Finite-Difference Time-Domain Calculations of a Liquid-Crystal-Based Switchable Bragg Grating

Transcription

1 Kent State University From the SelectedWorks of Philip J. Bos June, 2004 Finite-Difference Time-Domain Calculations of a Liquid-Crystal-Based Switchable Bragg Grating Bin Wang, Kent State University - Kent Campus Xinghua Wang, Kent State University - Kent Campus Philip Bos, Kent State University - Kent Campus Available at:

2 1066 J. Opt. Soc. Am. A/ Vol. 21, No. 6/ June 2004 Wang et al. Finite-difference time-domain calculations of a liquid-crystal-based switchable Bragg grating Bin Wang, Xinghua Wang, and Philip J. Bos Liquid Crystal Institute, Kent State University, Kent, Ohio Received October 15, 2003; revised manuscript received January 6, 2004; accepted January 12, 2004 A polymer-wall-confined transmissive switchable liquid crystal grating is proposed and investigated by twodimensional finite-difference time-domain optical calculation and liquid-crystal-director calculation, to our knowledge for the first time. The results show how to obtain optimized conditions for high diffraction efficiency by adjusting the liquid crystal parameters, grating geometric structure, and applied voltages. The light propagation direction and efficiency can be accurately calculated and visualized concurrently Optical Society of America OCIS codes: , , , , INTRODUCTION Switchable liquid crystal gratings play important roles in various optical systems, such as optical interconnects, optical data storage, optical spectral filters, dynamic color rendering for full-color displays, and color-image capture systems. Reference 1 shows three difference approaches to forming the liquid-crystal switchable gratings. The first approach consists of placement of a liquid crystal between two opposing electrodes, at least one of which has been patterned lithographically. Thus when a voltage is applied, a spatially modulated electric field is produced and electro-optically modulates the refractive index of the liquid crystal. A grating is thereby established that will diffract incident light. The advantages of such a device are that it is easy to manufacture and has low-voltage driving. The disadvantages include the slow switching time and the difficulty of forming a thick grating that operates in the Bragg regime. The second approach is a surface-relief pattern filled with liquid crystal and sandwiched between indium-tin-oxide electrodes. Advantages of this device are its manufacturability and low operating voltage. Its disadvantages are low switching speed and low diffraction efficiency. The third approach is a volume hologram, which has been shown to operate in the Bragg regime. One such hologram is a holographic polymer dispersed liquid crystal grating, in which the liquid crystal droplets are formed in periodic layers throughout the sample and are separated by dense polymer layers. This type of grating has been extensively studied. 2 5 The advantages of this system are that it yields true Bragg operation with high diffraction efficiency, has a simple single-step fabrication process, and provides fast response speed. Its disadvantage is driving voltage that relatively high compared with that of a bulk liquid crystal device. Coupled-wave theory 6 and rigorous coupled-wave analysis 7 are commonly used to investigate the diffraction efficiency of a Bragg grating. In this paper we propose another method to form a switchable liquid crystal grating that is based on a polymer-wall-forming technique. Assuming that both a complete liquid crystal and polymer phase separation occur, then high diffraction efficiency, low-voltage driving, and fast switching speed are expected. Instead of using coupled-wave theory or rigorous coupled-wave analysis, we will accurately analyze the electro-optical characteristics of the device by two-dimensional liquid-crystaldirector calculation and finite-difference time-domain (FDTD) optical calculation. 2. DEVICE DESIGN CONSIDERATIONS The multidimensional alignment method 8 can be used to form a periodic polymer wall; the liquid crystal structure and the procedure for forming the polymer wall are shown in Fig. 1. Before UV light exposure, the liquid crystal and the reactive liquid crystal monomers are homogeneously mixed. To obtain low voltage driving, the hybrid liquid-crystal-director configuration is employed. At the cell s bottom substrate the liquid crystal directors are aligned along the z direction, and at the top they are Fig. 1. Formation of polymer walls from aligned monomer units. The rubbing direction of the cell s bottom substrate is along the z direction; the top substrate has a homeotropic alignment layer /2004/ $ Optical Society of America

3 Wang et al. Vol. 21, No. 6/June 2004/J. Opt. Soc. Am. A 1067 Fig. 2. Two-dimensional hybrid director field at voltages of (a) 0.0 V, (c) 5.0 V, (e) 6.0 V, and (g) 10.0 V; the corresponding refractive-index profile of the device at those voltages are shown in (b), (d), (f), and (h). aligned perpendicular to the substrate. During UV light exposure, polymerization-induced phase separation occurs such that polymer walls are formed in UV-exposed areas and the liquid crystal directors are locked in their initial orientation in these walls, which do not response to the external field. But in the UV-non-exposed area, the liquid crystal directors can be reoriented by an external field. Thus a switchable liquid crystal grating is obtained. In order to simplify the liquid-crystal-director calculation, here we assume that the device has complete polymer and liquid crystal phase separation. But in the actual case, one usually cannot obtain this kind of ideal phase separation, which means liquid crystal will exist in the polymer wall and a small amount of reactive liquid crystal monomer will also be left in the liquid crystal region. Especially when the required polymer wall size is small, it is extremely difficult to obtain smooth polymer wall because of diffraction effects, and the driving voltage will increase if polymer cross linking takes place inside the liquid crystal region. In Ref. 8 a qualitative understanding of the multidimensional alignment device is provided that shows how to estimate the distribution polymer inside a liquid crystal pixel with a 100-m-wide surrounding polymer wall. In our recent fabrication process, we have obtained a smooth 25-m-wide polymer wall. 9

4 1068 J. Opt. Soc. Am. A/ Vol. 21, No. 6/ June 2004 Wang et al. 3. CHARACTERIZATION OF THE DEVICE BY COMPUTER SIMULATION A. Two-Dimensional Liquid-Crystal-Director Calculations The two-dimensional liquid crystal calculation is based on the LC3D software. 10 The liquid crystal directors at the boundaries, which include top and bottom substrates and surrounding polymer walls, are fixed. Therefore, when the relaxation method is used to calculate the director field, these director orientations at the boundaries will not change. The LC3D program is based on the Frank Oseen free-energy density given in Eq. (1) 11 : f g 1 2 K 11 n K 22 n n q K 33 n n D E. (1) Here n is the unit vector director; K 11, K 22, K 33 are the liquid-crystal splay, twist, and bend elastic constants respectively; q 0 is the chiral wave number (2/p); p is the intrinsic chiral pitch of the liquid crystal; D is the electric displacement; and E is the electric field. The relaxation method is used in our calculation, with the liquid-crystaldirector update formula given in Eq. (2), which can be derived from setting the viscous torque equal to the elastic torque 12 : n new i n old i t f g ni, i x, y, z. (2) 1 Here t is the numerical time step used in the simulation, 1 is the rotational viscosity of the liquid crystal material, n new i denotes the new value of the component of the director, n old i denotes the value at the previous time step, and f g ni is the functional derivative, given in Eq. (3): f g ni f g d f n i g dx dn i /dx d f g dy dn i /dy. (3) In determining the new voltage profile, the direct-solve method removes the requirement of extra iteration. The method is based on the fact that when Gauss s law ( D 0) is discretized, an equation that is linear in the values of the discretized voltage results. 13 This equation can then be solved for the new value of the voltage at the current grid point in terms of the values at the surrounding grid points. The parameters of liquid crystal BL006 (ne 1.816, no 1.530, 10.1) and reactive liquid crystal-monomer RM82 (ne 1.656, no 1.532) are used for the liquid-crystal director calculation. One specific example is considered in following discussion for cell thickness of d 15.0 m and width of liquid crystal and polymer wall 1.0 m and 0.5 m, respectively. The simulated liquid-crystal-director configuration results are shown in Figs. 2(a), 2(c), 2(e), and 2(g), which correspond to applied voltages of 0.0, 5.0, 6.0, and 10.0 V, respectively. Considering the case of incident light polarized along the z direction, the liquid crystal effective refractive index can be calculated by its director configuration. Figures 2(b), 2(d), 2(f), and 2(h) show the grating device s effective refractive-index profile at the corresponding voltages of 0.0, 5.0, 6.0, and 10.0 V. Since the boundaries of the polymer wall are assumed to have a strong anchoring condition, the external field will not affect the orientations of the liquid crystal director, and the refractive index is unchanged. B. Two-Dimensional In-Plane-Grating Equation In Ref. 14 a general three-dimensional conical diffraction geometry was studied by coupled-wave analysis, and the angle of diffraction for the ith propagation order was given. For the case of a two-dimensional in-plane grating shown in Fig. 3, the angle of diffraction can be reduced to the well-known grating equation, which is given by Eq. (4): n 1 sin i n 3 sin m m s, m 0, 1, 2,..., m m sin1 n 1 sin n 3 s n 3 i, where n 1 and n 3 are the refractive indices of the incident and the exiting media, respectively. Here n 1 and n 3 are both in air, so n 1 n 3 1.0, i is the incident angle of the input light, m is the mth-order diffraction angle of the output light, m is the diffraction order, is the wavelength of the incident light in free space, and s is the grating periodicity. Equation (5) will be used to verify the diffraction peak position for the FDTD computer simulation results. C. Finite-Difference Time-Domain Computer Simulation for Light Propagating in the Switchable Liquid Crystal Polymer-Wall Grating To accurately simulate the light propagating through the switchable liquid crystal polymer-wall grating, a twodimensional optical calculation is required. Here the FDTD calculation method was implemented This method is a numerical approach for directly solving Maxwell s time-dependent curl equations in a twodimensional or three-dimensional domain with no other assumptions involved. For sourceless inhomogeneous anisotropic media, Maxwell s equations can be written as Fig. 3. Geometrical setup of a two-dimensional in-plane transmissive diffraction grating. (4) (5)

5 Wang et al. Vol. 21, No. 6/June 2004/J. Opt. Soc. Am. A 1069 Fig. 4. Layout of the two-dimensional FDTD computational domain in the XY plane. A plane wave is incident at 30 from the bottom of the grating, and the beam width equals the width of the calculation domain. L1, Width of the liquid crystal; L2, width of the polymer wall. The periodicity s of the grating is L1 L2, and the grating thickness is d. Er r t Hr, (6) Hr 0 r t Er, (7) where (r) is the spatially varying dielectric tensor. the uniaxial liquid crystal case, For n x n x n x n y n x n z r n y n x n y n y n y n z n z n x n z n y n z n z, where n x, n y, and n z, are the liquid-crystal-director components in Cartesian coordinates and is the liquid crystal dielectric anisotropy at optical frequencies. Since the liquid crystal directors can be oriented only in the YZ plane, which is shown in Fig. 1, the incident light polarized along the z direction will experience the largest refractive-index change when voltage is applied. Therefore the following discussion will consider the optical wave in the s-polarized condition, and only the electric component E z and the magnetic components H x and H y exist in Maxwell equations (6) and (7). The electric and magnetic components time-step update formulas in the two-dimensional case are given in Eqs. (9) (11), 1 H y n1/2 H x n1/2 E n1 z E n z t zz, (9) x y H n1/2 x H n1/2 x t n E 0 z, (10) y H n1/2 y H n1/2 y t 0 E n z, (11) x where t is the wave-propagating time step, n is number of the time step, and 1 zz is the zz component in the inverse of the spatially varying dielectric tensor of (r), which here we define as 1 and, x, y, z. The space derivative /z 0, since we consider only twodimensional calculation in the XY plane. (8) Fig. 5. Geometry of the two-dimensional diffraction problem. The diffraction object lies in the y y 0 plane between x d and x d. The normal to the diffracting object is n y. Diffracted light is observed at (x far, y far ). In our computer simulations, the central-difference scheme is employed to calculate the derivative of the time and space. The discretization technique that we used provides fourth-order accuracy 19 to approximate the firstorder derivative in space and second-order accuracy to approximate the first-order derivative in time. The domain of the FDTD calculation is shown in Fig. 4. The perfectly-matched-layer technique is use for terminating the calculation grid. 20 Because of the limitations of computer speed and memory, it is difficult to realize the FDTD method to calculate the far field. However, the far-field result can be easily obtained by near-field to far-field transformation, which can be done by means of the integral theorem of Helmholtz and Kirchhoff, 21 far r 1 4 S n near r G G near rds, (12) where G exp(ikr)/r is known as Green s function and near and far represent the near and the far electric or magnetic field.

6 1070 J. Opt. Soc. Am. A/ Vol. 21, No. 6/ June 2004 Wang et al. For the two-dimensional case (in the XY plane), the near-field to far-field transformation scheme is shown in Fig. 5, and Eq. (12) can be approximated as 22 where R x x far 2 y 0 y far 2 1/2, k 2/. far x far, y far expi/4 8k d d expikr R y near x, y 0 ik y far y 0 near x, y 0 dx, (13) R Therefore, once near-field two-dimensional FDTD calculation is completed, the far-field diffraction pattern can be calculated by Eq. (13). The FDTD calculation domain shown in Fig. 4 is illuminated by the z-direction-polarized plane wave from the bottom of the grating with constant amplitude and 30 incident angle with respect to the gratings surface normal. The simulation wavelength in free space is 1.55 m, and Fig. 6. FDTD near-field calculation for i 30, 1.55 m, s 1.5 m at voltage of (a) 0.0 V, (c) 5.0 V, (e) 6.0 V, and (g) 10.0 V. The corresponding far-field calculations are shown in (b), (d), (f), and (h).

7 Wang et al. Vol. 21, No. 6/June 2004/J. Opt. Soc. Am. A 1071 the simulation grid space is /20. By mapping the calculated liquid-crystal-director field to the FDTD calculation domain and then performing the FDTD calculation, we obtain near-field calculation results for different applied voltages. Figures 6(a), 6(c), 6(e), and 6(g) show the images of the wave propagation of the electric field distributions at voltages of 0.0, 5.0, 6.0, and 10.0 V, respectively. By collecting the calculated near-field results and transforming them to the far field, we obtain the far-field distribution. Figures 6(b), 6(d), 6(f), and 6(g) are the light intensity distribution at the far field for the corresponding voltages. These calculations are normalized to the peak for light passing through an isotropic media with refractive index of 1.5 under the same light source and incident condition. These calculations show clearly that this switchable liquid crystal grating operates in the Bragg regime. If we plug 1.55 m, i 30, and s 1.50 m into Eq. (5), the zero- and first-order diffraction peak positions calculated are 30 and 32.23, respectively, which approximately agree with the FDTD simulation results of and The discrepancies between the FDTD and the theoretical calculation results may come from the different conditions: In the theoretical case, a perfect grating with an infinite number of periods is considered, whereas in the FDTD simulation, only a finite number of periods are considered. In addition, the boundary effect of the calculation domain shown in Fig. 6 may cause some numerical error. Therefore the more periods included in the calculation, the smaller the discrepancy between the simulation and the theoretical results. High diffraction efficiency occurs at 0.0 V, and at 10.0 V the diffraction effect vanishes, because the grating periodicity of s 1.5 m disappears. If the voltage is larger than 10.0 V, the diffraction effect reappears. If the thickness of the device changes, the driving voltages for high diffraction efficiency also change. The relation of the diffraction efficiency scale to the thickness of the device will be studied in future research. The far-field calculations also show that the intensity distribution between the zero-order and the first-order diffraction peaks can be easily adjusted by controlling the driving voltage. It is worth mentioning that for a periodic grating structure, a periodic boundary condition will give a more accurate result than an absorbing boundary condition. It will overcome the shortcomings of the absorbing boundary condition, the calculations can be limited to a single period, and the distortions from the left-hand side of the computational domain in Figs. 6(a), 6(c), 6(e), and especially 6(g) will not appear. The periodic boundary condition has already been carried out and been explicitly explained in Ref. 23. However, it will add more complexity in a coding simulation program, and we have not used that method in this preliminary study. 4. CONCLUSIONS We used the FDTD calculation method to study a new switchable liquid crystal/polymer Bragg grating. The FDTD method exhibits many advantages for characterizing these types of devices: (i) it can accurately calculate the optical effects of light propagating through the device with many different refractive-index profiles without other assumptions; (ii) it can calculate the gratings operated either in the Bragg regime or in the Raman Nath regime; and (iii) the light propagation can be visualized to aid understanding and to optimize the devices geometric dimensions and the material s physical and optical parameters to achieve high diffraction efficiency. We show that the new switchable Bragg grating has high efficiency and operates at low voltages. Corresponding author Philip Bos s address is pbos@kent.edu. REFERENCES 1. R. L. Sutherland, L. V. 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