## 2 Microwave Liquid Crystal Technology
### Orientation Mechanisms and Biasing Schemes  
Actually, the response time is the most critical parameter for tunable LC components, which is defined as the time interval required to reach an equilibrium state of LC devices [74; 143; 175; 185; 189; 191; 192; 196; 199]. This will be explained for an LC-filled parallel-plate capacitor according to Figure 9a, using the basic or hybrid biasing scheme, which is known from LCDs. Figure 8: Switch-off response time of a parallel-plate capacitor cell filled with LCs from Table 1 versus the LC layer height up to 5 \(\upmu\)m (**left**) and up to 140 \(\upmu\)m (**right**). The curves for TUD-566 and GT5-26001 are nearly identical. The lower plot is a zoom out. To make use of the LC's anisotropic nature, the orientation of the LC molecules must be controlled by means of electrostatic fields between the two plates (pair of electrodes) of the capacitor. Similar to LCDs, additional alignment layers are used, e.g., they are made of a spin-coated, cured polyimide film such as Nylon 6, having a thickness typically in between 50 and 100 nm, which is mechanically rubbed with a velvet cloth [185]. The result of this rubbing is a microscopic grooving of the polyimide surface, which anchors the LC molecules in the vicinity of this thin layer along the rubbing direction in parallel to the surface, according to Figure 10.  
Accompanied with the tendency of LC molecules to align in parallel to each other, this concept can be principally used for pre-orientation of the LC molecules in a preferred direction. This pre-alignment layer is particularly useful in thin planar structures with LC layer heights \(h_{LC}<\) 150 \(\upmu\)m, since with higher layers, more LC molecules are no longer in parallel to the surface, because of decreasing anchoring forces, reducing the orientation effectiveness. The effect of alignment layers--their thickness, orientation, and deepness of the grooves--on an LC device's tuning effectivity is analyzed in [2; 74; 175; 185; 224]. When a bias voltage \(V_{b}\) (AC typically at 1 kHz) is applied on the plates of the capacitor in Figure 9a, exceeding a certain threshold voltage \(V_{b}>V_{th}\), the LC directors \(\stackrel{{\rightarrow}}{{n}}\) start to change their orientation from the initial state parallel to the grooves of the alignment layers toward the field lines of the electrostatic bias field \(\vec{E}_{b}\). This threshold phenomenon is well-known as a _Fredericksz transition_[185; 191; 192; 199], which defines the voltage where the electric and elastic forces of the surface anchoring are in an equilibrium. In thin planar structures, the splay deformation is dominant, resulting in a threshold voltage of  
\[V_{th}=E_{th}h_{LC}=\pi\sqrt{\frac{K_{11}}{\epsilon_{0}\Delta\varepsilon_{r,1 \mathrm{kHz}}}} \tag{4}\]  
Figure 10: Schematic of the pre-alignment layer (blue), which anchors the LC molecules in the direction of the grooves being rubbed into the polyimide film with a velvet cloth. Figure 9: (**a**) Basic or hybrid biasing scheme of a parallel-plate capacitor filled with LC of a layer height \(h_{LC}\). An alignment layer on the plates is assumed to pre-align the LC molecules initially in parallel to the surface (left). When bias voltage \(V_{b}>V_{th}\) is applied on the plates of the capacitor, the tube-like LC directors \(\stackrel{{\rightarrow}}{{n}}\) start to change its orientation from the initial state slowly toward the field lines of the electrostatic bias field \(\vec{E}_{b}\) (three pictures in the middle). For \(V_{b}=V_{sat}\), LC directors \(\stackrel{{\rightarrow}}{{n}}\) are almost in parallel to the field \(\vec{E}_{b}\) (picture 5 from the left). (**b**) Fully or all-electrical biasing scheme for a parallel-plate capacitor without any pre-alignment layer by introducing a second pair of electrode, orthogonal to the plates of the capacitor cell. Hence, it is proportional to the square root of the elastic constant for the splay deformation \(K_{11}\) over the anisotropy \(\Delta\varepsilon_{r,1\text{kHz}}\) at the biasing frequency.