# Microwave Liquid Crystal Technology
## 2 How to Use LC at Microwave Frequencies? The mixtures used for microwave applications are thermotropic calamitic nematic LCs. The phase transitions occur at certain temperatures, where the liquid crystalline mesophase is defined thermodynamically stable between the melting point of the crystalline phase and the clearing point to the isotropic liquid phase. In this mesophase, liquid crystals show an orientational order but lack translational periodicity. In comparison to the smectic phase, the elongated molecules tend to align parallel to each other but are not organized into layers. The fundamental properties of nematic LCs versus temperature are shown in Figure 2. LC exhibits an anisotropy in the crystalline as well as in the nematic phase, whereas only the nematic phase can be used for electrically tunable RF circuits/devices. Figure 1: Working principle of a TN-cell used in LCDs. (**Right**) the unbiased LC rotates polarization by 90\({}^{\circ}\). The light can pass two twisted polarization grids and the pixel is on. When the LC is biased (**Left**), the polarization of light is not rotated. Hence, the pixel is off as the light cannot pass the second polarization grid. Figure 2: Overview on different phases of nematic liquid crystals. (**Left**) shows the material’s crystalline structure and the resulting permittivity for different orientations. In the (**Right**) picture the different phases can be differentiated by the LC’s optical properties. The molecules can be depicted with a rod-like shape where their uniaxial anisotropy is used to realize tunable devices. Due to the molecule's rotational symmetry, the permittivity tensor Equation (1) reduces to  
\[\bar{\epsilon}=\left[ \begin{array}{ccc}\varepsilon_{uu}&0&0\\ 0&\varepsilon_{vv}&0\\ 0&0&\varepsilon_{ww}\end{array} \right]\cdot\varepsilon_{0}=\left[ \begin{array}[ ]{ccc}\varepsilon_{\parallel}&0&0\\ 0&\varepsilon_{\parallel}&0\\ 0&0&\varepsilon_{\perp}\end{array} \right]\cdot\varepsilon_{0} \tag{2}\]  
with the (\(uvw\))-coordinate system aligned with the director \(\vec{n}\) of the LC volume, according to Figure 3. The director is a macroscopic unit vector giving the main direction of the molecules inside a unit volume element. The values of the permittivities \(\varepsilon_{\perp}\) and \(\varepsilon_{\parallel}\), and therefore, the LC's anisotropy \(n\)  
\[n=\sqrt{\varepsilon_{\perp}}-\sqrt{\varepsilon_{\parallel}} \tag{3}\]  
are material and temperature dependent, refer to Section 3. The electric displacement field \(\mathbf{D}\) interacts with an electric field \(\mathbf{E}\) according to  
\[\mathbf{D}=\bar{\epsilon}\cdot\mathbf{E}. \tag{4}\]  
As displacement and electric fields on one side and the permittivity tensor on the other side are defined in different coordinate systems, effective displacement must consider the rotation of the coordinate systems by introducing the rotation matrix \(\mathbf{R}\). \[\mathbf{D}=\mathbf{R}\bar{\epsilon}\mathbf{R}^{-1}\cdot\mathbf{E}. \tag{5}\]  
The degree of freedom to realize tunable devices is now this rotation matrix \(\mathbf{R}\), leading to a tunable effective permittivity  
\[\varepsilon_{\text{eff}}=\sqrt{\left[\varepsilon_{\perp}\cdot\sin\left( \Theta\right)\right]^{2}+\left[\varepsilon_{\parallel}\cdot\cos\left(\Theta \right)\right]^{2}}, \tag{6}\]  
with the tilt angle \(\Theta\) between both coordinate systems. For example, when \(\mathbf{E}=E_{0}\cdot\bar{\varepsilon}_{x}\), a continuously tunable effective permittivity \(\varepsilon_{\text{eff}}\) can be achieved as  
\[\varepsilon_{\text{eff}}= \begin{cases}\varepsilon_{\parallel}&\text{for }\bar{ \varepsilon}_{u}\parallel\bar{\varepsilon}_{x}\\ \varepsilon_{\perp}&\text{for }\bar{\varepsilon}_{u}\parallel\bar{ \varepsilon}_{y}\vee\bar{\varepsilon}_{u}\parallel\bar{\varepsilon}_{z}\\ \varepsilon_{\perp}\ldots\varepsilon_{\parallel}&\text{else}\end{cases}