Abstract

The finite-difference time-domain (FDTD) method is a powerful numerical algorithm used to directly solve Maxwell’s equations. We introduce the idea of the FDTD method and the techniques required for optical simulation of cholesteric liquid crystal (Ch-LC) devices. Bragg reflection characteristics of Ch-LC cells are investigated using the FDTD method. Three approaches to broadening the bandwidth of Bragg reflection are demonstrated: (1) using a higher birefringence LC, (2) using a cell with a gradient pitch length, and (3) using a cell with a new multidimensional structure of a Ch-LC.

© 2006 Optical Society of America

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  1. D. W. Berreman, "Optics in stratified and anisotropic media: 4 × 4-matrix formulation," J. Opt. Soc. Am. 62,502-510 (1972).
    [CrossRef]
  2. K. S. Yee, "Numerical solutions of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas. Propag. AP-14,302-307 (1966).
  3. A. Taflove, Computational Electromagnetic: The Finite-Difference Time-Domain Method (Artech House, 1995).
  4. A. Lien, "Extended Jones matrix representation for the twisted nematic liquid crystal display at oblique incidence," Appl. Phys. Lett. 57,2767-2769 (1990).
    [CrossRef]
  5. E. E. Kriezis and S. J. Elston, "Light wave propagation in liquid crystal displays by the 2-D finite-difference time-domain method," Opt. Commun. 177,69-77 (2000).
    [CrossRef]
  6. B. Witzigmann, P. Regli, and W. Fichtner, "Rigorous electromagnetic simulation of liquid crystal displays," J. Opt. Soc Am.  A 15,753-757 (1998).
    [CrossRef]
  7. C. M. Titus, P. J. Bos, J. R. Kelly, E. C. Gartland, "Two-dimensional optical simulation tool for microdisplay analysis," J. Appl. Phys. 38,1488-1494 (1999).
  8. E. E. Kriezis and S. J. Elston, "Light wave propagation in liquid crystal displays by the 2-D FDTD method," Opt. Commun. 177,69-77 (2000).
    [CrossRef]
  9. E. E. Kriezis, C. J. P. Newton, T. P. Spiller, and S. J. Elston, "Three-dimensional simulations of light propagation in periodic liquid-crystal microstructures," Appl. Opt. 41,25, 5346-5356 (2002).
    [CrossRef] [PubMed]
  10. D. K. Hwang and A. D. Rey, "Computational modeling of the propagation of light through liquid crystals containing twist disclinations based on the finite-difference time-domain method," Appl. Opt. 44,21, 4513-4522 (2005).
    [CrossRef] [PubMed]
  11. J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comp. Physiol. 114,185-200 (1994).
    [CrossRef]
  12. J. P. Berenger, "Perfectly matched layer for the FDTD solution of wave-structure interaction problems," IEEE Trans. Antennas Propag. 44,110-117 (1996).
    [CrossRef]
  13. S. T. Wu and D. K. Yang, Reflective Liquid Crystal Displays (John Wiley & Sons, 2001).
  14. D. K. Yang, J. L. West, L. C. Chien, and J. W. Doane, "Control of the reflectivity and bistability in displays based on cholesteric liquid crystals," J. Appl. Phys. 76,1331-1333 (1994).
    [CrossRef]
  15. D.-K. Yang, L.-C. Chien, and Y. K. Fung, "Polymer stabilized cholesteric textures: materials and applications," in Liquid Crystals in Complex Geometries, G. P. Crawford and S. Zumer, eds. (Taylor & Francis, 1996), pp. 103-143.
  16. M. Xu, F. D. Xu, and D.-K. Yang, "Effects of cell structure on the reflection of cholesteric liquid crystal display," J. Appl. Phys. 83,1938-1955 (1998).
    [CrossRef]
  17. T. X. Hong and S.-T. Wu, "Optical wave propagation in a cholesteric liquid crystal using the finite element method," Liq. Cryst. 30,367-375 (2003).
    [CrossRef]

2005 (1)

2003 (1)

T. X. Hong and S.-T. Wu, "Optical wave propagation in a cholesteric liquid crystal using the finite element method," Liq. Cryst. 30,367-375 (2003).
[CrossRef]

2002 (1)

2000 (2)

E. E. Kriezis and S. J. Elston, "Light wave propagation in liquid crystal displays by the 2-D FDTD method," Opt. Commun. 177,69-77 (2000).
[CrossRef]

E. E. Kriezis and S. J. Elston, "Light wave propagation in liquid crystal displays by the 2-D finite-difference time-domain method," Opt. Commun. 177,69-77 (2000).
[CrossRef]

1999 (1)

C. M. Titus, P. J. Bos, J. R. Kelly, E. C. Gartland, "Two-dimensional optical simulation tool for microdisplay analysis," J. Appl. Phys. 38,1488-1494 (1999).

1998 (2)

B. Witzigmann, P. Regli, and W. Fichtner, "Rigorous electromagnetic simulation of liquid crystal displays," J. Opt. Soc Am.  A 15,753-757 (1998).
[CrossRef]

M. Xu, F. D. Xu, and D.-K. Yang, "Effects of cell structure on the reflection of cholesteric liquid crystal display," J. Appl. Phys. 83,1938-1955 (1998).
[CrossRef]

1996 (1)

J. P. Berenger, "Perfectly matched layer for the FDTD solution of wave-structure interaction problems," IEEE Trans. Antennas Propag. 44,110-117 (1996).
[CrossRef]

1994 (2)

D. K. Yang, J. L. West, L. C. Chien, and J. W. Doane, "Control of the reflectivity and bistability in displays based on cholesteric liquid crystals," J. Appl. Phys. 76,1331-1333 (1994).
[CrossRef]

J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comp. Physiol. 114,185-200 (1994).
[CrossRef]

1990 (1)

A. Lien, "Extended Jones matrix representation for the twisted nematic liquid crystal display at oblique incidence," Appl. Phys. Lett. 57,2767-2769 (1990).
[CrossRef]

1972 (1)

1966 (1)

K. S. Yee, "Numerical solutions of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas. Propag. AP-14,302-307 (1966).

Berenger, J. P.

J. P. Berenger, "Perfectly matched layer for the FDTD solution of wave-structure interaction problems," IEEE Trans. Antennas Propag. 44,110-117 (1996).
[CrossRef]

J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comp. Physiol. 114,185-200 (1994).
[CrossRef]

Berreman, D. W.

Bos, P. J.

C. M. Titus, P. J. Bos, J. R. Kelly, E. C. Gartland, "Two-dimensional optical simulation tool for microdisplay analysis," J. Appl. Phys. 38,1488-1494 (1999).

Chien, L. C.

D. K. Yang, J. L. West, L. C. Chien, and J. W. Doane, "Control of the reflectivity and bistability in displays based on cholesteric liquid crystals," J. Appl. Phys. 76,1331-1333 (1994).
[CrossRef]

Doane, J. W.

D. K. Yang, J. L. West, L. C. Chien, and J. W. Doane, "Control of the reflectivity and bistability in displays based on cholesteric liquid crystals," J. Appl. Phys. 76,1331-1333 (1994).
[CrossRef]

Elston, S. J.

E. E. Kriezis, C. J. P. Newton, T. P. Spiller, and S. J. Elston, "Three-dimensional simulations of light propagation in periodic liquid-crystal microstructures," Appl. Opt. 41,25, 5346-5356 (2002).
[CrossRef] [PubMed]

E. E. Kriezis and S. J. Elston, "Light wave propagation in liquid crystal displays by the 2-D FDTD method," Opt. Commun. 177,69-77 (2000).
[CrossRef]

E. E. Kriezis and S. J. Elston, "Light wave propagation in liquid crystal displays by the 2-D finite-difference time-domain method," Opt. Commun. 177,69-77 (2000).
[CrossRef]

Fichtner, W.

B. Witzigmann, P. Regli, and W. Fichtner, "Rigorous electromagnetic simulation of liquid crystal displays," J. Opt. Soc Am.  A 15,753-757 (1998).
[CrossRef]

Gartland, E. C.

C. M. Titus, P. J. Bos, J. R. Kelly, E. C. Gartland, "Two-dimensional optical simulation tool for microdisplay analysis," J. Appl. Phys. 38,1488-1494 (1999).

Hong, T. X.

T. X. Hong and S.-T. Wu, "Optical wave propagation in a cholesteric liquid crystal using the finite element method," Liq. Cryst. 30,367-375 (2003).
[CrossRef]

Hwang, D. K.

Kelly, J. R.

C. M. Titus, P. J. Bos, J. R. Kelly, E. C. Gartland, "Two-dimensional optical simulation tool for microdisplay analysis," J. Appl. Phys. 38,1488-1494 (1999).

Kriezis, E. E.

E. E. Kriezis, C. J. P. Newton, T. P. Spiller, and S. J. Elston, "Three-dimensional simulations of light propagation in periodic liquid-crystal microstructures," Appl. Opt. 41,25, 5346-5356 (2002).
[CrossRef] [PubMed]

E. E. Kriezis and S. J. Elston, "Light wave propagation in liquid crystal displays by the 2-D FDTD method," Opt. Commun. 177,69-77 (2000).
[CrossRef]

E. E. Kriezis and S. J. Elston, "Light wave propagation in liquid crystal displays by the 2-D finite-difference time-domain method," Opt. Commun. 177,69-77 (2000).
[CrossRef]

Lien, A.

A. Lien, "Extended Jones matrix representation for the twisted nematic liquid crystal display at oblique incidence," Appl. Phys. Lett. 57,2767-2769 (1990).
[CrossRef]

Newton, C. J. P.

Regli, P.

B. Witzigmann, P. Regli, and W. Fichtner, "Rigorous electromagnetic simulation of liquid crystal displays," J. Opt. Soc Am.  A 15,753-757 (1998).
[CrossRef]

Rey, A. D.

Spiller, T. P.

Titus, C. M.

C. M. Titus, P. J. Bos, J. R. Kelly, E. C. Gartland, "Two-dimensional optical simulation tool for microdisplay analysis," J. Appl. Phys. 38,1488-1494 (1999).

West, J. L.

D. K. Yang, J. L. West, L. C. Chien, and J. W. Doane, "Control of the reflectivity and bistability in displays based on cholesteric liquid crystals," J. Appl. Phys. 76,1331-1333 (1994).
[CrossRef]

Witzigmann, B.

B. Witzigmann, P. Regli, and W. Fichtner, "Rigorous electromagnetic simulation of liquid crystal displays," J. Opt. Soc Am.  A 15,753-757 (1998).
[CrossRef]

Wu, S.-T.

T. X. Hong and S.-T. Wu, "Optical wave propagation in a cholesteric liquid crystal using the finite element method," Liq. Cryst. 30,367-375 (2003).
[CrossRef]

Xu, F. D.

M. Xu, F. D. Xu, and D.-K. Yang, "Effects of cell structure on the reflection of cholesteric liquid crystal display," J. Appl. Phys. 83,1938-1955 (1998).
[CrossRef]

Xu, M.

M. Xu, F. D. Xu, and D.-K. Yang, "Effects of cell structure on the reflection of cholesteric liquid crystal display," J. Appl. Phys. 83,1938-1955 (1998).
[CrossRef]

Yang, D. K.

D. K. Yang, J. L. West, L. C. Chien, and J. W. Doane, "Control of the reflectivity and bistability in displays based on cholesteric liquid crystals," J. Appl. Phys. 76,1331-1333 (1994).
[CrossRef]

Yang, D.-K.

M. Xu, F. D. Xu, and D.-K. Yang, "Effects of cell structure on the reflection of cholesteric liquid crystal display," J. Appl. Phys. 83,1938-1955 (1998).
[CrossRef]

Yee, K. S.

K. S. Yee, "Numerical solutions of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas. Propag. AP-14,302-307 (1966).

Appl. Opt. (2)

Appl. Phys. Lett. (1)

A. Lien, "Extended Jones matrix representation for the twisted nematic liquid crystal display at oblique incidence," Appl. Phys. Lett. 57,2767-2769 (1990).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

J. P. Berenger, "Perfectly matched layer for the FDTD solution of wave-structure interaction problems," IEEE Trans. Antennas Propag. 44,110-117 (1996).
[CrossRef]

IEEE Trans. Antennas. Propag. (1)

K. S. Yee, "Numerical solutions of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas. Propag. AP-14,302-307 (1966).

J. Appl. Phys. (3)

C. M. Titus, P. J. Bos, J. R. Kelly, E. C. Gartland, "Two-dimensional optical simulation tool for microdisplay analysis," J. Appl. Phys. 38,1488-1494 (1999).

D. K. Yang, J. L. West, L. C. Chien, and J. W. Doane, "Control of the reflectivity and bistability in displays based on cholesteric liquid crystals," J. Appl. Phys. 76,1331-1333 (1994).
[CrossRef]

M. Xu, F. D. Xu, and D.-K. Yang, "Effects of cell structure on the reflection of cholesteric liquid crystal display," J. Appl. Phys. 83,1938-1955 (1998).
[CrossRef]

J. Comp. Physiol. (1)

J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comp. Physiol. 114,185-200 (1994).
[CrossRef]

J. Opt. Soc Am.  A (1)

B. Witzigmann, P. Regli, and W. Fichtner, "Rigorous electromagnetic simulation of liquid crystal displays," J. Opt. Soc Am.  A 15,753-757 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

Liq. Cryst. (1)

T. X. Hong and S.-T. Wu, "Optical wave propagation in a cholesteric liquid crystal using the finite element method," Liq. Cryst. 30,367-375 (2003).
[CrossRef]

Opt. Commun. (2)

E. E. Kriezis and S. J. Elston, "Light wave propagation in liquid crystal displays by the 2-D FDTD method," Opt. Commun. 177,69-77 (2000).
[CrossRef]

E. E. Kriezis and S. J. Elston, "Light wave propagation in liquid crystal displays by the 2-D finite-difference time-domain method," Opt. Commun. 177,69-77 (2000).
[CrossRef]

Other (3)

A. Taflove, Computational Electromagnetic: The Finite-Difference Time-Domain Method (Artech House, 1995).

D.-K. Yang, L.-C. Chien, and Y. K. Fung, "Polymer stabilized cholesteric textures: materials and applications," in Liquid Crystals in Complex Geometries, G. P. Crawford and S. Zumer, eds. (Taylor & Francis, 1996), pp. 103-143.

S. T. Wu and D. K. Yang, Reflective Liquid Crystal Displays (John Wiley & Sons, 2001).

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Figures (18)

Fig. 1.
Fig. 1.

General layout of the computational space used in the FDTD method. An indicative Yee cell used to fill the computational space is also shown. PML: perfect matched layer, PBC: periodic boundary condition, Ch-LC: cholesteric liquid crystal, d: the thickness of Ch-LC cell.

Fig. 2.
Fig. 2.

Comparison of the simulated reflective spectra of a Ch-LC using the FDTD method and the 4×4 Berreman matrix method. ne=1.616, no=1.494, Po=338nm, and cell gap d=12Po.

Fig. 3.
Fig. 3.

Comparison between the measured reflective spectra of a Ch-LC [(a1)-(a3)] [13] and the FDTD simulation results [(b1)-(b3), (c1)-(c3)]. ne=1.616; no=1.494; P0=338nm; d=5µm. (a1)(b1)(c1) incident light: p-polarization; detection: s-polarization. (b1)(b2)(b3) incident light: p-polarization and perpendicular to the LC director on the entrance plane; detection: ppolarization. (c1)(c2)(c3) incident light: unpolarized; detection: unpolarized.

Fig. 4.
Fig. 4.

Reflective spectra of the planar Ch-LC cells with different pitch lengths, and ne=1.75;no=1.5231; d=10Po.

Fig. 5.
Fig. 5.

Blue shift of the Bragg reflection due to the different Θclc, and ne=1.75; no=1.5231; Po=340 nm; d=10Po.

Fig. 6.
Fig. 6.

Blue shift of Bragg reflection due to the increase of incident angle, and with ne=1.75; no=1.5231; Po=350 nm; d=10Po.

Fig. 7.
Fig. 7.

Bragg reflections of the Ch-LCs with different birefringences. no=1.5231, d=12P0, and P0=340 nm.

Fig. 8.
Fig. 8.

Comparison of reflective spectra of a Ch-LC having a gradient pitch length with P0=280 nm and P1=380 nm, and a uniform pitch length.

Fig. 9.
Fig. 9.

General layout of one unit of a 2-D Ch-LC structure; W is the width of one unit of the periodic structure, d is the cell gap, and θ a is the angle between the tangential plane of the substrates and the x-y plane.

Fig. 10.
Fig. 10.

Far-field distribution of reflection of the 2-D Ch-LC shown in Fig. 8 for λ=550 nm at normal incidence. The total reflectance Rtotal(550 nm)=0.2694.

Fig. 11.
Fig. 11.

Wavelength dependence of total reflectance at normal incidence.

Fig. 12.
Fig. 12.

The reflective spectra of 2-D Ch-LC at viewing angles (a) θview=0°; (b) 15°; (c) 30°; (d) 45° at normal incidence.

Fig. 13.
Fig. 13.

Far-field distribution of reflection of the 2-D Ch-LC shown in Fig.11 for λ=550 nm at oblique incidence θin=30°. The total reflectance for Rtotal(550 nm)=0.5366.

Fig. 14.
Fig. 14.

Wavelength dependence of total reflectance at oblique incidence θin=30°.

Fig. 15.
Fig. 15.

The reflective spectra of 2-D a Ch-LC at viewing angles (a) θview=0°, (b) 30°, (c) 45°, (d) 60° at oblique incidence θin=30°.

Fig. 16.
Fig. 16.

A diagram of light incident normally and obliquely onto the sample.

Fig. 17.
Fig. 17.

The reflective spectra of a 2-D Ch-LC under imitative natural environment at viewing angles (a) θview=0°; (b) 15°; (c) 30°; (d) 45°.

Fig. 18.
Fig. 18.

Comparison of the chromaticity between 1-D and 2-D Ch-LCs at various viewing angles. In this 1-D Ch-LC simulation, ne=1.75, no=1.5231, P0=350 nm, and d=8P0.

Equations (8)

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D n 1 2 t = ε E n E n 1 Δ t = σ E n 1 2 + × H n 1 2 , and
B n t = μ H n + 1 2 H n 1 2 Δ t = × E n ,
E n = 1 σ Δ t 2 ε 1 + σ Δ t 2 ε E n 1 + Δ t ε 1 + σ Δ t 2 ε × H n 1 2 ,
H n + 1 2 = H n 1 2 Δ t μ × E n .
ε = [ ε 11 ε 12 ε 13 ε 21 ε 22 ε 23 ε 31 ε 32 ε 33 ]
p ( t ) = α 2 exp ( α ( t t 0 ) 2 ) sin ( ω c ( t t 0 ) ) .
ϕ uni ( z ) = 2 π P o z ,
ϕ grad ( z ) = ϕ o + 2 π P 0 z + ( 2 π P 1 2 π P 0 ) 2 d z 2 ,

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