Monte Carlo simulation of the molecular distribution and optical properties of a nematic liquid crystal system with periodic surface gratings

Monte Carlo simulation of the molecular

distribution and optical properties of a nematic

liquid crystal system with periodic surface

gratings

C. Berlic and V. Barna*

Faculty of Physics, University of Bucharest, PO Box Mg-11, 077125, Bucharest, Romania

*[email protected]

Abstract: We report Monte Carlo simulations based on the Lebwohl-

Lasher model for characterizing the molecular director configuration in a

nematic liquid crystal cell presenting periodical boundary anchoring

conditions. We demonstrate the molecular orientation and spatial

behaviour, while profiling the local order parameter distribution for the

proposed confining geometry, as well as the boundary and interface

interaction fields propagation through the namatic bulk for various

temperatures in the proximity of the nematic-isotropic transition.

Simulations were also performed concerning with the light passing through

the planar and homeotropic periodical regions of the nematic cell and a

mapping of the transmitted intensity was obtained for several ambient

temperatures. The boundary constraints and the selected periodical

geometry of the simulated system play an extremely important role for the

demonstrated optical and orientational properties of the liquid crystalline

material.

© 2010 Optical Society of America

OCIS codes: (260.1440) Birefringence; (160.3710) Liquid crystals; (160.4760) Optical

properties; (230.3720) Liquid-crystal devices; (350.2770) Gratings; (160.1190) Anisotropic

optical materials.

References and links

1. P. G. de Gennes, and J. Prost, The Physics of Liquid Crystals (Oxford University Press, 1995).

2. S. Chandrasekhar, Liquid Crystals (Cambridge University Press, 1993).

3. P. Yeh, and C. Gu, Optics of Liquid Crystal Displays (Wiley, 2009).

4. A. L. Alexe-Ionescu, A. Th. Ionescu, E. S. Barna, N. Scaramuzza, and G. Strangi, “Role of Surface Order on the

Total Electric Conduction in NLC Samples,” J. Phys. Chem. B 107(23), 5487–5490 (2003).

5. N. Scaramuzza, C. Berlic, E. S. Barna, G. Strangi, V. Barna, and A. Th. Ionescu, “Molecular Simulation of the

Free Surface Order in NLC Samples,” J. Phys. Chem. B 108(10), 3207–3210 (2004).

6. D. W. Berremann, “Solid Surface Shape and the Alignment of an Adjacent Nematic Liquid Crystal,” Phys. Rev.

Lett. 28(26), 1683–1686 (1972).

7. M. Schadt, K. Schmitt, V. Kozinkov, and V. Chigrinov, “Surface-Induced Parallel Alignment of Liquid Crystals

by Linearly Polymerized Photopolymers,” Jpn. J. Appl. Phys. 31(Part 1, No. 7), 2155–2164 (1992).

8. F. C. Frank, “On the Theory of Liquid Crystals,” Discuss. Faraday Soc. 25, 19–28 (1958).

9. M. P. Allen, and D. J. Tildesley, Computer Simulation of Liquids (Oxford University Press, 1989)

10. D. Frenkel, and B. Smit, Understanding Molecular Simulation: From Algorithms to Applications (Academic

Press, 2001).

11. M. E. J. Newman, and G. T. Barkema, Monte Carlo Methods in Statistical Physics (Oxford University Press,

1999)

12. P. Pasini, C. Zannoni, and S. Zumer, Computer Simulations of Liquid Crystals and Polymers (Springer, 2005).

13. D. P. Landau, and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics (Cambridge University

Press, 2000).

14. E. Gatin, D. Alexandreanu, A. Popescu, C. Berlic, and I. Alexandreanu, “Correlations between permeability

properties and pore-size distribution of the porous media “hydron” useful as contact lenses,” Phys. Med. XVI,

13–19 (2000).

#131619 - $15.00 USD Received 13 Jul 2010; revised 28 Sep 2010; accepted 13 Oct 2010; published 27 Oct 2010(C) 2010 OSA 8 November 2010 / Vol. 18, No. 23 / OPTICS EXPRESS 23646

15. E. Berggren, C. Zannoni, C. Chiccoli, P. Pasini, and F. Semeria, “A Monte Carlo Simulation of a Twisted

Nematic Liquid Crystal Display,” Int. J. Mod. Phys. C 6(1), 135–141 (1995).

16. C. Chiccoli, P. Pasini, S. Guzzeti, and C. Zannoni, “A Monte Carlo Simulation of In-Plane Switching Liquid

Crystal Display,” Int. J. Mod. Phys. C 9(3), 409–419 (1998).

17. C. Chiccoli, S. Guzzeti, P. Pasini, and C. Zannoni, “Computer Simulations of Nematic Displays,” Mol. Cryst.

Liq. Cryst. (Phila. Pa.) 360(1), 119–129 (2001).

18. C. Chiccoli, P. Pasini, A. Sarlah, C. Zannoni, and S. Zumer, “Structures and transitions in thin hybrid nematic

films: a Monte Carlo study,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 050703 (2003).

19. A. M. Smondyrev, and R. A. Pelcovits, “Nematic Structures in Cylindrical Cavities,” Liq. Cryst. 26(2), 235–240

(1999).

20. E. Berggren, C. Zannoni, C. Chiccoli, P. Pasini, and F. Semeria, “Computer simulations of nematic droplets with

bipolar boundary conditions,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 50(4), 2929–

2939 (1994).

21. C. Berlic, E. Barna, and C. Ciucu, “Monte Carlo Simulation of a Nematic Liquid Crystal Cell with a

Hemispheric Defect on One Electrode,” J. Optoelectron. Adv. Mater. 9, 3854–3859 (2007).

22. C. Berlic, and V. Barna, “Nematic Director Distribution of a Liquid Crystalline System Presenting a Cylindrical

Defect,” Journal J. Optoelectron. Adv. Mater. 12, 1427–1432 (2010).

23. P. Pasini, and C. Zannoni, eds., Advances in the Computer Simulations of Liquid Crystals (Kluver, Dordrecht,

2000).

24. D. W. Berreman, “Liquid-Crystal Twist Cell Dynamics with Backflow,” J. Appl. Phys. 46(9), 3746–3751

(1975).

25. M. Schadt, and W. Helfrich, “Voltage-Dependent Optical Activity of a Twisted Nematic Liquid Crystal,” Appl.

Phys. Lett. 18(4), 127–128 (1971).

26. M. Schadt, H. Seiberle, and A. Schuster, “Optical Patterning of Multi-Domain Liquid-Crystal Displays with

Wide Viewing Angles,” Nature 381(6579), 212–215 (1996).

27. P. A. Lebwohl, and G. Lasher, “Nematic Liquid Crystal Order – A Monte Carlo Calculation,” Phys. Rev. A 6(1),

426–429 (1972).

28. J. A. Schellman, “Polarization Modulation Spectroscopy”, in Polarized Spectroscopy of Ordered Systems, B.

Samori’ and E.W. Thulstrup, eds. (Kluwer, Dordrecht, 1988).

29. G. W. Gray, K. J. Harrison, and J. A. Nash, “New Family of Nematic Liquid Crystals for Displays,” Electron.

Lett. 9(6), 130–131 (1973).

30. T. Scharf, Polarized Light in Liquid Crystals and Polymers (John Wiley & Sons, Inc., Hoboken, New Jersey,

2007), Chap. 8.

1. Introduction

Liquid crystals represent a state of matter whose characteristics are in-between those of an

isotropic liquid and those of a solid crystal. They have properties such as the flowing

characteristic of liquids, as well as the anisotropy of optical, electrical and magnetic

phenomena characteristic to the crystalline state, which are due to the partial ordering of

molecules [1–3].

It is well known that the main application of liquid crystals is that of display devices, due

to the possibility of manipulation of the molecular director via magnetic and electrical fields,

as well as by anchoring to the surface of contact [3–5], either by periodic topologies [6] or by

anisotropic Van der Waals forces [7].

The orientational order of liquid crystals is possible due to the fact that the interactions

between molecules are anisotropic, but this fact makes the task of theoretically describing this

class of materials in restrictive geometries much more difficult. Several theories based on the

phenomenological treatment of Frank [8], or on the minimization of the free energy have been

proposed, but all of them involve especially complex calculus, making the use of

approximations necessary.

Another way of describing the orientational order and the properties of liquid crystals is

by using computer simulations [9–12]. Since the advent of the computer technology and

programming languages, Monte Carlo and molecular dynamics simulations have become

widely used tools in describing properties of polymers and liquid crystals, including optical

properties of these classes of materials [12–14]. Much effort was also devoted to simulate the

behaviour of liquid crystals displays [15–18], liquid crystals in confined geometries [5,19–23]

and in using numerical methods of computing the light transmission [24]. Confined liquid


crystals, due to the anisotropy of the material, play a major role in liquid crystal displays,

based on the electro-optical effects which make light modulation.

Nowadays, the twisted nematic liquid crystal display, based on the pioneering work of

Schadt and Helfrich [25], is a widespread and relatively well known device. Nevertheless, it

presents some drawbacks, such as the viewing characteristics problems due to the asymmetric

alignment of the molecules [3]. One of the solutions for overcoming this issue consists in

using multi domain structures in which we have two or more orthogonal surface anchoring

directions obtained by treatment of the layer surfaces via rubbing or photo exposure [26]. In

order to investigate this behaviour for a liquid crystalline system having two opposite

anchoring directions, in our study we simulated an anchoring regime with half homeotropic

and half planar boundary conditions, the cell electrodes having parallel striped layers which

we denote as gratings.

2. Molecular model and simulation method

The simulation model was the well-known Lebwohl-Lasher model [27], where the liquid

crystal molecules are considered as unit vectors (versors or spins), which occupy fixed

positions in the sites of a cubic crystalline lattice. These versors are free to rotate in space and

interact with each other through an orientation dependent energy:

2ij ij ijU P (1)

where εij is a positive constant, ε, for nearest neighbors particles and zero otherwise. P2 is the

second rank Legendre polynomial and ij i j

s s is the angle between the versors.

An important aspect of the model is, because the energy of interaction is invariant under a

uniform rotation of all spins, the bend, splay and twist elastic constants of the liquid crystal

are all equal [5, 19].

The fact that the molecules’ centers of mass are arranged in an ordered fashion does not

contradict the fact that in a liquid crystal there is no position order of the molecules, because,

in a real liquid crystal, the molecules arrange themselves in ordered domains. In fact, each

spin represents an ordered domain encompassing many molecules whose centers of mass are

disordered distributed [17, 22].

Although the system is very simple, it was widely used to describe the properties of the

liquid crystals displays [12, 15–18], or nematic structures in various geometries [5, 12, 19–

23]. The main advantage of this simple model is the fact that spins' centers of mass are fixed,

saving computer time during the simulation, comparatively with other models considering

also translational degrees of freedom. On the other hand, the system is well studied and gives

a realistic representation of a nematic liquid crystal, showing a first order phase transition at

scaled temperature * / 1.1232 0.0006NI NIT kT [15, 18].

The nematic liquid crystal cell used in the simulation is of rectangular shape, having the

dimensions X Y ZN N N in lattice spacings and electrodes parallel with YOZ plane, situated

at 0z and 1ZN z as in Fig. 1. For the left half of the cell, 12

yNy , the electrodes

impose homeotropic conditions, and for the right one, 12

y

y

Ny N , planar conditions. In

all other directions, we consider periodic boundary conditions.


Fig. 1. Schematic representation of the initial state for the simulated liquid crystal system. The

nematic liquid crystal molecules are confined in a glass sandwich type cell. A system of

coordinates XYZ is assigned.

The spins which are not placed in the immediate vicinity of the walls interact with each

other through energy B and are free to rotate inside their cells following a standard Monte

Carlo procedure: a random spin is chosen and the interaction energy with its first neighbors is

computed. This spin rotates randomly with a small angle, following the procedure described

in [10], and then the interaction energy for the new state is computed. This new state is

accepted according to the Metropolis criterion [9–13].

The anchoring of the molecules to the walls is realized by introducing fixed spins [15–23].

These spins cannot rotate, but can interact with free spins through energy S . In our model,

the fixed spins are situated at 1z and ZNz .

As it can be seen from Fig. 1, the fixed spins situated in the left half of the cell, with y

between 1 and / 2yN , are perpendicular to the electrodes. The fixed spins situated in the

right side of the cell, with y between / 2 1yN and yN are parallel with the electrodes.

Because this cell geometry, there is a competition between tendency of the left spins to align

normally to the wall, and the right spins which tend to parallel align.

For such a system with complicated boundary conditions, instead of obtaining the bulk

order parameter [1, 2], it is preferably to use the local tensor parameter [5, 9, 18, 21–23]:

1

1 3 1

2 2

n

k k

k

Q s sn

(2)

where , , ,x y z , δαβ is the Kronecker delta, <…> is the ensemble average and n is the

number of spins. If n N , the usual order parameter of the system is obtained from the

largest positive eigenvalue of the order tensor Qαβ and the corresponding eigenvector is the

liquid crystal director [9]. If we choose 1n , the average is performed on Monte Carlo

cycles only and Qαβ describes the local order [5, 21].

From the way in which the tensor order parameter is defined, it is noticeable that it has

two important properties [21]: it is a symmetric tensor, Q Q , and it has a zero trace,

0XX YY ZZQ Q Q , meaning that only 5 of its 9 components are independent.


The diagonal components of the tensor order parameter represent the degree of order with

respect the coordinate axis. The off-diagonal components Q represent the bending of the

director field in the corresponding plane O [5, 18]. The values of the tensor order

parameter are between 0.5 , the director is perpendicular to the corresponding direction, and

1, when the director in perfectly parallel with the direction [1, 18]. Because the components

of the tensor are obtained as statistical averages for each cell, their values are between these

limits, with the value of 0 meaning the total disorder with respect the direction.

The system described above was used to simulate the behaviour of a liquid crystal cell

with boundary order imposed by electrodes. The number of spins used during the simulation

was kept at values 16xN , 60yN and 24N z , which means that we had 23,040 spins.

Because spins located at electrodes are fixed and, because we have used periodic conditions

along the OX and OY directions, the number of free spins is 21,120 and a Monte Carlo cycle

consisted of 21,120 attempted moves.

The Monte Carlo procedure was a standard one: we randomly picked a spin and rotated it.

We calculated the energies in the old and in the new state and the move was accepted using

the Metropolis acceptance criterion [8–13]. The mechanism was repeated for 160,000 Monte

Carlo cycles with 60,000 of them used for equilibration.

The big numbers of spins and Monte Carlo cycles were chosen in order to avoid finite

geometrical effects and to have good statistical averages.

The coupling interaction constant between free spins was 1B and the interaction

between a free spin and a fixed one was 1.5S . It is known the fact that a liquid crystal

display is commanded as an electro-optical device, using an electric field [15–23]. However,

in our work we were interested in only the effects of the boundary conditions on the liquid

crystal cell microscopic and optical properties, so we have chosen a somewhat big value for

this parameter, meaning a strong anchoring regime.

In order to obtain the optical properties of the simulated liquid crystal display, we used the

standard Müeller matrix approach which has been used in other numerical calculations [24]

and Monte Carlo simulations of optical textures [15–18, 20]. The main assumptions we did

were that we may use ray optics and each cell of the simulated liquid crystal is a simple linear

retarder described by a Müeller matrix which mainly depends on the orientation of the

director, as well as optical properties of the molecule [20,28].

Thus, the light beam crossing the liquid crystal display is then retarded by a matrix

resulting for a product of Müeller matrix corresponding to each cell. Due to the geometry of

our system, we considered the light traveling in the OZ direction. The light modulated by the

liquid crystal display is retarded and polarized and it is represented by the Stokes vector [15–

18, 20]:

1

2

zN

OUT i IN IN

i

S P M P S

(3)

where SIN is the Stokes vector for the incoming unpolarized light, S is the Stokes vector for

the light passed through liquid crystal display, Mi is the Müeller matrix of the site i in a

column of cell parallel with OZ. As in real liquid crystal optics experiments, the light is

observed with the help of crossed polarizers placed on each side of the cell, PIN and POUT.

In the above relation, the product is taken only for spins with 2z to 1N zz , because

these spins are mobile, the spins standing on the electrode being only an artifact to simulate

anchoring.

During the Monte Carlo simulation, beside tensorial order parameter calculation, we

performed the product from relation (3) and we recorded the first component of the Stokes


vector of the light passed thorough the liquid crystal display, quantity that represents the

intensity of the light [20, 28].

3. Results and discussions

3.1 Director distribution

Because the periodic boundary condition along OX axes, we have obtained the components of

the tensor order parameter by averaging in OX direction.

We performed a first set of Monte Carlo simulations at reduced temperature * 0.9T

which is very deep in the nematic phase. The components of the tensor order parameter in the

middle of the cell, 12z , far enough from the electrodes, are depicted in Fig. 2. From this

graph, we may conclude that Qxx has a value close to 0.5 , which means that the director

does not lies along the OX axis, as it was normal, because the system's boundaries impose

direction parallel with OZ in the left side of the cell, and with OY in the right ones. This

observation is sustained by the facts that the values for Qxy and Qxz are approximately 0,

meaning that we have no rotations of the director in the XOY and XOZ planes.

On the other hand, the Qzz and Qyy have a more interesting behaviour. In the middle of the

left side of the cell, Qzz is approximately 0.7 , which is an excellent agreement with the value

of the bulk order parameter found for the Lebwohl-Lasher model [15–20]. In the same point,

the value of Qyy is 0.5 . We may thus conclude that, in the left side of the cell, the director is

mainly parallel with the OZ axis, as it was imposed by boundary conditions. In the right side

of the cell, the situation is absolutely symmetric: in the middle of this region 0.7yyQ

and 0.5Q zz, meaning that the director is in that case parallel with the OY axis, as it was

imposed by boundary conditions for 60y .

The rotation of the director take place in the YOZ plane, as it can be noticed from the

behaviour of the Qyz, which minimum is reached in the points were Qzz and Qyy have their

own maximum points and the maximum is for 0y , where YY ZZQ Q . As it is expected, the

maximum value for the components of tensor order parameter that describe the rotation is

achieved near the central region of the display, where boundary conditions jumps from

hometropic to planar.

Fig. 2. Components of the tensor order parameter for Z=12 at T*=9. Error bars sizes are of

dimension of the symbols and were omitted. Lines are only guide to the eye.


From the behaviour of tensor order parameter, we noticed that the director rotation take

place in the YOZ plane, thus we considered that it is interesting to plot in Fig. 3 the complete

map of Qyy and Qzz in this plane.

We may notice the left-right and up-down symmetry of order parameters in each half of

the cell, symmetry that is imposed by boundary conditions. From these two maps, we may

also discern the presence of the transition region situated in the middle of the cell and to

margins, because of the periodic conditions. It also worth to mention the left-right mirroring

of QYY and QZZ. For example, the behaviour of the QYY in right region of the cell, where we

have planar boundary conditions, it is similar with the behavior of QZZ in the left region of the

cell, where are homeotropic ones.

Fig. 3. (a) Map of QYY in the YOZ plane at T*=0.9.

Fig. 3. (b) Map of Qzz in the YOZ plane at T*=0.9.

In Fig. 4, we show the Qyy component as function of y, at different temperatures. Once

more, in order to avoid the values for layers too close to the electrodes, we used the layer

situated in the middle of the cell, for which 12z .


From Fig. 4, we may notice that the shape of Qyy is maintained, but the order decrease

with temperature, as it is expected, because of the increased thermal fluctuations.

For * 1.3T , temperature situated in the isotropic phase, 0yyQ , as well the all other

components of the tensor order parameter, meaning the absence of the nematic order, as it is

expected.

For all the temperatures used for simulation we systematically check if

0XX YY ZZQ Q Q for all the cells in the liquid crystal display, and we found a total

agreement with the theory.

Fig. 4. Qyy components of the tensor order parameter for z=12 at different temperatures. For

T*=1.3 we have QYY=0, meaning that there is no order. Error bars sizes are of dimension of the

symbols and were omitted. Lines are only guide to the eye.

3.2 Light traveling simulations

We performed simulation of light traveling through a liquid crystal cell filled with a nematic

liquid crystal having ordinary refractive index 0 1.5n and the extraordinary refractive

index 1.7en , similar of the 5CB liquid crystal [13, 18, 29].

Because we have considered that the light beam is parallel with the OZ axis, we recorded

the intensity of light after it passed the liquid crystal display, obtaining the map of intensities

as in Figs. 5a and 5b, where each pixel of the display correspond to a lattice point in the XOY

plane. For this set of simulations, we used a wavelength of 545 nm, as in reference [20]. The

temperatures used in the simulations where * 0.9T to * 1.1T . As expected, we notice a

sharper transition region corresponding to the lower temperature, due to the higher order

degree at * 0.9T . This statement is also sustained by the higher values we found for the

order tensor components at this temperature, Fig. 4.

Analyzing the intensity map of the transmitted light, it is easy to see that the intensity is,

with good approximation, constant along the OX axis, because of the imposed boundary

conditions.

The small values of the intensity of the transmitted light in the region where homeotropic

boundary conditions are imposed are easy to understand, because it is known that that under

crossed polarizers for normal incidence, the homeotropic texture is always black, which is a

characteristic of this texture [30]. The deviation from zero of the intensity may be explained

by the presence of the thermal fluctuations, which, distorting the perfect homeotropic


alignment, altered the angle of incidence, and we finally have be transmission due to the

birefringence at off-normal angles.

Fig. 5. (a) Map of the intensity of the transmitted light for T*=0.9 and λ=545nm.

Fig. 5. (b) Map of the intensity of the transmitted light for T*=1.1 and λ=545nm.

For the region with planar boundary conditions, the intensity is somewhat bigger than in

the region with homeotropic ones, but also small in absolute value. In [30] is stated that for a

nematic slab of thickness d, pretilt angle θ and planar conditions, the transmitted intensity is

2 20

2 2 2 2sin 2 sin

2 sin cos

e o

o

e o

I n ndI n

n n

(4)

where φ is the angle between the projection of the director in the plane perpendicular to the

direction of light and the OX axis. Since in our simulation the light was traveling in the OZ

direction and the planar structure imposed by the boundary condition is parallel with the OY

axis, the φ angle is close to 2

. In that situation, 2sin 2 is close to zero and the intensity

given by Eq. (4) should be very small, as we actually obtained from Monte Carlo simulations.


Another important feature is that the maps of intensities also change with temperature. In

Fig. 6, we plotted the transmitted light averaged along the OX axis.

Fig. 6. Intensity of the transmitted light averaged along the OX axis.

We observe that the intensity of the transmitted light is always bigger in the region where

planar boundary conditions are imposed. As temperature increases, the intensity in the

homeotropic region increases, because larger fluctuations mean a higher depart of the director

from a perfect hometropic alignment. Same explanation can be given for the region with

planar boundary conditions: at small temperatures, the projection of the director is mainly

parallel with the OY direction, but with the temperature increasing it deviates from this

position. Note that, if the system temperature exceeds the nematic-isotropic transition

temperature, the intensity decreases, so that we may expect a more complex behaviour. It is

also interesting to see that the homeotropic region is more sensitive to temperature changes.

In all the simulations, we noticed that the transition between hometropic and planar

conditions was both smooth in light intensities and tensor order parameter components. From

Fig. 6, we may also notice that in the hometropic region, the influence of the temperature is

more pronounced than in the planar one.

4. Conclusions

In the present paper, we have investigated a nematic liquid crystal cell having periodic surface

gratings by means of Monte Carlo simulations for the well known Lebwohl-Lasher model.

We have obtained a complete mapping for the tensor order parameter components of the

liquid crystal at various temperatures below the isotropic-nematic transition temperature. We

found that for a relatively strong anchoring regime, the order imposed by the gratings situated

on the electrodes propagates deep in the nematic bulk resulting in a complex behaviour for the

components of the molecular tensor. The order level of the nematic liquid crystal has a strong

influence on the optical properties as well. By using the Müeller matrix approach during the

simulations, we acquired the optical textures of the nematic system, while recording the

intensity of the light transmitted through the cell at normal incidence. Because we have not

used an electric or magnetic field to orientate the molecules, the intensity was fairly small,

whereas all the optical effects are the result of the interplay between thermal fluctuations and

propagation in the bulk of the order imposed by the complex boundary conditions. We also

found that the system temperature plays an extremely important role in tailoring the optical

characteristics of the simulated cell. We also believe that the investigated model is useful not


only in understanding the operation mode of liquid crystal cells with complex geometries but

also in optimizing the design for novel LCD displays.

Acknowledgements

This research work was supported by CNCSIS – UEFISCSU PNII grant “IDEI”, no. 450-

1902 /2008.


Documents

Monte Carlo simulation of the molecular distribution and optical properties of a nematic liquid crystal system with periodic surface gratings