# Helix-Free Ferroelectric Liquid Crystals: Electro Optics and Possible Applications
## 5 Theoretical Model of the Ferroelectric Liquid Crystal Deformation and Director Reorientation
### Modulation of Light Transmission  
The modulation of light transmission was observed when an electro-optic cell of 1.7 um thickness (achromatic for visible light) was placed between crossed polarizers and bipolar voltage of the rectangular shape (meander) was applied. The FLC director reorientation due to motion of solitons made it possible to reduce the optical response time in weak fields to 25 us and to reach the light modulation frequency of 7 kHz at the control voltage of \(\pm\)1.5 V [3] (Figure 8). The transition to the soliton mode can occur not only with increasing frequency of the electric field change, but also with increasing field strength at fixed frequency. This transition is accompanied by a sharp decrease in the optical response time when a certain threshold value of the field strength is reached (Figure 9). With increasing frequency, the threshold value of the field for the transition to the soliton mode decreases, and the time of the optical response \(\tau_{0.1-0.9}\) decreases (Figure 9, curve 2) at a lower frequency. Figure 8: Oscillogram of the bipolar control voltage (zero level—digit 3) and optical response (zero level—digit 1) for an electro-optical cell with composition HF-32C at the voltage amplitude of \(\pm\)1.5 V and frequency of 7.0 kHz. The upper level of the optical response is the closed state; the lower one is the light transmission state. After the transition to the soliton mode, a section in the dependence \(\tau_{0.1-0.9}\) (\(E\)) appears where the optical response time is independent of the electric field strength (Figure 9). This means that the FLC dissipative coefficient is the shear viscosity. Constant change in the director position along the FLC smectic layers provides the substantially hysteresis-free dependence of the light transmission of the electro-optical cell on the control voltage amplitude when both increasing and decreasing in a wide frequency range. Experiments have shown that the hysteresis-free modulation characteristic I (V) for both positive and negative voltage occurs if the following two conditions are satisfied: First, the frequency control voltage corresponds to the frequency interval of existence of the soliton mode (for the HF-32C composition, it extends from 100 Hz to 7 kHz). Second, this frequency does not correspond to the static portions (Figure 10) of the frequency dependence of the FLC birefringence \(\Delta n\) (f). This means that the frequency of the hysteresis-free modulation does not exceed 6 kHz (Figure 11). This maximum value was realized experimentally [25]. Figure 10: Frequency dependence of the birefringence of the HF-32C composition. Inset: a low-frequency part of this dependence. The thickness of the electro-optical cell is 1.7 \(\upmu\)m. The amplitude of the bipolar control voltage (meander) is \(\pm\)1.5 V.  
Figure 9: Field dependences of the optical response time for a cell with the HF32F composition. The thickness of the electro-optical cell is 1.7 \(\upmu\)m. The voltage frequencies (meander) are 200 Hz (curve 1) and 3 kHz (curve 2). Note that the halftone modulation characteristic in such a uniquely wide frequency range is realized only on the basis of the physical properties of new FLC materials. In addition, unlike the bistable (two-level) characteristic of the known FLCs, possible applications of novel helix-free FLC do not require additional electronic modulation of a signal, which reduces the frame rate of color image formation. It is important that the predominance of shear viscosity in a soliton mode results in weakening the temperature dependence of the electro-optical response time in a wide temperature interval (Figure 12). The higher the frequency of light modulation (and the frequency of the control voltage), the wider the temperature interval in which the time \(\tau_{0.1\text{-}0.9}\) is almost constant.