Abstract

The finite-difference time-domain (FDTD) method is used to compute propagation of light through textured uniaxial nematic-liquid crystal (NLC) films containing various types of twist disclination (defect) lines. Computational modeling by the FDTD method provides an accurate prediction of the optical response in multidimensional and multiscale heterogeneities in NLC films in which significant spatial optic axis gradients are present. The computations based on the FDTD method are compared with those of the classic Berreman matrix-type method. As expected, significant deviations between predictions from the two methods are observed near the twist disclination line defects because lateral optic axis gradients are ignored in the matrix Berreman method. It is shown that the failure of Berreman’s method to take into account lateral optic axis gradient effects leads to significant deviations in optical output. In addition, it is shown that the FDTD method is able to distinguish clearly different types of twist disclination lines. The FDTD optical simulation method can be used for understanding fundamental relationships between optical response and complex NLC defect textures in new liquid-crystal applications including liquid-crystal-based biosensors and rheo-optical characterization of flowing liquid crystals.

© 2005 Optical Society of America

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  1. E. Lueder, Liquid Crystal Displays: Addressing Schemes and Electro-Optical Effects (Wiley, 2001).
  2. J. J. Skaife, N. L. Abbott, “Influence of molecular-level interactions on the orientations of liquid crystals supported on nano-structured surfaces presenting specifically bound proteins,” Langmuir 17, 5595–5604 (2001).
    [CrossRef]
  3. J. J. Skaife, N. L. Abbott, “Quantitative interpretation of the optical texture of liquid crystals caused by specific binding of immunoglobulins to surface-bound antigens,” Langmuir 16, 3529–3536 (2000).
    [CrossRef]
  4. M. Kleman, O. D. Lavrentovich, Soft Matter Physics: An Introduction (Springer-Verlag, 2002).
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    [CrossRef]
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    [CrossRef]
  7. D. K. Yang, X. D. Mi, “Modelling of the reflection of cholesteric liquid crystals using the Jones matrix,” J. Phys. D 33, 672–676 (2000).
    [CrossRef]
  8. C. Gu, P. Yeh, “Extended Jones matrix method and its application in the analysis of compensators for liquid crystal displays,” Displays 20, 237–257 (1999).
    [CrossRef]
  9. K. H. Yang, “Elimination of the Fabry–Perot effect in the 4 × 4 matrix method for inhomogeneous uniaxial media,” J. Appl. Phys. 68, 1550–1554 (1990).
    [CrossRef]
  10. J. R. Park, G. Ryu, J. Byun, H. Hwang, S. T. Kim, I. Kim, “Numerical modeling and simulation of a cholesteric liqud crystal polarizer,” Opt. Rev. 9, 207–212 (2002).
    [CrossRef]
  11. E. E. Kriezis, 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]
  12. E. E. Kriezis, S. K. Filippov, S. J. Elston, “Light propagation in domain walls in ferroelectric liquid crystal devices by the finite-difference time-domain method,” J. Opt. A Pure Appl. Opt. 2, 27–33 (2000).
    [CrossRef]
  13. A. Taflove, “Review of the formaulation and applications of the FDTD for numerical modeling of electromagnetic wave interactions with arbitrary structures,” Wave Motion 10, 547–582 (1998).
    [CrossRef]
  14. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic-waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  15. E. E. Kriezis, S. J. Elston, “Finite-difference time domain method for light wave propagation within liquid crystal devices,” Opt. Commun. 165, 99–105 (1999).
    [CrossRef]
  16. D. K. Hwang, A. D. Rey, “Computational modeling of light propagation in textured liquid crystals based on the finite-difference time-domain (FDTD) method,” Liq. Cryst. 32, 483–497 (2005).
    [CrossRef]
  17. A. F. Konstantinova, K. K. Konstantinov, B. V. Nabatov, E. A. Evdishchenko, “Modern application packages for rigorous solution of problems of light propagation in anisotropic media,” Crystallogr. Rep. 47, 645–652 (2002).
    [CrossRef]
  18. H. Wohler, G. Hass, M. Fritsch, D. A. Mlynski, “Faster 4 × 4 matrix method for uniaxial inhomogeneous media,” J. Opt. Soc. Am. A 5, 1554–1557 (1988).
    [CrossRef]
  19. A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).
  20. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
    [CrossRef]
  21. D. M. Sullivan, “An unsplit step 3-D PML for use with the FDTD method,” IEEE Microwave Guided Wave Lett. 7, 184–186 (1997).
    [CrossRef]
  22. D. M. Sullivan, “A simplified PML for use with the FDTD method,” IEEE Microwave Guided Wave. Lett. 6, 97–99 (1996).
    [CrossRef]
  23. J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 127, 363–379 (1996).
    [CrossRef]
  24. P. G. de Gennes, J. Prost, The Physics of Liquid Crystals (Oxford U. Press, 1993).
  25. V. K. Gupta, N. L. Abbott, “Using droplets of nematic liquid crystal to probe the microscopic and mesoscopic structure of organic surfaces,” Langmuir 15, 7213–7223 (1999).
    [CrossRef]
  26. A. M. J. Spruijt, “Twist-disclination line in planar oriented samples of liquid-crystals,” Solid State Commun. 13, 1919–1922 (1973).
    [CrossRef]
  27. J. Nehring, “Calculation of structure and energy of nematic threads,” Phy. Rev. A. 7, 1737–1748 (1973).
    [CrossRef]

2005 (1)

D. K. Hwang, A. D. Rey, “Computational modeling of light propagation in textured liquid crystals based on the finite-difference time-domain (FDTD) method,” Liq. Cryst. 32, 483–497 (2005).
[CrossRef]

2002 (2)

A. F. Konstantinova, K. K. Konstantinov, B. V. Nabatov, E. A. Evdishchenko, “Modern application packages for rigorous solution of problems of light propagation in anisotropic media,” Crystallogr. Rep. 47, 645–652 (2002).
[CrossRef]

J. R. Park, G. Ryu, J. Byun, H. Hwang, S. T. Kim, I. Kim, “Numerical modeling and simulation of a cholesteric liqud crystal polarizer,” Opt. Rev. 9, 207–212 (2002).
[CrossRef]

2001 (1)

J. J. Skaife, N. L. Abbott, “Influence of molecular-level interactions on the orientations of liquid crystals supported on nano-structured surfaces presenting specifically bound proteins,” Langmuir 17, 5595–5604 (2001).
[CrossRef]

2000 (4)

J. J. Skaife, N. L. Abbott, “Quantitative interpretation of the optical texture of liquid crystals caused by specific binding of immunoglobulins to surface-bound antigens,” Langmuir 16, 3529–3536 (2000).
[CrossRef]

E. E. Kriezis, 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]

E. E. Kriezis, S. K. Filippov, S. J. Elston, “Light propagation in domain walls in ferroelectric liquid crystal devices by the finite-difference time-domain method,” J. Opt. A Pure Appl. Opt. 2, 27–33 (2000).
[CrossRef]

D. K. Yang, X. D. Mi, “Modelling of the reflection of cholesteric liquid crystals using the Jones matrix,” J. Phys. D 33, 672–676 (2000).
[CrossRef]

1999 (3)

C. Gu, P. Yeh, “Extended Jones matrix method and its application in the analysis of compensators for liquid crystal displays,” Displays 20, 237–257 (1999).
[CrossRef]

V. K. Gupta, N. L. Abbott, “Using droplets of nematic liquid crystal to probe the microscopic and mesoscopic structure of organic surfaces,” Langmuir 15, 7213–7223 (1999).
[CrossRef]

E. E. Kriezis, S. J. Elston, “Finite-difference time domain method for light wave propagation within liquid crystal devices,” Opt. Commun. 165, 99–105 (1999).
[CrossRef]

1998 (1)

A. Taflove, “Review of the formaulation and applications of the FDTD for numerical modeling of electromagnetic wave interactions with arbitrary structures,” Wave Motion 10, 547–582 (1998).
[CrossRef]

1997 (1)

D. M. Sullivan, “An unsplit step 3-D PML for use with the FDTD method,” IEEE Microwave Guided Wave Lett. 7, 184–186 (1997).
[CrossRef]

1996 (2)

D. M. Sullivan, “A simplified PML for use with the FDTD method,” IEEE Microwave Guided Wave. Lett. 6, 97–99 (1996).
[CrossRef]

J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 127, 363–379 (1996).
[CrossRef]

1994 (1)

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

1990 (1)

K. H. Yang, “Elimination of the Fabry–Perot effect in the 4 × 4 matrix method for inhomogeneous uniaxial media,” J. Appl. Phys. 68, 1550–1554 (1990).
[CrossRef]

1988 (1)

1982 (1)

1973 (2)

A. M. J. Spruijt, “Twist-disclination line in planar oriented samples of liquid-crystals,” Solid State Commun. 13, 1919–1922 (1973).
[CrossRef]

J. Nehring, “Calculation of structure and energy of nematic threads,” Phy. Rev. A. 7, 1737–1748 (1973).
[CrossRef]

1972 (1)

1966 (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[CrossRef]

Abbott, N. L.

J. J. Skaife, N. L. Abbott, “Influence of molecular-level interactions on the orientations of liquid crystals supported on nano-structured surfaces presenting specifically bound proteins,” Langmuir 17, 5595–5604 (2001).
[CrossRef]

J. J. Skaife, N. L. Abbott, “Quantitative interpretation of the optical texture of liquid crystals caused by specific binding of immunoglobulins to surface-bound antigens,” Langmuir 16, 3529–3536 (2000).
[CrossRef]

V. K. Gupta, N. L. Abbott, “Using droplets of nematic liquid crystal to probe the microscopic and mesoscopic structure of organic surfaces,” Langmuir 15, 7213–7223 (1999).
[CrossRef]

Berenger, J. P.

J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 127, 363–379 (1996).
[CrossRef]

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

Berreman, D. W.

Byun, J.

J. R. Park, G. Ryu, J. Byun, H. Hwang, S. T. Kim, I. Kim, “Numerical modeling and simulation of a cholesteric liqud crystal polarizer,” Opt. Rev. 9, 207–212 (2002).
[CrossRef]

de Gennes, P. G.

P. G. de Gennes, J. Prost, The Physics of Liquid Crystals (Oxford U. Press, 1993).

Elston, S. J.

E. E. Kriezis, 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]

E. E. Kriezis, S. K. Filippov, S. J. Elston, “Light propagation in domain walls in ferroelectric liquid crystal devices by the finite-difference time-domain method,” J. Opt. A Pure Appl. Opt. 2, 27–33 (2000).
[CrossRef]

E. E. Kriezis, S. J. Elston, “Finite-difference time domain method for light wave propagation within liquid crystal devices,” Opt. Commun. 165, 99–105 (1999).
[CrossRef]

Evdishchenko, E. A.

A. F. Konstantinova, K. K. Konstantinov, B. V. Nabatov, E. A. Evdishchenko, “Modern application packages for rigorous solution of problems of light propagation in anisotropic media,” Crystallogr. Rep. 47, 645–652 (2002).
[CrossRef]

Filippov, S. K.

E. E. Kriezis, S. K. Filippov, S. J. Elston, “Light propagation in domain walls in ferroelectric liquid crystal devices by the finite-difference time-domain method,” J. Opt. A Pure Appl. Opt. 2, 27–33 (2000).
[CrossRef]

Fritsch, M.

Gu, C.

C. Gu, P. Yeh, “Extended Jones matrix method and its application in the analysis of compensators for liquid crystal displays,” Displays 20, 237–257 (1999).
[CrossRef]

Gupta, V. K.

V. K. Gupta, N. L. Abbott, “Using droplets of nematic liquid crystal to probe the microscopic and mesoscopic structure of organic surfaces,” Langmuir 15, 7213–7223 (1999).
[CrossRef]

Hass, G.

Hwang, D. K.

D. K. Hwang, A. D. Rey, “Computational modeling of light propagation in textured liquid crystals based on the finite-difference time-domain (FDTD) method,” Liq. Cryst. 32, 483–497 (2005).
[CrossRef]

Hwang, H.

J. R. Park, G. Ryu, J. Byun, H. Hwang, S. T. Kim, I. Kim, “Numerical modeling and simulation of a cholesteric liqud crystal polarizer,” Opt. Rev. 9, 207–212 (2002).
[CrossRef]

Kim, I.

J. R. Park, G. Ryu, J. Byun, H. Hwang, S. T. Kim, I. Kim, “Numerical modeling and simulation of a cholesteric liqud crystal polarizer,” Opt. Rev. 9, 207–212 (2002).
[CrossRef]

Kim, S. T.

J. R. Park, G. Ryu, J. Byun, H. Hwang, S. T. Kim, I. Kim, “Numerical modeling and simulation of a cholesteric liqud crystal polarizer,” Opt. Rev. 9, 207–212 (2002).
[CrossRef]

Kleman, M.

M. Kleman, O. D. Lavrentovich, Soft Matter Physics: An Introduction (Springer-Verlag, 2002).

Konstantinov, K. K.

A. F. Konstantinova, K. K. Konstantinov, B. V. Nabatov, E. A. Evdishchenko, “Modern application packages for rigorous solution of problems of light propagation in anisotropic media,” Crystallogr. Rep. 47, 645–652 (2002).
[CrossRef]

Konstantinova, A. F.

A. F. Konstantinova, K. K. Konstantinov, B. V. Nabatov, E. A. Evdishchenko, “Modern application packages for rigorous solution of problems of light propagation in anisotropic media,” Crystallogr. Rep. 47, 645–652 (2002).
[CrossRef]

Kriezis, E. E.

E. E. Kriezis, 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]

E. E. Kriezis, S. K. Filippov, S. J. Elston, “Light propagation in domain walls in ferroelectric liquid crystal devices by the finite-difference time-domain method,” J. Opt. A Pure Appl. Opt. 2, 27–33 (2000).
[CrossRef]

E. E. Kriezis, S. J. Elston, “Finite-difference time domain method for light wave propagation within liquid crystal devices,” Opt. Commun. 165, 99–105 (1999).
[CrossRef]

Lavrentovich, O. D.

M. Kleman, O. D. Lavrentovich, Soft Matter Physics: An Introduction (Springer-Verlag, 2002).

Lueder, E.

E. Lueder, Liquid Crystal Displays: Addressing Schemes and Electro-Optical Effects (Wiley, 2001).

Mi, X. D.

D. K. Yang, X. D. Mi, “Modelling of the reflection of cholesteric liquid crystals using the Jones matrix,” J. Phys. D 33, 672–676 (2000).
[CrossRef]

Mlynski, D. A.

Nabatov, B. V.

A. F. Konstantinova, K. K. Konstantinov, B. V. Nabatov, E. A. Evdishchenko, “Modern application packages for rigorous solution of problems of light propagation in anisotropic media,” Crystallogr. Rep. 47, 645–652 (2002).
[CrossRef]

Nehring, J.

J. Nehring, “Calculation of structure and energy of nematic threads,” Phy. Rev. A. 7, 1737–1748 (1973).
[CrossRef]

Park, J. R.

J. R. Park, G. Ryu, J. Byun, H. Hwang, S. T. Kim, I. Kim, “Numerical modeling and simulation of a cholesteric liqud crystal polarizer,” Opt. Rev. 9, 207–212 (2002).
[CrossRef]

Prost, J.

P. G. de Gennes, J. Prost, The Physics of Liquid Crystals (Oxford U. Press, 1993).

Rey, A. D.

D. K. Hwang, A. D. Rey, “Computational modeling of light propagation in textured liquid crystals based on the finite-difference time-domain (FDTD) method,” Liq. Cryst. 32, 483–497 (2005).
[CrossRef]

Ryu, G.

J. R. Park, G. Ryu, J. Byun, H. Hwang, S. T. Kim, I. Kim, “Numerical modeling and simulation of a cholesteric liqud crystal polarizer,” Opt. Rev. 9, 207–212 (2002).
[CrossRef]

Skaife, J. J.

J. J. Skaife, N. L. Abbott, “Influence of molecular-level interactions on the orientations of liquid crystals supported on nano-structured surfaces presenting specifically bound proteins,” Langmuir 17, 5595–5604 (2001).
[CrossRef]

J. J. Skaife, N. L. Abbott, “Quantitative interpretation of the optical texture of liquid crystals caused by specific binding of immunoglobulins to surface-bound antigens,” Langmuir 16, 3529–3536 (2000).
[CrossRef]

Spruijt, A. M. J.

A. M. J. Spruijt, “Twist-disclination line in planar oriented samples of liquid-crystals,” Solid State Commun. 13, 1919–1922 (1973).
[CrossRef]

Sullivan, D. M.

D. M. Sullivan, “An unsplit step 3-D PML for use with the FDTD method,” IEEE Microwave Guided Wave Lett. 7, 184–186 (1997).
[CrossRef]

D. M. Sullivan, “A simplified PML for use with the FDTD method,” IEEE Microwave Guided Wave. Lett. 6, 97–99 (1996).
[CrossRef]

Taflove, A.

A. Taflove, “Review of the formaulation and applications of the FDTD for numerical modeling of electromagnetic wave interactions with arbitrary structures,” Wave Motion 10, 547–582 (1998).
[CrossRef]

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).

Wohler, H.

Yang, D. K.

D. K. Yang, X. D. Mi, “Modelling of the reflection of cholesteric liquid crystals using the Jones matrix,” J. Phys. D 33, 672–676 (2000).
[CrossRef]

Yang, K. H.

K. H. Yang, “Elimination of the Fabry–Perot effect in the 4 × 4 matrix method for inhomogeneous uniaxial media,” J. Appl. Phys. 68, 1550–1554 (1990).
[CrossRef]

Yee, K. S.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[CrossRef]

Yeh, P.

C. Gu, P. Yeh, “Extended Jones matrix method and its application in the analysis of compensators for liquid crystal displays,” Displays 20, 237–257 (1999).
[CrossRef]

P. Yeh, “Extended Jones matrix method,” J. Opt. Soc. Am. 72, 507–513 (1982).
[CrossRef]

Crystallogr. Rep. (1)

A. F. Konstantinova, K. K. Konstantinov, B. V. Nabatov, E. A. Evdishchenko, “Modern application packages for rigorous solution of problems of light propagation in anisotropic media,” Crystallogr. Rep. 47, 645–652 (2002).
[CrossRef]

Displays (1)

C. Gu, P. Yeh, “Extended Jones matrix method and its application in the analysis of compensators for liquid crystal displays,” Displays 20, 237–257 (1999).
[CrossRef]

IEEE Microwave Guided Wave Lett. (1)

D. M. Sullivan, “An unsplit step 3-D PML for use with the FDTD method,” IEEE Microwave Guided Wave Lett. 7, 184–186 (1997).
[CrossRef]

IEEE Microwave Guided Wave. Lett. (1)

D. M. Sullivan, “A simplified PML for use with the FDTD method,” IEEE Microwave Guided Wave. Lett. 6, 97–99 (1996).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[CrossRef]

J. Appl. Phys. (1)

K. H. Yang, “Elimination of the Fabry–Perot effect in the 4 × 4 matrix method for inhomogeneous uniaxial media,” J. Appl. Phys. 68, 1550–1554 (1990).
[CrossRef]

J. Comput. Phys. (2)

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

J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 127, 363–379 (1996).
[CrossRef]

J. Opt. A Pure Appl. Opt. (1)

E. E. Kriezis, S. K. Filippov, S. J. Elston, “Light propagation in domain walls in ferroelectric liquid crystal devices by the finite-difference time-domain method,” J. Opt. A Pure Appl. Opt. 2, 27–33 (2000).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

J. Phys. D (1)

D. K. Yang, X. D. Mi, “Modelling of the reflection of cholesteric liquid crystals using the Jones matrix,” J. Phys. D 33, 672–676 (2000).
[CrossRef]

Langmuir (3)

J. J. Skaife, N. L. Abbott, “Influence of molecular-level interactions on the orientations of liquid crystals supported on nano-structured surfaces presenting specifically bound proteins,” Langmuir 17, 5595–5604 (2001).
[CrossRef]

J. J. Skaife, N. L. Abbott, “Quantitative interpretation of the optical texture of liquid crystals caused by specific binding of immunoglobulins to surface-bound antigens,” Langmuir 16, 3529–3536 (2000).
[CrossRef]

V. K. Gupta, N. L. Abbott, “Using droplets of nematic liquid crystal to probe the microscopic and mesoscopic structure of organic surfaces,” Langmuir 15, 7213–7223 (1999).
[CrossRef]

Liq. Cryst. (1)

D. K. Hwang, A. D. Rey, “Computational modeling of light propagation in textured liquid crystals based on the finite-difference time-domain (FDTD) method,” Liq. Cryst. 32, 483–497 (2005).
[CrossRef]

Opt. Commun. (2)

E. E. Kriezis, 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]

E. E. Kriezis, S. J. Elston, “Finite-difference time domain method for light wave propagation within liquid crystal devices,” Opt. Commun. 165, 99–105 (1999).
[CrossRef]

Opt. Rev. (1)

J. R. Park, G. Ryu, J. Byun, H. Hwang, S. T. Kim, I. Kim, “Numerical modeling and simulation of a cholesteric liqud crystal polarizer,” Opt. Rev. 9, 207–212 (2002).
[CrossRef]

Phy. Rev. A. (1)

J. Nehring, “Calculation of structure and energy of nematic threads,” Phy. Rev. A. 7, 1737–1748 (1973).
[CrossRef]

Solid State Commun. (1)

A. M. J. Spruijt, “Twist-disclination line in planar oriented samples of liquid-crystals,” Solid State Commun. 13, 1919–1922 (1973).
[CrossRef]

Wave Motion (1)

A. Taflove, “Review of the formaulation and applications of the FDTD for numerical modeling of electromagnetic wave interactions with arbitrary structures,” Wave Motion 10, 547–582 (1998).
[CrossRef]

Other (4)

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).

P. G. de Gennes, J. Prost, The Physics of Liquid Crystals (Oxford U. Press, 1993).

E. Lueder, Liquid Crystal Displays: Addressing Schemes and Electro-Optical Effects (Wiley, 2001).

M. Kleman, O. D. Lavrentovich, Soft Matter Physics: An Introduction (Springer-Verlag, 2002).

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