# Helix-Free Ferroelectric Liquid Crystals: Electro Optics and Possible Applications
## 5 Theoretical Model of the Ferroelectric Liquid Crystal Deformation and Director Reorientation
### Spatially Inhomogeneous Modulation of Light Phase Delay  
When the duration of voltage pulses supplied simultaneously to the electro-optical cell corresponds to different maxima of light scattering efficiency, the transitions between light-scattering modes (which correspond to light scattering maxima) result in the most chaotic changes in the position of a scattering indicatrix. As a result of a short-term switching on of light scattering (less than for 50 \(\upmu\)s), structures with an almost random distribution of refractive index gradients are formed in the entire volume of the FLC. They cause a spatially inhomogeneous phase modulation of the light beam (over its cross section) passing through the electro-optical cell.  
Spatially inhomogeneous modulation with a phase delay of the order of and more than \(\pi\) allows one to destroy the phase relations in a laser beam passing through the electro-optical cell and, further, to suppress the speckle noise in images formed due to the ability of laser rays to interfere.  
This approach was used to develop the first electro-optical despeckler, the device that reduces the contrast of speckles and, due to this, suppresses the speckle noise in laser images. In [28; 29], helix FLCs were used for this aim, and an alternating electric field was applied across the cell simultaneously at low and high frequencies, which caused spatial deformations of the helicoid in an FLC layer. Unfortunately, such a despeckler had serious drawbacks. First, the deformed helix structure of molecules changed the spectral composition of the laser radiation. Besides this, after the electric field was switched off, the residual scattering caused by the helices remained. Also, the light modulation frequency at the electric field strength of \(\sim\)2 V/\(\upmu\)m was limited by the value of 500 Hz, which hampers possible applications of this despeckler.  
These drawbacks were eliminated after using new helix-free FLC in the experimental samples of a despeckler. Figure 15 illustrates the operation of one of the best samples. Signals of the control voltage and the modulated optical response are shown in Figure 15a. To create the different nonrepeating distributions of refractive index gradients in the volume of the FLC layer, two-frequency voltage pulses (meander) were supplied to the electro-optical cell, and pulses of low frequency (2 kHz) were modulated by pulses of high frequency (10 kHz).  
In Figure 15b,c, the speckle intensity distributions in the cross section of the laser beam behind the FLC electro-optical cell are presented for when the control voltage on the cell electrodes is not applied and is applied, respectively. The optical data were recorded using a CCD (charge-coupled device) camera and were processed using a special software product. The reduction in the contrast of the speckle pattern was calculated from the data in Figure 15b,c as the ratio R between the contrast \(C_{0}\) of a speckle pattern measured without the despeckler sample and the contrast \(C_{1}\) of a speckle pattern measured with the despeckler sample: R = 10 \(\log_{10}\) (\(C_{0}/C_{1}\)) = 10 \(\log_{10}\) (0.82/0.07) = 10.2 dB [30; 31].  
The experiments confirmed the simplicity and high efficiency of the electro-optical despeckler prototype based on an electro-optical cell with the helix-free FLC. Such a device can be widely used in holography and in laser projection displays.