Optical diffraction from a liquid crystal phase grating

JOURNAL OF APPLIED PHYSICS VOLUME 91, NUMBER 6 15 MARCH 2002 Optical diffraction from a liquid crystal phase grating C. V. Brown, a) Em. E. Kriezis, and S. J. Elston Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, United Kingdom Received 30 August 2001; accepted for publication 27 November 2001 The finite-difference time-domain method has been used in the numerical analysis of the optical diffraction properties of a liquid crystal phase grating. The grating is formed using a nematic material that is switched using striped electrodes with a unity mark-space ratio. Three different surface pretilts have been investigated: 0, 30 parallel, and 30 antiparallel. The tilt geometry determines the degree of suppression of the zero order and the asymmetry of the diffraction. Highly efficient beam-steering devices are shown to be possible using the antiparallel tilt alignment. 2002 American Institute of Physics. DOI: 10.1063/1.1446216 I. INTRODUCTION Liquid crystal LC phase grating devices have been reported for applications that include switchable zero-order diffraction filters, color control in two-dimensional displays, large screen projection display systems, beam steering, and for holographic optical elements in a number of optical computing applications. 1 5 The optical properties of some of these liquid crystal phase grating structures have been previously described theoretically by solving the wave equation using a number of approximations. The theory developed by Kogelnick is valid in the thick-grating regime and not the Raman Nath regime. 6,7 Rigorous coupled-wave analysis has been able to describe the full range of thickness regimes but still relies on dividing the device structure into layers. This requires some averaging of the liquid crystal orientation within particular regions of the device. 8,9 The simplest possible phase grating device structure consists of a linear nematic LC sandwiched between a continuous ground plane electrode and a periodic striped electrode. 10,11 A single period of such a device is shown in Fig. 1. The stripes of the periodic electrode on the upper plate are in the plane that is perpendicular to the plane of the figure. The LC layer is aligned such the molecular director lies always in the x z plane. A voltage applied to the upper periodic electrode causes the nematic LC in the regions below the electrodes to rotate towards the homeotropic orientation. A periodic phase pattern is then presented to normally incident light that is polarized in the x direction. The actual operation of the device is more complicated because field fringing occurs at the electrode edges, in exact analogy to the fringing field effects seen at the open edge of a parallel plate capacitor. These nonuniform fields couple to the director distortion and in turn give rise to complicated switched structures in the LC director configuration. 11,12 This decreases the available phase modulation depth for a given device thickness and causes a departure from the simple square wave profile. Methods that have been reported in order to reduce these effects include the fabrication of polymer In this article variations in the orientation of the molecular director will be constrained to the two-dimensional x z plane that is shown in Fig. 1. The orientation at every point in space is therefore described solely by the zenithal tilt angle (x,z). The bulk elastic deformation energy in the nematic liquid crystal layer is given by the usual Frank Oseen free energy in Eq. 1. 17 The elastic constants K 11, K 22, and K 33 correspond to splay, twist, and bend deformations, respectively, and the director n is a unit vector describing the ava Electronic mail: carl.brown@eng.ox.ac.uk walls in the regions between the striped electrodes or by the use of interdigitated grounding electrodes. 3,4,13 In the current work these fringing field effects have been exploited in order to produce a device that is capable of switching optical power into higher diffraction orders with high efficiency. Asymmetry is introduced into the structure by using high tilt alignment of the nematic LC at the upper and lower supporting plates. A theoretical analysis of the device is presented using nematic continuum theory in order to predict the nematic LC tilt profile, and a rigorous optical wave method is employed to predict the optical diffraction properties. The finite-difference time-domain FDTD method was chosen for the optical wave calculations due to its generality in solving Maxwell s equations in anisotropic and inhomogeneous media, without the introduction of restricting approximations. This method has been successfully applied in predicting the optics of small-sized LC structures, such as twisted nematic pixel edges 14 and ferroelectric domain walls. 15 Application of the earlier method provides an accurate description of the optical field profile in two dimensions, as it propagates through the thickness of the device. One further advantage of the FDTD method over other already used approaches is that it is possible to accurately determine the differences between the diffraction patterns for illumination on the two different faces of the device. This asymmetry between the two different illumination directions is well known in studies of transparent dielectric surface relief structures on flat substrates. 16 II. THEORY AND DEVICE GEOMETRY 0021-8979/2002/91(6)/3495/6/$19.00 3495 2002 American Institute of Physics

3496 J. Appl. Phys., Vol. 91, No. 6, 15 March 2002 Brown, Kriezis, and Elston the upper plate. This is a standard approximation, which is used in order to avoid the requirement to solve for the potential as z. Having obtained the molecular director orientation and thus the corresponding optical tensor optical, we then proceed to the study of the optical wave propagation through the device in order to determine its diffractive properties. A rigorous analysis should be based on the solution of Maxwell s equations for the optical field vectors (e,h): Ãh optical x,z e t, 4 FIG. 1. Single period of the liquid crystal phase grating under study. eraged local orientation of the liquid crystal molecules 2W B K 11 "n 2 K 22 n" n 2 K 33 n n 2 E"E. For the geometry in Fig. 1 the director is given by n (cos,0,sin ) and the term in K 22 disappears. The static configuration of n is found by minimization of the Euler Lagrange equation W B W B 0. The electric field vector E(x,z) is given by the gradient of the potential V(x,z), which is calculated using the Laplace Eq. 3 for an anisotropic medium. The permittivity tensor follows from the definition given earlier for the director n, V 0; 1 2 cos2 0 sin cos 0 0 3 sin cos 0 sin. 2 Equations 2 and 3 have been discretized and solved self-consistently using a numerical relaxation algorithm on a rectangular mesh that fills the region of the cell between the glass plates in Fig. 1. Dirichlet fixed boundary conditions have been used for the tilt angle for the full width of the upper and lower plates, and for the potential V on the regions of these plates where an electrode exists. Periodic boundary conditions have been used at the boundaries in the x direction for both V and. Periodic boundary conditions have also been used for V in the regions where there is no electrode on h Ãe 0 t. 5 The FDTD method provides a general explicit solution of Maxwell s Eqs. 4 and 5, both in space and in time. It has been successfully introduced to the optics of LC display devices for a number of characteristic geometries and device configurations. 14,15 The reader is referred to the earlier articles and to general FDTD method literature 18 for particular details of the method. In short the FDTD method solves Maxwell s equations without introducing any approximations apart from the numerical discretization error, and therefore, rigorously models the optical tensor variation, which is here dominated by the liquid crystal director orientation in the fringing fields of the striped electrodes. Time stepping in the FDTD algorithm is undertaken until the steady-state harmonic response of the device has been reached. Periodic boundary conditions are imposed along the x direction consistent with the device structure, while appropriate absorbing boundary conditions the perfectly matched layer PML are applied along the z direction in order to provide reflectionless computational space termination. At the end of the FDTD modeling the emerging propagating fields will be known along a line parallel to the x axis spanning the extent of the grating pitch, at the exit face of the device. Due to the periodicity this will naturally lead into a decomposition of the propagating fields in terms of Floquet type plane-wave modes, which are the diffracted orders. As the director orientation involves tilt only, combined with the fact that the illuminating plane wave is polarized along the x direction p-polarized, this will result in solely p-polarized diffracted orders. The plane wave spectrum representation for the forward propagating transmitted field acquires the typical form M e x,z m M e p m pˆ m exp j m x m z, 6a m 2m L, m k 2 2 m. 6b In Eq. 6a the plane wave amplitudes e p m are now easily obtained from the recorded FDTD data e(x,z) making use of the Floquet modes orthogonality properties. The normalized optical power carried by each individual p-polarized diffracted order diffraction efficiency can be calculated through the following formula, 16 which has been adapted for the case of normally incident electric field of unit amplitude

J. Appl. Phys., Vol. 91, No. 6, 15 March 2002 Brown, Kriezis, and Elston 3497 TABLE I. LC material parameters used throughout the simulations. Parameter Numerical value Elastic constant K 11 11.7 pn Elastic constant K 33 19.5 pn Parallel permittivity 1 5.17 Dielectric anisotropy 14.33 Refractive index n o 1.49 9.65 10 15 / 2 Refractive index n e 1.65 30.94 10 15 / 2 P m e m p 2 cos m ; m tan 1 m m. Equation 7 contains the cos( m ) term, m being the angle of propagation of order m, which corrects the diffracted beam cross-section propagating at the oblique angle m.a similar plane wave expansion to Eq. 6a can also be written for the backwards propagating reflected spectrum and the corresponding diffraction efficiencies are also defined in accordance to Eq. 7. The sum of all of the diffraction efficiencies for the transmitted and reflected parts will be equal to one, due to the energy conservation principle for a losses device. 7 III. NUMERICAL RESULTS A single period of the liquid crystal grating is shown in Fig. 1. As noted earlier a transparent electrode covers the whole extent of the lower glass plate, whereas stripe electrodes are patterned on the upper plate. The liquid crystal material parameters that have been used in all following simulations are shown in Table I. The values given correspond to room temperature measurements on the nematic material E7. 19 Note that these values would lead to a Frederikz threshold of 0.95 V in a cell with continuous electrodes and zero surface pretilt. 20 The geometrical dimensions that have been used for the cell geometry of Fig. 1 are grating pitch L 9.6 m, stripe electrode width w 4.8 m, and thickness of the nematic LC layer d 4.8 m. Figure 2 shows director orientation profiles that were calculated for the case of an applied voltage of V 0 3Von the conducting stripes keeping the continuous electrode grounded, with zero surface pretilts 2a, parallel pretilts ( s 30 /30 ) 2b, and antiparallel pretilts ( s 30 / 30 ) 2c. These particular pretilt values were chosen because they can be readily obtained with silicon oxide alignment layers in an experimental device. 21 The tilt orientations are shown in the figure on a 300 nm 300 nm grid for clarity. The numerical calculations were carried out with a finer grid spacing of 100 nm 100 nm. In Fig. 2 a the tilt profile is completely symmetrical about the x 4.8 m line at the center of the electrode. The director tilts are in opposite directions on either side of the striped electrodes and there is a line with no switching at the center of the electrode. This switching profile occurs at voltages above the Frederikz threshold because the shape of the fringing electric fields at the electrode edges leads to a competition between the two switching directions. For still higher voltages this would lead to the formation of defects FIG. 2. Director orientation profiles for the LC phase grating of Fig. 1 at V 0 3V. a Zero pretilt; b s 30 /30 ; c s 30 / 30. under the electrodes, which could only be correctly modeled by allowing for variations in the nematic order parameter. In Fig. 2 the switching remains in the elastic regime and can therefore be accurately described using continuum theory. The effect of the surface pretilts and the fringing fields in Figs. 2 b and 2 c lead to an asymmetry in the switching profiles. The surface tilts on the upper and lower glass plates are parallel in Fig. 2 b antiparallel rubbing/evaporation and so the whole director field is already tilted at 30 with no voltage applied. The director field therefore switches more readily and in the direction of initial tilt under an applied voltage. The asymmetry in the switched director profile is far more significant in Fig. 2 c where surface tilts on the upper and lower glass plates are antiparallel. At zero applied voltage the director profile is predistorted due to the pretilt but there is no bias of the direction of tilt.

3498 J. Appl. Phys., Vol. 91, No. 6, 15 March 2002 Brown, Kriezis, and Elston When a plane wave impinges on the device it is diffracted giving rise to a number of propagating modes. The optical power, which is carried by each propagating mode, will be determined in the following calculations. Two different alternatives exist for illumination at normal incidence: one is from the side of the continuous electrode and we will refer to this as the lower side illumination, whereas the other is from the patterned electrodes side and it is referred to as the upper side illumination in consistence with Fig. 1. Optical calculations are performed for a free space wavelength of 633 nm. Tilt angles are interpolated over a finer grid with a cell size of 20 nm 20 nm. The latter grid is used in conjunction with the FDTD method for all calculations at 633 nm. Liquid crystal refractive indices for E7 material at this particular wavelength see Table I are: n o 1.518, n e 1.735. Lower and upper glass plates are assumed to have a common refractive index n glass 1.5. Figure 3 plots the diffraction efficiencies of the lower order modes corresponding to m 3 versus the applied voltage V 0.In Fig. 3 and in subsequent Figs. 3 and 4 the diffraction efficiency of the zero order mode is measured using the left vertical scale ranging from 0 to 1, whereas the efficiencies of all other diffracted orders (m 0) are measured using the right vertical scale ranging from 0.0 to 0.4. Illumination is polarized along the xˆ direction and at normal incidence. Figures 3 a and 3 b correspond to the lower side and upper side illumination, respectively, when zero pretilt ( s 0 /0 ) has been assigned to both supporting surfaces. At the low voltage regime the diffraction efficiency out of the zero order mode is approaching its maximum value, which is only limited by the reflection losses at the air/gas and glass/lc interfaces. For the practical material parameters considered here this limiting value is close to 0.9, as shown in the figures, because some light is inevitably reflected back at the earlier interfaces where a refractive index mismatch is found. It is noticeable that under normal operating voltages above the threshold i.e., V 0 2 3 V the directly transmitted field zero order mode is very weak, leading to selective redistribution of the optical power amongst the various diffracted orders. In particular, the second order (m 2) mode appears to be the strongest in this case. Due to the zero pretilt the structure is symmetric, and therefore, diffracted orders corresponding to negative values of m carry the same optical power as positive ones. Also the side of illumination has a noticeable effect: it leads in the case of the upper side illumination to an enhancement of the second order mode at the expense of other non-zero orders for instance m 3, 1,1,3) as clearly seen in Fig. 3 b, while leaving the zero order mode unaffected. The sum of all diffraction efficiencies for the earlier forward propagating modes will result in a value around 0.9 due to the presence of reflections, the exact value depends on the distribution of optical power amongst the various diffracted orders, where each one experiences a different reflection coefficient. As previously mentioned asymmetry is introduced in the device by assigning nonzero pretilt angles. Figures 4 a and 4 b correspond to the lower side and upper side illumination, respectively, when the two supporting surfaces have been treated with antiparallel evaporation leading to the FIG. 3. Diffraction efficiencies of the lower order modes versus the applied voltage V 0 for a device with zero pretilt on both surfaces. a Lower side illumination; b Upper side illumination. same pretilt angles ( s 30 /30 ). As can be inferred from Fig. 2 b, due to the identical nonzero pretilt on both surfaces the director profile is less distorted compared to the one of Fig. 2 a, and thus demonstrates a smoother variation. This has a direct impact on the optical response of the device: it is now less diffracting compared to the zero pretilt case and the power carried by the zero order mode does not drop below 0.4 for all voltages considered. However, the nonzero pretilt has introduced an asymmetry to the power carried by the positive and negative diffracted orders, and this asymmetry can be quite significant. In order to quantify this effect we define an asymmetry factor as m 100 P m P m /(P m P m ). As the strongest diffracted orders for this director profile are those corresponding to m 1, 1,

J. Appl. Phys., Vol. 91, No. 6, 15 March 2002 Brown, Kriezis, and Elston 3499 FIG. 4. Diffraction efficiencies of the lower order modes versus the applied voltage V 0 for a parallel pretilt device ( s 30 /30 ). a Lower side illumination; b Upper side illumination. FIG. 5. Diffraction efficiencies of the lower order modes versus the applied voltage V 0 for an antiparallel pretilt device ( s 30 / 30 ). a Lower side illumination; b Upper side illumination. in both types of illumination, we tabulate the values of the asymmetry factor 1 in Table II in order to facilitate the comparisons. As can be seen in Table II, in the case of the lower side illumination the recorded asymmetry factors are rather low, reaching a peak value of 10%. On the other hand, reversing the side of illumination to the upper side results in an asymmetry of almost 50% at V 0 4 V. Although the low TABLE II. Asymmetry factor 1 vs applied voltage V 0 in the case of a parallel pretilt device ( s 30 /30 ). 1V 2V 3V 4V Lower side illumination 2.11 9.50 5.15 10.00 Upper side illumination 1.05 0.46 27.82 47.66 operating voltage is preferred in order to prevent the generation of defects, even at V 0 3 V an appreciable asymmetry of 27% exist, which can be used advantageously if preferential light steering to one side of the device is sought. Far more interesting device behavior is observed when antiparallel pretilt angles s 30 / 30 are used. As clearly seen from Fig. 2 c the corresponding director profiles are steeper compared to the parallel tilt case and they are expected to be more diffracting. In Figs. 5 a and 5 b the diffraction efficiencies of the lower diffracted orders are plotted versus the applied voltage for both illumination conditions. Even an applied voltage of V 0 2 V generates strong diffracted orders at the expense of the directly transmitted field. The presence of a nonzero pretilt is again introducing asymmetry to the higher diffracted orders. Table III records the asymmetry factor 1 in the case of antiparallel pretilt de-

3500 J. Appl. Phys., Vol. 91, No. 6, 15 March 2002 Brown, Kriezis, and Elston TABLE III. Asymmetry factor 1 vs applied voltage V 0 in the case of an antiparallel tilt device ( s 30 / 30 ). 1V 2V 3V 4V Lower side illumination 33.33 45.22 31.34 68.59 Upper side illumination 28.20 39.61 14.01 49.67 vices. It is noticeable that in the intended region of operating voltages (V 0 2 3 V strong asymmetry exists, and this is observable in both types of illumination. In particular, at V 0 2 V very strong first order diffraction occurs together with an asymmetry factor in the order of 40% 45%. To complete the optical studies we have plotted in Fig. 6 the spectral response of the lower order diffraction efficiencies for a representative device. The device under study is the one with antiparallel pretilt surfaces subject to an applied voltage of V 0 3 V, and the LC refractive indices are set to those predicted by the E7 dispersion curve of Table I. Lower side illumination has been considered. Figure 6 demonstrates a continuous exchange of optical power between the various FIG. 6. Spectral response of the antiparallel pretilt ( s 30 / 30 ) LC phase grating operating at V 0 3 lower side illumination. diffracted orders. In also predicts that for particular wavelengths it is possible to extinguish the directly transmitted light, and therefore distribute light only amongst the higher diffracted orders. IV. CONCLUSIONS We have studied the optical diffraction from a LC phase grating constructed using striped electrodes with unity markspace ratio. Effects arising from the fringing electric fields were fully incorporated in the director orientation calculation, which in turn was used in the optical analysis by the FDTD method. It was found that the surface pretilt angles greatly influence the optical power distribution amongst the zero and higher order diffracted modes. Highly diffracting devices with a high degree of asymmetry in the diffraction pattern can be obtained for antiparallel pretilt angles s 30 /30 operating in the elastic regime (V 0 2 3 V. It has also been demonstrated that illumination from the continuous electrode side is not equivalent to illumination from the patterned electrode side. 1 K. Knop and J. Kane, U.S. Patent No. 4,251,137 1981. 2 H. Sakata, M. Nishimura, M. Yamamnobe, and K. Matsumoto, J. Opt. Soc. Am. A 4, 32 1987. 3 H. Murai, Liq. Cryst. 15, 627 1993. 4 J. Chen, P. J. Bos, H. Vithana, and D. L. Johnson, Appl. Phys. Lett. 67, 2588 1995. 5 M. Stadler and M. Schadt, Proc. 16th Int. Conf. Disp. Res., 1996, p. 434. 6 H. Kogelnick, Bell Syst. Tech. J. 48, 2909 1969. 7 R. W. Boyd, Nonlinear Optics Academic, San Diego, 1992. 8 T. Gaylord and M. Moharum, Proc. IEEE 73, 894 1985. 9 M. Moharum, D. Pommet, E. Grann, and T. Gaylord, J. Opt. Soc. Am. A 12, 1068 1995. 10 J. Prost and P. S. Persham, J. Appl. Phys. 47, 2298 1976. 11 R. G. Lindquist, J. H. Kulick, G. P. Nordin, J. M. Jarem, S. T. Kowel, and M. Friends, Opt. Lett. 19, 670 1994. 12 G. Haas, H. Wohler, M. W. Fritsh, and D. A. Mlynski, Mol. Cryst. Liq. Cryst. 198, 15 1991. 13 W. M. Gibbons and S. T. Sun, Appl. Phys. Lett. 65, 2542 1994. 14 E. E. Kriezis and S. J. Elston, Opt. Commun. 177, 69 2000. 15 E. E. Kriezis and S. J. Elston, J. Opt. A, Pure Appl. Opt. 2, 27 2000. 16 Electromagnetic Theory of Gratings, edited by R. Petit Springer, Berlin, 1980. 17 C. W. Oseen, Trans. Faraday Soc. 29, 883 1933 ; H.Zöcher, ibid. 29, 945 1933 ; F. C. Frank, Discuss. Faraday Soc. 25, 19 1958. 18 A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method Artech House, Boston, 2000. 19 Material Licrilte BL001, Merck Eurolab Ltd., Poole, UK. 20 H. Gruler, T. J. Scheffer, and G. Meier, Z. Naturforsch. A 27A, 966 1972. 21 E. Guyon, P. Pieranski, and M. Boix, Lett. Appl. Eng. Sci. 1, 19 1973.

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