, subscripts of \(e\) or \(o\) mean the extraordinary and ordinary refractive indices, respectively. The \(\mathbf{R}(\Psi)\) and \(\Gamma\) are given as  
\[\mathbf{R}(\Psi)=\left[ \begin{array}{cc}\cos\Psi&-\sin\Psi\\ \sin\Psi&\cos\Psi\end{array} \right], \tag{4}\]  
\[\mathbf{\Gamma}=\frac{2\pi|n_{e}^{\prime}-n_{o}^{\prime}|d}{\lambda}, \tag{5}\]  
Figure 3: Experiment setup for terahertz measurements. Figure 2: Structure of electrically tunable phase shifter for a terahertz wave.