Abstract: The photonic band structure and transmission properties of a cholesteric blue phase II liquid crystal, which is elongated in the  direction by electrostriction, are analyzed by finite-difference time-domain method. The simple cubic lattice deforms into a tetragonal lattice under the influence of an electric field, resulting in a change of the photonic band structure. Moreover, we show that the circular polarization dependence of the transmittance spectrum changes in an electric field, a behavior that has yet to be observed in experiment.
© 2014 Optical Society of America
Cholesteric blue phases (BPs) are liquid crystal phases which typically appear between the cholesteric and isotropic phases of a chiral liquid crystal. BPs spontaneously form three-dimensionally periodic helical structures with a periodicity of the order of a few 100 nms [1, 2], and are classified into three types, BP I, BP II and BP III, based on the lattice symmetry. BP I has body-centered cubic symmetry, BP II has simple cubic symmetry, and BP III is said to have an amorphous structure. BPs have attracted much attention due to their potential applicability to next generation displays and other optical devices because of their attractive properties such as fast electro-optic response (< 1 ms) and macroscopic optical isotropy in the longer wavelength than the Bragg reflection [3–6].
BPs have also been studied as tunable photonic crystal and metamaterials, since the three-dimensional structure of BPs can be deformed by applying moderate electric field [7–9], and guest materials such as dyes, polymers and nano-particles can be embedded in the lattice [10, 11]. Understanding the optical properties of BPs, both pristine (cubic) and modulated, is therefore important to corroborate experiments or predict performance of optical devices. There are several studies that are available on the optical properties of pristine BPs. Techniques such as plane-wave expansion, Berreman’s 4 × 4 matrix, and finite-difference time-domain (FDTD) method have been used to simulate the photonic band structure, Kossel diagram, and transmission properties of BPs [12–15]. However, there are no reports on the optical properties of BPs under electrostriction. Here, we analyze the photonic band structure and the transmission properties of BPII, elongated in the  direction by electrostriction. We describe the orientational order of the BPII under an electric field using the Landau-de Gennes (L-dG) theory, and calculate the optical properties using the FDTD method.
2. Dielectric constant distribution model and calculation method
The lattice structure and local orientational order of BPII under an electric field was calculated using the L-dG theory, which represents the local orientational order of liquid crystals by a second-rank symmetric and traceless tensor Q . In the L-dG theory, the free energy density of a chiral liquid crystal is described by the following equations.
The dielectric tensor ε and scalar order parameter S are calculated from χ according to the following equations.17].
The photonic band structure and transmission properties of BPII were calculated by the FDTD method, which directly solves Maxwell’s equations in real space using spatial and temporal discretization . To calculate the photonic band structure of BPII, we used a unit cell of BPII as the computational region. Bloch’s periodic boundary condition was imposed on the boundaries of the unit cell. Linearly and circularly polarized light was given as the initial field in the unit cell, and then eigenmodes were calculated for each wavenumber. To calculate the transmission properties of BP II in parallel and perpendicular to the direction of applied electric field, we assumed that BPII were aligned with the (100) or (001) plane between glass substrates with 13.6 μm thickness. The direction of the applied electric field was in the  direction of the BPII. Periodic boundary conditions were imposed on the planes perpendicular to the light propagation direction, and perfect matching layer absorbing boundary conditions  were applied in the light propagation direction. Right-handed circularly (RC) and left-handed circularly (LC) polarized Gaussian pulses were excited at one of the glass substrates and the transient response of the electric field was measured at the other substrate. The transmission spectrum was obtained by applying a Fourier transform to the time signal.
In all our calculations, the unit cells of BPII were divided into 32 × 32 × 32 grids. The BP was assumed to have a right-handed helix. Ordinary and extraordinary dielectric constants of the liquid crystal at optical frequencies were εo = 1.52 and εe = 1.72, respectively, and the dielectric constant of glass substrates was εglass = εave. The direction of the applied electric field was assumed in the  direction of the BPII, and the rescaled strengths of the applied electric field were Ẽ2 = 0, 0.05, 0.10, and 0.20, respectively. At zero electric field (Ẽ2 = 0), the spatial discretizations used in the FDTD calculation were Δx = Δy = Δz = 4.25 nm, and changes in the lattice constants of BPII due to the applied electric field were implemented by changing Δx, Δy, and Δz. The time discretization was Δt = 4 × 10−18 s. Figure 1(a) shows the scheme of the lattice deformation of BPII having Δεele > 0 by an applied electric field in the  direction. The lattice symmetry of BPII was transformed from simple cubic to simple tetragonal symmetry, and the relation between the lattice constants changed from Lx = Ly = Lz to Lx ≠ Ly = Lz. Figure 1(b) shows the first Brillouin zone of the simple tetragonal structure.
3. Results and discussion
Figure 2 shows the calculated photonic band structure of the BPII unit cell under an electric field applied in the  direction. Although BPII has a simple cubic structure under zero electric field (Ẽ2 = 0), we described the photonic band structure of BPII over the Brillouin zone for a simple tetragonal structure, to compare with that under the electric field (Ẽ2 = 0.05, 0.1, and 0.2). The frequency of the eigenmodes is normalized by L, which is the lattice constant of BPII at Ẽ2 = 0. As the strength of the electric field increases, the first group of the eigenmodes at the M, X, R, and A points in the photonic band structure shifted to lower energies; in contrast, those at the Z point shifted to a higher energy. At the Γ point, degeneracy breaking of the eigenmodes was observed. Contrasting effects were observed when we focus on the Γ-X and Γ-Z directions in the photonic band structure, which correspond to light propagating along and perpendicular to the direction of applied field, respectively. The photonic bands in the Γ-X and Γ-Z directions were the same at Ẽ2 = 0 because of simple cubic symmetry.
When an electric field was applied to the BPII in the  direction, however, the photonic band in the Γ-X direction shifted to lower energies while that in the Γ-Z direction shifted to higher energies, indicating that the reflection bands shift in opposite directions. This is observed more clearly when the transmittances of the BPII are calculated. Figures 3(a) and 3(c) show the transmission spectrum in the Γ-X and Γ-Z directions of BP II. As the strength of the electric field increases, the reflection band for light propagating in the  direction shifted toward longer wavelengths, whereas that in the  direction shifted toward shorter wavelengths. The reflection peak wavelength changed in proportion to the lattice constant of the BPII along the direction parallel to the light propagation direction (Lx, and Lz), as shown in Figs. 3(b) and 3(d). Thus, the main factor for reflection band shift is the deformation of the lattice of BPII. In the case of BPII having Δεele > 0, it has been experimentally reported that the reflection peak wavelength along the direction parallel to the applied electric field shifted to longer wavelengths in proportion to the strength of the field [8, 20]. Our calculations agreed with these reports.
We also calculated the polarization dependence of the photonic band structure and transmission spectra of deformed BPII. Figure 4 shows the circular polarization dependence of the photonic band structure near the X point in the Γ-X direction, and transmission spectra of BPII in the  direction. In pristine BPII, there are two different photonic bandgaps, which correspond to RC and LC polarized light, in the Γ-X direction of the photonic band structure. Since the bandgap for LC polarized light is very small, the reflection band in the transmission spectrum was barely observed in this case, as shown in Fig. 4(e). When the strength of the applied electric field increased, the photonic bandgap for LC polarized light expanded. Therefore, the reflectance for LC polarized light increased and the reflection band for LC polarized light was observed in the transmission spectrum. Figure 5 shows the circular polarization dependence of the photonic band structure near the Z point in the Γ-Z direction, and transmission spectra of deformed BPII in the  direction. Unlike the Γ-X direction, the photonic bands were not separated by circular polarization and the same photonic band structures were observed for both RC and LC polarized light. When the electric field was applied to BPII, in the direction perpendicular to light propagation, the component of the dielectric constant tensor parallel to the electric field increased, and that perpendicular to the electric field decreased due to the reorientation of liquid crystal molecules. Therefore, BPII acted as a uniaxial material and the polarization state of the eigenmodes propagating in the Γ-Z direction changed from circular polarization. As a result, the transmittance increased for RC polarized light, and decreased for LC polarized light. These calculation results indicate that the circular polarization selectivity of the reflection band of deformed BPII becomes different, depending on the relationship between the applied electric field direction and the propagation direction of the light.
We calculated the photonic band structures and the transmission spectra of BPII, which are deformed by electrostriction, using the FDTD method. The reflection band shift due to an applied electric field was in agreement with a previous experimental report. Moreover, it was shown that circular polarization selectivity of the reflection band becomes different, depending on the relation between the direction of the applied electric field and that of the propagating light. In experiment, the orientation of the plane of BPs can be controlled by applying electric fields during isotropic-BPs transition . The case in which the directions of the applied field and the light propagation are parallel corresponds to vertical field switching of liquid crystal devices, and the case in which they are perpendicular corresponds to in-plane switching of liquid crystal devices. Therefore, if we would like to use the selective reflection of BPs for optical devices, we have to consider the driving system of the liquid crystal.
This work was supported by Grant-in-Aid from Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists (11J00829). H. Yoshida gratefully acknowledges support from the JST PRESTO Program.
References and links
1. D. L. Johnson, J. H. Flack, and P. P. Crooker, “Structure and properties of the cholesteric blue phases,” Phys. Rev. Lett. 45(8), 641–644 (1980). [CrossRef]
2. D. C. Wright and N. D. Mermin, “Crystalline liquids the blue phases,” Rev. Mod. Phys. 61(2), 385–432 (1989). [CrossRef]
3. Y. Hisakado, H. Kikuchi, T. Nagamura, and T. Kajiyama, “Large electro-optic Kerr effect in polymer-stabilized liquid-crystalline blue phases,” Adv. Mater. 17(1), 96–98 (2005). [CrossRef]
5. J. Yan, L. Rao, M. Jiao, Y. Li, H. C. Cheng, and S. T. Wu, “Polymer-stabilized optically isotropic liquid crystals for next-generation display and photonics applications,” J. Mater. Chem. 21(22), 7870–7877 (2011). [CrossRef]
6. J. Yan, Z. Luo, S. T. Wu, J. W. Shiu, Y. C. Lai, K. L. Cheng, S. H. Liu, P. J. Hsieh, and Y. C. Tsai, “Low voltage and high contrast blue phase liquid crystal with red-shifted Bragg reflection,” Appl. Phys. Lett. 102(1), 011113 (2013). [CrossRef]
7. G. Heppke, B. Jérôme, H. S. Kitzerow, and P. Pieranski, “Electrostriction of BPI and BPII for blue phase systems with negative dielectric anisotropy,” J. Phys. France 50(5), 549–562 (1989). [CrossRef]
8. G. Heppke, B. Jérôme, H. S. Kitzerow, and P. Pieranski, “Electrostriction of the cholesteric blue phases BPI and BPII in mixtures with positive dielectric anisotropy,” J. Phys. France 50(19), 2991–2998 (1989). [CrossRef]
9. H. S. Kitzerow, “The effect of electric fields on blue phases,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 202(1), 51–83 (1991). [CrossRef]
10. M. Ravnik, G. P. Alexander, J. M. Yeomans, and S. Zumer, “Mesoscopic modelling of colloids in chiral nematics,” Faraday Discuss. 144, 159–169, discussion 203–222, 467–481 (2009). [CrossRef] [PubMed]
11. M. Ravnik, G. P. Alexander, J. M. Yeomans, and S. Žumer, “Three-dimensional colloidal crystals in liquid crystalline blue phases,” Proc. Natl. Acad. Sci. U.S.A. 108(13), 5188–5192 (2011). [CrossRef] [PubMed]
12. R. M. Hornreich, S. Shtrikman, and C. Sommers, “Photonic bands in simple and body-centered-cubic cholesteric blue phases,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 47(3), 2067–2072 (1993). [CrossRef] [PubMed]
13. D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4-matrix formulation,” J. Opt. Soc. Am. 62(4), 502–510 (1972). [CrossRef]
14. M. Ojima, Y. Ogawa, R. Ozaki, H. Moritake, H. Yoshida, A. Fujii, and M. Ozaki, “Finite-difference time-domain analysis of polarization dependent transmission in cholesteric blue phase II,” Appl. Phys. Express 3(3), 032001 (2010). [CrossRef]
15. Y. Ogawa, J. Fukuda, H. Yoshida, and M. Ozaki, “Finite-difference time-domain analysis of cholesteric blue phase II using the Landau-de Gennes tensor order parameter model,” Opt. Lett. 38(17), 3380–3383 (2013). [CrossRef] [PubMed]
16. J. Fukuda, M. Yoneya, and H. Yokoyama, “Simulation of cholesteric blue phases using a Landau-de Gennes theory: effect of an applied electric field,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80(3), 031706 (2009). [CrossRef] [PubMed]
17. S. Urban, B. Gestblom, W. Kuczynski, S. Pawlus, and A. Würflinger, “Nematic order parameter as determined from dielectric relaxation data and other methods,” Phys. Chem. Chem. Phys. 5(5), 924–928 (2003). [CrossRef]
18. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antenn. Propag. 14(3), 302–307 (1966). [CrossRef]
19. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994). [CrossRef]
21. P. Pieranski, P. E. Cladis, T. Garel, and R. Barbetmassin, “Orientation of crystals of blue phases by electric fields,” J. Phys. 47(1), 139–143 (1986). [CrossRef]