# Microwave Liquid Crystal Technology
## 2 How to Use LC at Microwave Frequencies?
, \tag{7}\]  
Figure 3: Definition of the director \(\vec{n}\) for a single LC unit volume element in the device’s coordinate system (\(xyz\)) and relation to the permittivities defined in the material’s coordinate system (\(uvw\)) with the material properties \(\varepsilon_{m\parallel}\), \(\varepsilon_{m\perp}\) and the macroscopic unit volume properties \(\varepsilon_{\parallel}\), \(\varepsilon_{\perp}\). This example can be visualized with a simple parallel-plate capacitor. The LC material is sandwiched between two electrodes and an electric field is applied as shown in Figure 4.  
The LC material in tunable devices is described by the continuum theory [3]. The system's energy is defined based on the Gibbs Free Energy, which is affected from the elastic deformations, external electric and magnetic fields as well as anchoring effects. Thus, an increase in the Gibbs Free Energy is minimized by changing the orientation of the director to reach equilibrium. \[W_{f}=\iiint\limits_{V}df=\iiint\limits_{V}df_{\text{elastic}}+df_{\text{ electric}}+df_{\text{magnetic}}+df_{\text{surface}} \tag{8}\]  
From this, three different methods can be derived for the practical implementation of alignment of LCs. In the presence of an external electric field, the energy  
\[df_{\text{electric}}=-\frac{1}{2}\epsilon_{0}\left[\epsilon_{r,\perp}+\Delta \epsilon_{r}\left(\cos\Theta\right)^{2}\right]\left\langle\mathbf{E}^{2} \right\rangle. \tag{9}\]  
is causing a re-orientation of the director \(\vec{n}\) to achieve minimum system energy, if a certain threshold energy is overcome. It should be noted that \(\Delta\epsilon\) is positive for millimeter wave optimized nematic LCs. Therefore, the director tends to align in a way that is parallel to the electric field lines. In principle, the same is valid for magnetic fields, where energy is defined as  
\[df_{\text{magnetic}}=-\frac{1}{2}\mu_{0}\left[\mu_{r,\perp}+\Delta\mu_{r} \left(\cos\Theta\right)^{2}\right]\left\langle\mathbf{H}^{2}\right\rangle \tag{10}\]  
At last, surface forces (which have been neglected in Figure 4) can also be used to align LC molecules. Any physical impurities on the boundaries of the cavity result in interaction between the boundary and the LC molecules. According to the Rapini-Papoular model [4; 5; 6], this interaction energy density is given as  
\[df_{\text{surface}}=\frac{1}{2}\left[W_{p}\left(\sin\left(\Theta-\Theta_{0} \right)\right)^{2}+W_{a}\left(\sin\left(\Phi-\Phi_{0}\right)\right)^{2}\right] \tag{11}\]  
with the director and the preferred alignment for the polar and azimuthal angles specified by (\(\Theta\), \(\Phi\)) and (\(\Theta_{0}\), \(\Phi_{0}\)). The energy depends on the anchoring strength \(W_{a}\) and \(W_{p}\), which are LC material and anchoring surface-dependent parameters. In practical applications, different orientation mechanisms are combined according to Table 1. For example, the combination of surface anchoring and electrical biasing concepts as it is used in the LCD technology, compare Figure 5a. Without the electric field, the directors are aligned in state I and with applied field they change their orientation towards state II, at which the numbering of the states has been arbitrarily chosen. This mechanism is preferred in planar devices. One reason for the limitation to planar devices is that the forces of the surface anchoring are weak compared to electric  
Figure 4: Idealized parallel-plate capacitor with LC as dielectric filling for the three cases defined in Equation (7), (**Left**) \(\vec{\epsilon}_{u}\parallel\vec{\epsilon}_{x}\), (**Right**) \(\vec{\epsilon}_{u}\parallel\vec{\epsilon}_{y}\), and (**Center**) “else”. Note: Interface effects between LC and electrodes are neglected for simplification. At this point, it is clear how the alignment of the director can be technically realized. For application in tunable components, the RF signal and the control (bias) must be discriminated. For magnetic alignment, the bias field and the RF signal's E-field are decorrelated by definition.