This work studies the polarization characteristics of diffracted beams from a liquid crystal polarization grating. The grating is fabricated by exploiting the photo-alignment effect on a substrate that is coated with an azo dye-doped polyvinyl alcohol (PVA) film. The mechanism is induced by the irradiation of this film with suitably polarized light, which reorients the dyes. The reoriented dyes then align the liquid crystals (LCs). An LC polarization grating is fabricated using this approach. The LC alignment of the grating on one substrate is uni-directionally parallel to the surface, while that on the other is rotated. The polarization and the intensity of the diffracted beams are measured. A simulation based on the finite-difference time-domain (FDTD) method is performed and is very consistent with the experimental results.
©2007 Optical Society of America
Holographic gratings are divided into two categories - intensity-modulated gratings (IGs)  and polarization-modulated gratings (PGs). The former are typically generated using two coherent beams with identical polarization, while the latter are generated using two orthogonally polarized beams. PGs based on liquid crystals are interesting because most of them exhibit unique diffraction characteristics, such as polarization-independence, low scattering and high diffractive efficiencies [2–6]. Therefore, these devices have attracted much interest because of their great potential for various applications.
In this experiment, a PG is recorded in an LC cell (Fig. 2). One substrate is coated with a photosensitive azo dye-doped PVA film, whereas the other is a homogeneous PVA surface made by rubbing. Two coherent, orthogonally circularly polarized beams are incident onto the cell from the command substrate, and produce a polarization modulation on the command surface. The periodically modulated polarization induces periodical re-orientation of dyes via the photo-isomerization effect, which then realign the LCs in contact with the command surface, giving rise to a liquid crystal PG. A simulation that is based on the finite-difference time-domain method, which accounts for the structure, demonstrates the propagation of waves in the LC grating and explains the experimental results.
The Jones matrix formalism is commonly used in analyzing the wave behavior through a LC cell that exhibits one-dimensional (1D) director change . However, in a complex structure that exhibits two-dimensional (2D) or three-dimensional (3D) director variation, the finite-difference time-domain (FDTD) approach is commonly adopted [8–10]. The director orientation in the PG herein is three-dimensional. Therefore, the polarization state and intensity of the diffracted beams are calculated using the FDTD method. A magnified image of a PG under a polarized optical microscope (POM) is used to infer the LC directors profile. The simulated results agree closely with the experimental results.
The LCs and azo-dyes used herein are K15 (from Merck) and methyl orange (MO, from Showa). Figure 1 presents the absorption spectrum of the MO/PVA (dissolved in dimethyl sulfoxide). Two indium-tin-oxide- (ITO-) coated glass slides, separated by two 25 µm-thick plastic spacers, are adopted to produce an empty cell. An alignment film PVA is coated onto one of the two ITO glass slides and rubbed. This surface is called the reference surface. The other is coated with an azo dye-doped PVA film, and is called the command surface. The concentration of MO doped in the PVA solution is ~0.75 wt%. The results for MO dye-doped PVA regarding photo-alignment on LCs are similar to those obtained from dye-doped polyimide (PI) . However, the baking temperature of a PVA film is about 120°C, which is much lower than that of a PI film (about 250 °C). Therefore, the PVA film is adopted to prevent the degradation of the dyes at high temperatures. Drops of LCs are then injected into the empty cell to produce a homogeneous sample. The LC molecules align with each other near the rubbed surface and extend through the bulk of the sample to another surface without the rubbed alignment of the film. The homogeneous alignment of the LC cell is confirmed using a conoscope.
With reference to Fig. 2, a linearly polarized Ar-ion laser (λ=514.5 nm) is split into two equally intense beams with a total intensity of 1 W/cm2. The two beams, each passing through a quarter-wave plate, become orthogonally circularly polarized, and are incident onto the cell from the command substrate with an intersection angle α of ~2°. Therefore, the grating spacing is theoretically ~14.7 µm. The cell is illuminated for 1 h to fabricate a PG.
Photo-induced birefringence and dichroism are associated with the trans-cis-trans photoisomerization alignment of the azo dye. The mechanism of reorientation is associated with the photoisomerization alignment of the MO azo dyes, such that their long axes are perpendicular to the polarization of the incident light after many trans-cis photoisomerization cycles. The LC molecules near the command surface are then reoriented by the dyes such that the director of the LCs is perpendicular to the polarization of the local exciting light , and the LCs close to the reference surface retain their orientation because of the strong anchoring force that is exerted by the rubbed PVA surface. Since the interference between two coherent orthogonally circularly polarized beams generates the exciting light on the command surface, the light-intensity is constant as the polarization is periodically modulated, as shown in Fig. 2. Therefore, a PG is fabricated. Figures 3(a) and (b) present images observed under a POM with crossed and parallel polarizers, respectively. Figure 3 indicates that the grating spacing is ~15.4 µm, which value was theoretically predicted. Notably, obvious disclination lines were present in the spacings in Fig. 3(a). The brightness in the center between the two adjacent disclination lines is low, and increases slowly toward the lines. This result is reasonable, since the LCs of the formed PG (Fig. 2) on one substrate are uni-directionally parallel to the surface, while the other substrate is rotated. The parallel alignment regions are dark, and become brighter as the twisted angle increases under cross-polarizers.
Figure 4(a) magnifies part of Fig. 3(a). The disclination lines are clearly observed. Figure 4(b) presents the corresponding LC director profile on the command surface. The cause of this disclination line can be understood with reference to Fig. 5. Figure 5 indicates that the twist directions from the two sides approaching the disclination line oppose each other. A disclination line must therefore exist at the boundary, consistent with the continuum theory. The simulation considers the director profile on the command surface.
An He-Ne laser is diffracted from the formed PG to investigate the diffraction characteristics, as presented in Fig. 6. The sample is placed between two polarizers such that the first polarizer is placed before the reference surface with its transmission axis parallel to the rubbing direction of the sample. Figure 7 displays the diffraction patterns of the formed PG; (a) no polarizer or analyzer is placed before or after the cell, (b) a polarizer (P) and an analyzer (A) are present with P‖ A, and (c) a polarizer (P) and an analyzer (A) are present with P⊥A, based on the setup in Fig. 6. In Fig. 7(b), with P‖ A, the zeroth-order diffraction is bright, while weak first-order diffractions are also present, indicating that the polarization of the first-order diffraction is elliptic. In Fig. 7(c), with P⊥A, the zeroth-order diffraction disappears, while the first-order diffractions become bright, indicating that the polarization of the zeroth-order diffraction equals that of the incident probe beam.
3. Simulation, experimental results and discussion
The simulation is implemented using the FDTD method, which is a direct method for solving the Maxwell curl equations. The most commonly used FDTD formulations are based on the Yee grid . The Maxwell curl Eqs. (1) and (2) are rewritten in discrete forms as Eqs. (3) and (4), respectively, where n and Δt represent the time steps and the time difference, respectively.
Equations (5) and (6) describe the derived electric and magnetic fields, respectively. Notably, the electric and magnetic fields are updated from previous electric and magnetic fields (such as in deriving e.g. En is derived from En-1 and Hn-1/2).
Since the LC is an anisotropic medium, its dielectric tensor ε is expressed as a 3×3 tensor 
where the refractive indices ne and no are the extra-ordinary and ordinary light, respectively, in the LC material. θc is the angle between the z-axis and the director of LC, and φc is the angle between the x-axis and the projection of the director of LC on the x-y plane. Apply Eqs. (5), (6) and (7) in the computer code and applying the electric and magnetic fields at different time steps, yield the wave propagation in the LC. Since an LC grating has periodic replicas, only one period of the grating has to be modeled by applying periodic boundary conditions at the boundaries y=0 and y=yp, as presented in Fig. 8(a) .
The perfect matching layer (PML) is an artificial medium with fictitious exponential conductivities that vary from the normal to the tangential directions of electric components . Therefore, any arbitrary wave that enters the PML is absorbed exponentially without reflection. A PML is adopted at the front and back of the LC cell absorbs the scattered wave. The electric fields that emerge from the LC cell are integrated in Fourier transformation, as given by Eq. (9), and then the intensities and polarizations of the diffracted beams are then obtained.
In the FDTD simulation model, the Yee cubic grid size Δ, which is the unit distance between two nearby fields, should be defined; in this case, it is set to 1/10 of the incident wavelength to yield the required simulation accuracy. The Courant number, defined as c 0Δt/Δ, is 0.5; thus, the time difference Δt can be derived from the Courant number. The fabricated PG has a spacing of 15.4 µm and a cell gap of 25 µm, which correspond to, respectively, 243Δ along y-axis and 395Δ along z-axis. The experimental LCs director profile, presented in Fig. 4(b), is considered in the simulation. Figure 8 plots the simulated results. Figure 8(a) presents the pattern of the wave propagating in the LC cell. Figure 8(b) presents the simulated image under a crossed POM, where the dark lines on the two sides correspond to the disclination lines, which are shown in Fig. 4(a). Between the two dark lines, the image exhibits a gradient in brightness, which is consistent with the experimental result obtained under the crossed POM (Fig. 4(a)). Figures 8(c) and 8(d) plot the diffraction efficiencies and the polarization states of the diffraction beams, respectively. Comparing Figs. 7(c) with 8(d) reveals that the experimental results agree well with the theoretical predictions.
To confirm that the experimental and simulated results are mutually consistent, the diffraction efficiencies of the zeroth- and first-orders are measured using the setup presented in Fig. 6 as the analyzer is rotated to make an angle with the polarizer axis. The diffraction efficiency is defined as the ratio of the diffracted intensity to the intensity of the incident beam. Figures 9(a) and 9(b) plot the measured results and the theoretically fitted results, respectively, for the zeroth- and first-order diffracted beams. Again, the experimental results clearly agree closely with the theoretical predictions. Additionally, Fig. 9(a) indicates that the efficiency is maximal at an analyzer angle of 0°, and the minimum is approximately zero. This result reveals that the zeroth-order beam has the same polarization as the incident probe beam. For first-order diffraction (Fig. 9(b)), both the experimental data and the simulation reveal that the polarization is elliptic, and the long axis of the ellipse is perpendicular to the polarization of the zeroth-order beam. Figure 9(b) indicates that the experimental and simulated results are slightly mismatched. The experimental results concerning the first-order diffractions reveal that the polarization state is more linear than indicated by the simulation. This discrepancy is associated with the slight distortion of the LC orientation in the bulk of the sample. The simulation assumes the ideal distribution of the LC directors as they rotate continuously from the reference surface to the command surface in the z-direction. For a PG with a ratio of the cell gap to the spatial periodicity of (d/Λ)≤1, the LC director is modulated by the anchoring conditions, yielding a perfect bulk replica of the polarization gratings at the aligning substrates [5,15]. However, the actual LC directors in this cell with a cell gap (25 µm) that exceeds the grating spacing (14.7µm) are expected to deviate slightly from the ideal distribution, causing distortion in the bulk.
A PG is fabricated using photoalignment on an azo dye-doped PVA film. The FDTD approach is adopted to simulate the diffraction characteristics of the formed PG. The experimental results are consistent with the simulated results.
The authors would like to thank the National Science Council (NSC) of the Republic of China (Taiwan) for financially supporting this research under Grant No. NSC 95-2112-M-006-022-MY3.
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