# Helix-Free Ferroelectric Liquid Crystals: Electro Optics and Possible Applications
## 1 Introduction  
It is known that the minimum-time optical response is achieved in some smectic liquid crystals, called smectics C*, which possess ferroelectric properties and high sensitivity to electric fields [1; 2; 3]. The principle of electro-optical modulation in ferroelectric liquid crystals (FLCs), like in widely used nematic liquid crystals (NLCs), is the electrically controlled birefringence or light scattering. A distinctive feature of smectic crystals is the layered structure formed as a result of ordering the centers of mass of FLC molecules along the direction of orientation of their long axis (director), with a pitch of the order of the molecules' length. Among the smectic crystals, the helix FLCs are the most well known, in which the polar axes of various smectic layers are rotated relative to each other, forming a helix (spiral) twist of the FLC director in the absence of an external electric field (Figure 1). In each layer, the position of the director \(n\) is determined by the polar angle \(\Theta_{0}\) and the azimuth angle \(\varphi\), which varies from 0 to \(2\pi\) at a distance equal to the pitch \(p_{0}\) of the helix. Under the action of the electric field \(E\), which is parallel to smectic layers (along the coordinate \(x\)), the vector \(P_{S}\) of spontaneous polarization is oriented in all layers along the field direction. As a result, the director acquires one direction in the entire volume of the FLC layer--the direction of the FLC main optical axis. When a sign of the field \(\mathbf{E}\) changes, the orientation of the vector \(\mathbf{P_{S}}\) changes by \(180^{\circ}\), and the long axes of molecules unfold along the cone with the generatrix \(2\Theta_{0}\), resulting in a change of the angle \(\varphi\) by \(180^{\circ}\). The reorientation of the director determining the main optical axis of the ellipsoid of the FLC refractive indices results in a change in the angle between the polarization plane of the incident light (\(I_{0}\)) and the main optical axis of the ellipsoid. This leads to the modulation of the phase delay between the ordinary and extraordinary rays, or to light intensity modulation, if the electro-optic cell is placed between crossed polarizers [1; 2; 3]. Under certain conditions, electrically controlled light scattering on so-called transient domains is observed in the helix FLC due to the formation of a spatially inhomogeneous structure of refractive index gradients in the FLC layer [4; 5]. Helix-free (no spiral) FLCs are also known [6]. In them, the helicoidal twist of the director in the FLC volume is compensated or suppressed by the interaction of chiral optically active additives with opposite signs of optical activity. The coincidence of the same signs of spontaneous polarization in chiral additives makes it possible to obtain a value of spontaneous polarization of \(100\) nC/cm\({}^{2}\) or higher for FLCs with a compensated helicoid. In contrast to NLC, the electro-optical effect in FLC is linear relative to the field [7; 8], and since FLC reacts to the sign of the applied electric voltage, the values of the on and off times of the optical response are the same and are proportional to  
\[\tau_{R}\sim\gamma_{\varphi}/P_{S}\times E, \tag{1}\]  
if the dissipative coefficient is the rotational viscosity \(\gamma_{\varphi}\). The FLC is returned to its original state by a reverse-polarity pulse, i.e., forcedly, not as a result of relaxation due to elastic forces (like in the NLC). Therefore, the optical response at switching on and off is symmetrical and very short (in the sub-millisecond region), especially for FLC with low viscosity and large spontaneous polarization. However, it is impossible to significantly reduce \(\tau_{R}\) by increasing the spontaneous polarization, since this usually leads to an increase in the FLC rotational viscosity. This circumstance limits the frequency of light modulation if the applied voltage does not exceed several volts.