# Dynamic and Photonic Properties of Field-Induced Gratings in Flexoelectric LC Layers

## Abstract

**:**

_{1}–e

_{3}) and low dielectric anisotropy, electric fields exceeding certain threshold values result in transitions from the homogeneous planarly aligned state to the spatially periodic one. Field-induced grating is characterized by rotation of the LC director about the alignment axis with the wavevector of the grating oriented perpendicular to the initial alignment direction. The rotation sign is defined by both the electric field vector and the sign of the (e

_{1}–e

_{3}) difference. The wavenumber characterizing the field-induced periodicity is increased linearly with the applied voltage starting from a threshold value of about π/d, where d is the thickness of the layer. Two sets of properties of the field-induced gratings are studied in this paper using numerical simulations: (i) the dynamics of the grating appearance and relaxation; (ii) the transmittance and reflectance spectra, showing photonic stop bands in the waveguide mode. It is shown that under ideal conditions, the characteristic time of formation for a spatially limited grating is determined by the amplitude of the electric voltage and the size of the grating itself in the direction of the wave vector. For large gratings, this time can be drastically reduced via spatial modulation of the LC anchoring on one of the alignment surfaces. In the last case, the time is defined not by the grating size, but the period of the spatial modulation of the anchoring. The spectral structure of the field-induced stop bands and their use in LC photonics are also discussed.

## 1. Introduction

## 2. Results of Numerical Simulations and Discussion

#### 2.1. Simulations of the Flexoelectric Instability

**n**= (n

_{x}, n

_{y}, n

_{z})) distribution are as follows:

_{1}, K

_{2}and K

_{3}are splay, twist and bend elastic coefficients, respectively;

**P**is the flexoelectric polarization; ε

_{f}_{0}≅ 8.85 × 10

^{−12}F/m is the vacuum dielectric constant;

**E**is the electric field vector. The elastic parameters used in this work are variable but close to those of the experimental LC (K

_{1}= 15 pN, K

_{2}= 7 pN) [3,4]. A value of 0.1 Pa s for the rotational viscosity (γ) used in dynamics simulations is within the range of quite typical values for nematic LCs (typical range is 0.03–0.2 Pa·s [31]). The flexoelectricity is of principal significance in this work, and it is assumed that e

_{1}= 10 pC/m and e

_{3}= 30 pC/m, so the difference |e

_{1}− e

_{3}| = 20 pC/m is close to that found in [3,4]. As was already mentioned, in the case described here, the low frequency dielectric anisotropy (ε

_{a}) is zero, so the dielectric tensor

**ε**is reduced to a scalar value ε and no corresponding dielectric torque appears. Our virtual LC is nonchiral and the corresponding term responsible for the natural helix pitch is omitted in (3).

_{x}and n

_{z}, which is reversed if we change either direction of the electric field or the sign of the flexoelectric coefficient difference (e

_{1}–e

_{3}).

_{th}.

_{1}and K

_{2}, but doesn’t depend on K

_{3}. A decrease in K

_{2}leads to an increase in b (compare also curves 1 and 2, Figure 4), and accordingly a greater sensitivity of the spatial frequency to the voltage. In the case of the one-constant approximation, when K

_{1}= K

_{2}= K

_{3}= K, analytical expressions were obtained in [1,2] for both the observed optical period W

_{th}= p

_{th}/2 of the induced grating and for the threshold voltage U

_{th}of the appearance of the grating:

_{a}= 0, then μ = 0, and from Equation (5) one can get W

_{th}= d = 2 μm and U

_{th}≅ 4.5 V, which is in good agreement with the simulations (curve 1, Figure 4).

_{1}and K

_{2}are quite pronounced. For example, an increase in K

_{1}from 10 pN to 30 pN leads to an increase in the period by about 1.5 times. It is interesting that the period p does not depend on K

_{3}. The dynamics of the appearance and relaxation of the grating at different amplitudes of the voltage pulse is illustrated by the data in Figure 6. These data were obtained for the x-size of the modeled domain L

_{x}= 20 μm, which of course coincides with the size of the induced grating. The analysis shows that the dynamics does not depend on the y-size as long as there are no inhomogeneities in the distribution of the director in the y-direction (I remind that the y-direction is for the field-off alignment). To reduce the computation time, as well as to exclude the influence of inhomogeneities in the y-direction, the data were obtained for a y-size of 100 nm, which in fact corresponds to the two-dimensional case.

_{d}). The second time (t

_{on}) is for the equilibrium state over the entire computational domain, which includes the time t

_{d}, and in fact is the time taken to induce an equilibrium grating. After switching off the electric field, very fast relaxation occurs with a characteristic time t

_{off}. As can be seen from the inset in Figure 6, this time lies in the sub-millisecond range and decreases as the amplitude of the voltage pulse increases; thus, this time can be associated with the period of the induced grating and can be estimated as:

_{off}≅ 0.7 ms, which is in good agreement with the numerical data for the lowest voltage in the inset to Figure 6. Note that the relaxation occurs uniformly over the entire simulated domain, with the exception of small near-boundary regions, where the relaxation time is found to be somewhat longer.

_{d}is well illustrated by the data in Figure 7. As can be seen, the dwell time increases linearly with increases in the size of the simulated domain (the size of the induced grating). This is because the dwell time is due to the propagation time of the deformation from the edges of the simulated domain to its center.

_{on}), which includes the dwell time, is determined by its size is very important for understanding the low speed of formation of the periodic texture in the experiments, where static electric fields [3] had to be used. At the same time, understanding the nature of the dwell time allows us to offer a liquid crystal system with a high response speed. Indeed, the boundaries of the modeled domain are, in fact, the region of nucleation of an elastic deformation wave, which propagates to the central part of the simulated domain; therefore, if on the second boundary of the alignment surface, a periodic structure is created from artificial anchoring “defects”, where the conditions for the vertical alignment of the LC director are achieved at short intervals (Figure 8, t = 0), then as a result we can obtain a system for which the total time t

_{on}of the grating induction is determined not by its total size, but the period of modulation of the anchoring conditions. In the example in Figure 8, the anchoring modulation period at the upper boundary of the LC layer is 10 μm and the corresponding time t

_{on}is about 6 ms, regardless of the overall grating size. Experimentally, the modulation of the anchoring boundary conditions can be carried out using a local focused ion beam treatment [27] of a rubbed polyimide film, which is usually used for the LC alignment.

#### 2.2. Photonic Properties (FDTD Simulations)

_{||}= 3.06, ε

_{⊥}= 2.31). The last values of the dielectric tensor components correspond to the principal refractive indices n

_{||}= 1.75 and n

_{⊥}= 1.52, which are close to typical for many LC materials (for example, the E7-LC from Merck).

^{o}with respect to the y-axis; thus, both TM- and TE-polarized modes are excited in the LC layer. The magnitude of the pulse is constant in the z-direction. The pulse represents a sine wave (λ = 550 nm) modulated by the Gaussian waveform with a 1/e-height duration of ~1 fs. This results in a rather wide spectrum of generated light, which allows for calculations of the reflectance spectra in a wavelength range of 500 to 4000 nm. The sensor (4) registers across time the components of the electromagnetic field. To obtain the reflectance and transmittance spectra, the ratio of the energy flux in the x-direction (P

_{x}) of the electromagnetic field at sensors (4) and (5) to the energy flux irradiated by light source (P

_{ls}) is calculated. The values P

_{x}and P

_{ls}are calculated versus the wavelength by taking the Fourier transform of the field registered by the sensors and finding x-components of the Poynting vector at the sensors and light source (3) positions. Because the x-components of the Poynting vector for the reflected light are negative, the reflectance magnitudes in spectra shown below are also negative. The spectral resolution in the calculated spectra is defined by total registration time for the electromagnetic field. In our calculations, the total registration time is about 2000 fs, which corresponds to resolutions better than 1 nm in the visible spectral range and a few nanometers in the near-infrared range.

_{1}, λ

_{2}) of these stop bands can be estimated from the Bragg law equation:

_{⊥}and n

_{||}are principal refractive indices of the LC material for the diffraction order m = 2. Actually, at m = 2, exactly the same situation occurs as in the cholesteric liquid crystals. Nevertheless, there is a principal difference from the cholesteric LCs, which is connected to the tilt of the director in the yz-plane when the director rotates around the y-axis (Figure 3). Because of the tilt, the director state with the maximum positive value of the n

_{z}director component is not equivalent to the state with the minimal negative n

_{z}component. The last property results in that the Bragg reflection is allowed not only for m = 2, but also for m = 1. Moreover, the reflections of the higher orders (m = 3, 4, 5) are very visible, especially in the reflectance spectra at the highest voltage (Figure 11c).

## 3. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Bobylev, Y.P.; Pikin, S.A. Threshold piezoelectric instability in a liquid crystal. Sov. Phys. JETP
**1977**, 72, 369. [Google Scholar] - Pikin, S.A. Structural Transformations in Liquid Crystal; Gordon and Breach: New York, NY, USA, 1991. [Google Scholar]
- Barnik, M.I.; Blinov, L.M.; Trufanov, A.N.; Umanski, B.A. Flexoelectric domains in nematic liquid crystals. Sov. Phys. JETP
**1977**, 73, 1936–1943. [Google Scholar] - Barnik, M.I.; Blinov, L.M.; Trufanov, A.N.; Umanski, B.A. Flexo-electric domains in liquid crystals. J. Phys.
**1978**, 39, 417–422. [Google Scholar] [CrossRef] [Green Version] - Umansky, B.A.; Chigrinov, V.G.; Blinov, L.M.; Podyachev, Y.B. Flexoelectric effect in liquid crystal twisted structures. Sov. Phys. JETP
**1981**, 81, 1307–1317. [Google Scholar] - Vistin, L.K. A new electrosructural phenomenon in liquid crystals of nematic type. Dokl. Akad. Nauk SSSR
**1970**, 194, 1318–1321. [Google Scholar] - Williams, R. Domains in liquid crystals. J. Chem. Phys.
**1963**, 39, 384. [Google Scholar] [CrossRef] - Delev, V.A.; Skaldin, O.A. Electrooptics of hybrid aligned nematics in the regime of flexoelectric instability. Tech. Phys. Lett.
**2004**, 30, 679–681. [Google Scholar] [CrossRef] - Palto, S.P.; Mottram, N.J.; Osipov, M.A. Flexoelectric instability and a spontaneous chiral-symmetry breaking in a nematic liquid crystal cell with asymmetric boundary conditions. Phys. Rev. E
**2007**, 75, 061707. [Google Scholar] [CrossRef] - Marinov, Y.G.; Hinov, H.P. On the threshold characteristics of the flexoelectric domains arising in a homogeneous electric field: The case of anisotropic elasticity. Eur. Phys. J. E
**2010**, 31, 179–189. [Google Scholar] [CrossRef] [PubMed] - Krekhov, A.; Pesch, W.; Buka, A. Flexoelectricity and pattern formation in nematic liquid crystals. Phys. Rev. E
**2011**, 83, 051706. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hinov, H.P. On the Coexistence of the Flexo-Dielectric Walls–Flexoelectric Domains for the Nematic MBBA—A New Estimation of the Modulus of the Difference between the Flexoelectric Coefficients of Splay and Bend |e
_{1z}−e_{3x}|. Mol. Cryst. Liq. Cryst.**2010**, 524, 26–35. [Google Scholar] [CrossRef] - Delev, V.A.; Skaldin, O.A.; Timirov, Y.I. The method for determination of flexoelectric coefficients of nematic liquid crystals. Proc. Mavlyutov Inst. Mech.
**2017**, 12, 101–108. (In Russian) [Google Scholar] [CrossRef] [Green Version] - Dolganov, P.V.; Ksyonz, G.S.; Dmitrienko, V.E.; Dolganov, V.K. Description of optical properties of cholesteric photonic liquid crystals based on Maxwell equations and Kramers-Kronig relations. Phys. Rev. E
**2013**, 87, 032506. [Google Scholar] [CrossRef] - Belyakov, V.A. From liquid crystals localized modes to localized modes in photonic crystals. J. Lasers Opt. Photonics
**2017**, 4, 153. [Google Scholar] - Belyakov, V.A.; Semenov, S.V. Localized conical edge modes of higher orders in photonic liquid crystals. Crystals
**2019**, 9, 542. [Google Scholar] [CrossRef] [Green Version] - Vetrov, S.Y.; Timofeev, I.V.; Shabanov, V.F. Localized modes in chiral photonic structures. Phys. Uspekhi
**2020**, 63, 33–56. [Google Scholar] [CrossRef] - Nys, I.; Beeckman, J.; Neyts, K. Voltage-controlled formation of short pitch chiral liquid crystal structures based on high resolution surface topography. Opt. Expr.
**2019**, 2, 11492–11502. [Google Scholar] [CrossRef] [PubMed] - Ahn, S.; Ko, M.O.; Kim, J.H.; Chen, Z.; Jeon, M.Y. Characterization of Second-Order Reflection Bands from a Cholesteric Liquid Crystal Cell Based on a Wavelength-Swept Laser. Sensors
**2020**, 20, 4643. [Google Scholar] [CrossRef] [PubMed] - Il’chishin, I.P.; Tikhonov, E.A.; Shpak, M.T.; Doroshkin, A.A. Stimulated emission lasing by organic dyes in a nematic liquid crystal. JETP Lett.
**1976**, 24, 303–306. [Google Scholar] - Kopp, V.I.; Zhang, Z.Q.; Genacka, A.Z. Lasing in chiral photonic structures. Prog. Quantum. Electron.
**2003**, 27, 369–416. [Google Scholar] [CrossRef] - Coles, H.; Morris, S. Liquid-Crystal lasers. Nat. Photonics
**2010**, 4, 676–685. [Google Scholar] [CrossRef] - Inoue, Y.; Yoshida, H.; Inoue, K.; Fujii, A.; Ozaki, M. Improved lasing threshold of cholesteric liquid crystal lasers with in-plane helix alignment. Appl. Phys. Express
**2010**, 3, 102702. [Google Scholar] [CrossRef] - Xiang, J.; Varanytsia, A.; Minkowski, F.; Paterson, D.A.; Storey, J.M.D.; Imrie, C.T.; Lavrentovich, O.D.; Palffy-Muhoray, P. Electrically tunable laser based on oblique heliconical cholesteric liquid crystal. Proc. Natl. Acad. Sci. USA
**2016**, 113, 12925–12928. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ortega, J.; Folcia, C.L.; Etxebarria, J. Upgrading the Performance of Cholesteric Liquid Crystal Lasers: Improvement Margins and Limitations. Materials
**2018**, 11, 5. [Google Scholar] [CrossRef] [Green Version] - Brown, C.M.; Dickinson, D.K.E.; Hands, P.J.W. Diode pumping of liquid crystal lasers. Opt. Laser Technol.
**2021**, 140, 107080. [Google Scholar] [CrossRef] - Shtykov, N.M.; Palto, S.P.; Geivandov, A.R.; Umanskii, B.A.; Simdyankin, I.V.; Rybakov, D.O.; Artemov, V.V.; Gorkunov, M.V. Lasing in liquid crystal systems with a deformed lying helix. Opt. Lett.
**2020**, 45, 4328–4331. [Google Scholar] [CrossRef] - Palto, S.P. The Field-Induced Stop-Bands and Lasing Modes in CLC Layers with Deformed Lying Helix. Crystals
**2019**, 9, 469. [Google Scholar] [CrossRef] [Green Version] - Blinov, L.M.; Palto, S.P. Cholesteric Helix: Topological Problem, Photonics and Electro-optics. Liq. Cryst.
**2009**, 36, 1037–1047. [Google Scholar] [CrossRef] - Palto, S.P.; Barnik, M.I.; Geivandov, A.R.; Kasyanova, I.V.; Palto, V.S. Spectral and polarization structure of field-induced photonic bands in cholesteric liquid crystals. Phys. Rev. E
**2015**, 92, 032502. [Google Scholar] [CrossRef] [PubMed] - Peng, F.; Huang, Y.; Gou, F.; Hu, M.; Li, J.; An, Z.; Wu, S.-T. High performance liquid crystals for vehicle displays. Opt. Mat. Express
**2016**, 6, 717–726. [Google Scholar] [CrossRef] - Rumpf, R.C. Electromagnetic Analysis Using Finite-Difference Time-Domain; EMPossible: El Paso, TX, USA; Available online: https://empossible.net/academics/emp5304/ (accessed on 28 June 2021).

**Figure 1.**Schematic illustration of the LC director distribution for different domains: Williams domains (

**a**); flexoelectric domains (

**b**). For the Williams domains, the initial alignment is along the x-axis, while for the flexoelectric domains the alignment is along the y-axis.

**Figure 2.**In-plane director distribution in the middle (z = 0.5 μm) of the LC layer at voltages of 15 V (bottom plane) and 30 V (top plane) applied across the LC layer of a thickness of one micrometer. The color scale is for the z-component of the LC director. The inset shows the director distribution in the xz-plane for a fraction of the calculation domain in the middle at 15 V.

**Figure 3.**Director distribution in the center of a 2-μm-thick layer at a voltage of 20 V after switching on an electric voltage with an amplitude of 20 V.

**Figure 4.**Dependence of the spatial frequency (1/p) of the induced grating on the applied voltage for two series of elastic coefficients: (1) K

_{1}= K

_{2}= K

_{3}= 15 pN; (2) K

_{1}= 15 pN, K

_{2}= 8 pN, K

_{3}= 30 pN. The thickness of the LC layer d = 2 μm.

**Figure 5.**Dependence of the spatial period p on one of the three elastic coefficients at fixed and equal values of the other two elastic coefficients for U = 13 V.

**Figure 6.**Dynamic response of the director state (n

_{y}-component, top) to the applied voltage pulse (bottom).

**Figure 7.**Dependence of the dwell time on the x-size of the calculate domain (grating size): 1 − U = 10V; 2 − U = 15 V.

**Figure 8.**Dynamics of the grating induction in case of the modulated (planar–vertical) anchoring at the top alignment surface. The anchoring is modulated by the pulse function, with a period of 10 μm and pulse width of 0.4 μm defining the vertical alignment. The layer thickness is 2 μm and the driving voltage is 15 V. The color scale is for the z-component of the LC director.

**Figure 9.**Scheme of the FDTD simulated domain. (1) LC grating layer; (2) the medium with a refractive index of 1.46; (3) unidirectional light source; (4) sensor for the reflected light; (5) sensor for the transmitted light; (6) the uniaxial perfectly matched layers.

**Figure 10.**(

**a**–

**c**) Transmittance spectra for the z- (1) and y-polarized (2) components of the light at different voltages U. The inset in (

**c**) shows a fraction of the same spectrum in visible range with a higher spectral resolution.

**Figure 11.**(

**a**–

**c**) Reflectance spectra for the z- (1) and y-polarized (2) components of the light at different voltages U. The inset in (

**c**) shows a fraction of the same spectrum in visible range with a higher spectral resolution.

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**MDPI and ACS Style**

Palto, S.P.
Dynamic and Photonic Properties of Field-Induced Gratings in Flexoelectric LC Layers. *Crystals* **2021**, *11*, 894.
https://doi.org/10.3390/cryst11080894

**AMA Style**

Palto SP.
Dynamic and Photonic Properties of Field-Induced Gratings in Flexoelectric LC Layers. *Crystals*. 2021; 11(8):894.
https://doi.org/10.3390/cryst11080894

**Chicago/Turabian Style**

Palto, Serguei P.
2021. "Dynamic and Photonic Properties of Field-Induced Gratings in Flexoelectric LC Layers" *Crystals* 11, no. 8: 894.
https://doi.org/10.3390/cryst11080894