## Photonic band structure and transmission analysis of cholesteric blue phase II: electrostriction in the [100] direction |

Optics Express, Vol. 22, Issue 4, pp. 3766-3772 (2014)

http://dx.doi.org/10.1364/OE.22.003766

Acrobat PDF (1439 KB)

### 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

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]

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]

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]

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]

## 2. Dielectric constant distribution model and calculation method

**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]

*f*is the local free energy,

_{local}*f*is the free energy due to the inhomogeneity of liquid crystal, and

_{grad}*f*is the free energy due to the electric field.

_{ele}*a*and

*b*are positive constants and

*c*is assumed to vary with temperature.

*K*

_{0}and

*K*

_{1}are the elastic constants and

*q*

_{0}is the strength and sign of the chirality.

**E**is the applied electric field and Δ

*ε*is the dielectric anisotropy at the frequency of

_{ele}**E**. To reduce relevant parameters, the rescaled free energy density φ ≡ (

*a*

^{3}/

*b*

^{4})

*f*is given by the rescaled tensor order parameter

**χ**= (

*a*

^{3}/

*b*

^{4})

**Q**. The contribution of the electric field to the rescaled free energy is given byHere, ȇ is a unit vector specifying the direction of electric field

**E**, and

*Ẽ*

^{2}≡ (

*a*

^{2}/

*b*

^{3})Δ

*ε*

_{ele}E^{2}is the rescaled strength of the electric field. Dielectric anisotropy was assumed Δ

*ε*> 0. The director distribution at a particular applied field intensity is the

_{ele}**χ**which minimizes the free energy density of the system.

**ε**and scalar order parameter

*S*are calculated from

**χ**according to the following equations. Here,

*ε*( = (

_{ave}*ε*+ 2

_{e}*ε*)/3) is the average dielectric constant, Δ

_{o}*ε*is the dielectric anisotropy, and

*ε*and

_{o}*ε*are ordinary and extraordinary dielectric constants of the liquid crystal at optical frequencies. In our calculation, the maximum scalar order parameter

_{e}*S*in the BP II unit cell was normalized to 0.7, which is a typical value found in the nematic phase [17

_{max}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]

*ε*= 1.5

_{o}^{2}and

*ε*= 1.7

_{e}^{2}, respectively, and the dielectric constant of glass substrates was

*ε*=

_{glass}*ε*. 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

_{ave}*Ẽ*

^{2}= 0, 0.05, 0.10, and 0.20, respectively. At zero electric field (

*Ẽ*

^{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 Δ

*ε*> 0 by an applied electric field in the [100] direction. The lattice symmetry of BPII was transformed from simple cubic to simple tetragonal symmetry, and the relation between the lattice constants changed from

_{ele}*L*=

_{x}*L*=

_{y}*L*to

_{z}*L*≠

_{x}*L*=

_{y}*L*. Figure 1(b) shows the first Brillouin zone of the simple tetragonal structure.

_{z}## 3. Results and discussion

*Ẽ*

^{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 (

*Ẽ*

^{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

*Ẽ*

^{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

*Ẽ*

^{2}= 0 because of simple cubic symmetry.

*L*, and

_{x}*L*), 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 Δ

_{z}*ε*> 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

_{ele}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. 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]

## 4. Conclusions

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]

## Acknowledgments

## 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. |

2. | D. C. Wright and N. D. Mermin, “Crystalline liquids the blue phases,” Rev. Mod. Phys. |

3. | Y. Hisakado, H. Kikuchi, T. Nagamura, and T. Kajiyama, “Large electro-optic Kerr effect in polymer-stabilized liquid-crystalline blue phases,” Adv. Mater. |

4. | H. Kikuchi, M. Yokota, Y. Hisakado, H. Yang, and T. Kajiyama, “Polymer-stabilized liquid crystal blue phases,” Nat. Mater. |

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. |

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. |

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 |

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 |

9. | H. S. Kitzerow, “The effect of electric fields on blue phases,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) |

10. | M. Ravnik, G. P. Alexander, J. M. Yeomans, and S. Zumer, “Mesoscopic modelling of colloids in chiral nematics,” Faraday Discuss. |

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. |

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 |

13. | D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4-matrix formulation,” J. Opt. Soc. Am. |

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 |

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. |

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. |

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. |

18. | K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antenn. Propag. |

19. | J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. |

20. | N. R. Chen and J. T. Ho, “Electric-field-induced phase diagrams of blue-phase systems,” Phys. Rev. A |

21. | P. Pieranski, P. E. Cladis, T. Garel, and R. Barbetmassin, “Orientation of crystals of blue phases by electric fields,” J. Phys. |

**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

Sort: Year | Journal | Reset

### References

- 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]
- D. C. Wright, N. D. Mermin, “Crystalline liquids the blue phases,” Rev. Mod. Phys. 61(2), 385–432 (1989). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- H. S. Kitzerow, “The effect of electric fields on blue phases,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 202(1), 51–83 (1991). [CrossRef]
- 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]
- 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]
- 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]
- D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4-matrix formulation,” J. Opt. Soc. Am. 62(4), 502–510 (1972). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- 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]
- J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994). [CrossRef]
- 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]
- 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]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.