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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 4 — Feb. 24, 2014
  • pp: 3766–3772
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Photonic band structure and transmission analysis of cholesteric blue phase II: electrostriction in the [100] direction

Yasuhiro Ogawa, Jun-ichi Fukuda, Hiroyuki Yoshida, and Masanori Ozaki  »View Author Affiliations


Optics Express, Vol. 22, Issue 4, pp. 3766-3772 (2014)
http://dx.doi.org/10.1364/OE.22.003766


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Abstract

Abstract: The photonic band structure and transmission properties of a cholesteric blue phase II liquid crystal, which is elongated in the [100] 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

1. Introduction

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

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]

9

9. H. S. Kitzerow, “The effect of electric fields on blue phases,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 202(1), 51–83 (1991). [CrossRef]

], and guest materials such as dyes, polymers and nano-particles can be embedded in the lattice [10

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

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]

]. 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

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]

15

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]

]. 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 [100] 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 [16

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]

]. In the L-dG theory, the free energy density of a chiral liquid crystal is described by the following equations.
f=flocal+fgrad+fele,
(1)
flocal=cTrQ26bTrQ3+a(TrQ2)2,
(2)
fgrad=14K1[(×Q)αβ+2q0Qαβ]2+14K0[(Q)α]2,
(3)
fele=ΔεeleEαEβQαβ.
(4)
Here, flocal is the local free energy, fgrad is the free energy due to the inhomogeneity of liquid crystal, and fele is the free energy due to the electric field. a and b are positive constants and c is assumed to vary with temperature. K0 and K1 are the elastic constants and q0 is the strength and sign of the chirality. E is the applied electric field and Δεele is the dielectric anisotropy at the frequency of E. To reduce relevant parameters, the rescaled free energy density φ ≡ (a3/b4)f is given by the rescaled tensor order parameter χ = (a3/b4)Q. The contribution of the electric field to the rescaled free energy is given by
φele=E˜2e^αe^βχαβ.
(5)
Here, ȇ is a unit vector specifying the direction of electric field E, and 2 ≡ (a2/b3εeleE2 is the rescaled strength of the electric field. Dielectric anisotropy was assumed Δεele > 0. The director distribution at a particular applied field intensity is the χ which minimizes the free energy density of the system.

The dielectric tensor ε and scalar order parameter S are calculated from χ according to the following equations.
εαβ=23Δε×χαβ+εaveδαβ.
(6)
S2=32Tr(χ2).
(7)
Here, εave ( = (εe + 2εo)/3) is the average dielectric constant, Δε is the dielectric anisotropy, and εo and εe are ordinary and extraordinary dielectric constants of the liquid crystal at optical frequencies. In our calculation, the maximum scalar order parameter Smax in the BP II unit cell was normalized to 0.7, which is a typical value found in the nematic phase [17

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]

].

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 [18

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]

]. 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 [100] 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 [19

19. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994). [CrossRef]

] 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 [100] direction of the BPII, and the rescaled strengths of the applied electric field were 2 = 0, 0.05, 0.10, and 0.20, respectively. At zero electric field (2 = 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 [100] direction.
Fig. 1 (a) Scheme of the lattice deformation of BPII having Δεele > 0 by an applied electric field parallel to the [100] direction, and (b) first Brillouin zone of the simple tetragonal structure.
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 LxLy = 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 [100] direction.
Fig. 2 Photonic band structure of the BPII under electric field (a) 2 = 0, (b) 2 = 0.05, (c) 2 = 0.10, and (d) 2 = 0.20.
Although BPII has a simple cubic structure under zero electric field (2 = 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 (2 = 0.05, 0.1, and 0.2). The frequency of the eigenmodes is normalized by L, which is the lattice constant of BPII at 2 = 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 2 = 0 because of simple cubic symmetry.

When an electric field was applied to the BPII in the [100] 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.
Fig. 3 Transmission spectrum of deformed BPII for RC polarized light (a) in the [100] direction and (c) in the [001] direction. (b) and (d) shows the applied electric field dependence of the reflection peak wavelength and lattice constant of the BPII along the light propagation direction.
As the strength of the electric field increases, the reflection band for light propagating in the [100] direction shifted toward longer wavelengths, whereas that in the [001] 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

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]

, 20

20. N. R. Chen and J. T. Ho, “Electric-field-induced phase diagrams of blue-phase systems,” Phys. Rev. A 35(11), 4886–4888 (1987). [CrossRef] [PubMed]

]. 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 [100] direction.
Fig. 4 Circular polarization dependence of the photonic band structure in the Γ-X direction ((a)-(d)), and transmission spectra of BPII in the [100] direction ((e)-(h)). The normalized strengths of the electric field are 2 = 0 ((a), (e)), 2 = 0.05 ((b), (f)), 2 = 0.10 ((c), (g)), and 2 = 0.20 ((d), (h)).
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 [001] direction.
Fig. 5 Circular polarization dependence of the photonic band structure in the Γ-Z direction ((a)-(d)), and transmission spectra of BPII in the [001] direction ((e)-(h)). The normalized strengths of the electric field are 2 = 0 ((a), (e)), 2 = 0.05 ((b), (f)), 2 = 0.10 ((c), (g)), and 2 = 0.20 ((d), (h)).
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.

4. Conclusions

Acknowledgments

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]

4.

H. Kikuchi, M. Yokota, Y. Hisakado, H. Yang, and T. Kajiyama, “Polymer-stabilized liquid crystal blue phases,” Nat. Mater. 1(1), 64–68 (2002). [CrossRef] [PubMed]

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]

20.

N. R. Chen and J. T. Ho, “Electric-field-induced phase diagrams of blue-phase systems,” Phys. Rev. A 35(11), 4886–4888 (1987). [CrossRef] [PubMed]

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]

OCIS Codes
(160.3710) Materials : Liquid crystals
(230.3720) Optical devices : Liquid-crystal devices
(160.5293) Materials : Photonic bandgap materials

ToC Category:
Physical Optics

History
Original Manuscript: January 6, 2014
Revised Manuscript: January 31, 2014
Manuscript Accepted: January 31, 2014
Published: February 10, 2014

Citation
Yasuhiro Ogawa, Jun-ichi Fukuda, Hiroyuki Yoshida, and Masanori Ozaki, "Photonic band structure and transmission analysis of cholesteric blue phase II: electrostriction in the [100] direction," Opt. Express 22, 3766-3772 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-4-3766


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References

  1. D. L. Johnson, J. H. Flack, P. P. Crooker, “Structure and properties of the cholesteric blue phases,” Phys. Rev. Lett. 45(8), 641–644 (1980). [CrossRef]
  2. D. C. Wright, N. D. Mermin, “Crystalline liquids the blue phases,” Rev. Mod. Phys. 61(2), 385–432 (1989). [CrossRef]
  3. Y. Hisakado, H. Kikuchi, T. Nagamura, T. Kajiyama, “Large electro-optic Kerr effect in polymer-stabilized liquid-crystalline blue phases,” Adv. Mater. 17(1), 96–98 (2005). [CrossRef]
  4. H. Kikuchi, M. Yokota, Y. Hisakado, H. Yang, T. Kajiyama, “Polymer-stabilized liquid crystal blue phases,” Nat. Mater. 1(1), 64–68 (2002). [CrossRef] [PubMed]
  5. J. Yan, L. Rao, M. Jiao, Y. Li, H. C. Cheng, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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]
  20. N. R. Chen, J. T. Ho, “Electric-field-induced phase diagrams of blue-phase systems,” Phys. Rev. A 35(11), 4886–4888 (1987). [CrossRef] [PubMed]
  21. P. Pieranski, P. E. Cladis, T. Garel, R. Barbetmassin, “Orientation of crystals of blue phases by electric fields,” J. Phys. 47(1), 139–143 (1986). [CrossRef]

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