Abstract

We analyze the optical behavior of two-dimensionally periodic structures that occur in electrohydrodynamic convection (EHC) patterns in nematic sandwich cells. These structures are anisotropic, locally uniaxial, and periodic on the scale of micrometers. For the first time, the optics of these structures is investigated with a rigorous method. The method used for the description of the electromagnetic waves interacting with EHC director patterns is a numerical approach that discretizes directly the Maxwell equations. It works as a space-grid–time-domain method and computes electric and magnetic fields in time steps. This so-called finite-difference–time-domain (FDTD) method is able to generate the fields with arbitrary accuracy. We compare this rigorous method with earlier attempts based on ray-tracing and analytical approximations. Results of optical studies of EHC structures made earlier based on ray-tracing methods are confirmed for thin cells, when the spatial periods of the pattern are sufficiently large. For the treatment of small-scale convection structures, the FDTD method is without alternatives.

© 2005 Optical Society of America

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References

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  1. P. G. de Gennes, The Physics of Liquid Crystals (Clarendon, 1993).
  2. C. M. Titus, P. J. Bos, J. R. Kelly, E. C. Gartland, “Comparison of analytical calculations to finite-difference time-domain simulations of one-dimensional spatially varying anisotropic liquid crystal structures,” J. Appl. Phys. 38, 1188–1190 (1999).
    [CrossRef]
  3. T. Scharf, C. Bohley, “Light propagation through alignment-patterned liquid crystal gratings,” Mol. Cryst. Liq. Cryst. Sci. Technol., Sect. A 375, 491–500 (2002).
    [CrossRef]
  4. C. Bohley, Polarization Optics of Periodic Media (UFO-Verlag, 2004).
  5. 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]
  6. G. Mur, “Absorbing boundary conditions for the finite-difference approximation of time-domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat. 23, 377–382 (1981).
    [CrossRef]
  7. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 184–208 (1994).
    [CrossRef]
  8. A. Taflove, S. C. Hagness, Computational Electrodynamics (Artech House, 2000).
  9. B. Witzigmann, P. Regli, W. Fichtner, “Rigorous electromagnetic simulation of liquid crystal displays,” J. Opt. Soc. Am. A 15, 753–757 (1998).
    [CrossRef]
  10. 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]
  11. E. Kriezis, “A comparative study of light scattering from liquid crystal droplets,” Microwave Opt. Technol. Lett. 35, 437–441 (2002).
    [CrossRef]
  12. R. Williams, “Domains in liquid crystals,” J. Chem. Phys. 39, 384–388 (1963).
    [CrossRef]
  13. L. Kramer, W. Pesch, “Electrohydrodynamic instabilities,” in Pattern Formation in Liquid Crystals, A. Buka and L. Kramer, eds. (Springer, 1996), pp. 221–255.
    [CrossRef]
  14. T. John, R. Stannarius, “Preparation of subharmonic patterns in nematic electroconvection,” Phys. Rev. E 70, 025202 (2004).
    [CrossRef]
  15. T. John, J. Heuer, R. Stannarius, “Influence of excitation wave forms and frequencies on the fundamental time symmetry of the system dynamics, studied in nematic electroconvection,” Phys. Rev. E 71, 056307 (2005).
    [CrossRef]
  16. S. Rasenat, G. Hartung, B. L. Winkler, I. Rehberg, “The shadowgraph method in convection experiments,” Exp. Fluids 7, 412–420 (1989).
    [CrossRef]
  17. T. O. Carroll, “Liquid crystal diffraction grating,” J. Appl. Phys. 43, 767–770 (1972).
    [CrossRef]
  18. H. M. Zenginoglou, J. A. Kosmopoulos, “Geometrical optics approach to the obliquely illuminated nematic liquid crystal diffraction grating,” Appl. Opt. 27, 3898–3901 (1988).
    [CrossRef] [PubMed]
  19. H. M. Zenginoglou, J. A. Kosmopoulos, “Linearized wave-optical approach to the grating effect of a periodically distorted nematic liquid crystal layer,” J. Opt. Soc. Am. A 14, 669–675 (1997).
    [CrossRef]
  20. M. Bouvier, T. Scharf, “Analysis of nematic liquid crystal binary gratings with high spatial frequency,” Opt. Eng. 39, 2129–2137 (2000).
    [CrossRef]
  21. S. Trainoff, D. Cannell, “Physical optics treatment of the shadowgraph,” Phys. Fluids 14, 1340–1363 (2002).
    [CrossRef]
  22. T. John, U. Behn, R. Stannarius, “Laser diffraction by periodic dynamic patterns in anisotropic fluids,” Eur. Phys. J. B 35, 267–278 (2003).
    [CrossRef]
  23. H. Amm, M. Grigutsch, R. Stannarius, “Optical characterization of electroconvection in nematics,” Mol. Cryst. Liq. Cryst. Sci. Technol., Sect. A 320, 11–27 (1998).
    [CrossRef]
  24. H. Amm, M. Grigutsch, R. Stannarius, “Spatiotemporal analysis of electroconvection in nematics,” Z. Naturforsch., A: Phys. Sci. 53a, 117–126 (1998).
  25. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  26. R. Dändliker, P. Blattner, C. Rockstuhl, H. P. Herzig, “Phase singularities generated by optical microstructures: Theory and experimental results,” in Singular Optics (Optical Vortices) Fundamentals and Applications, M. S. Soskin and M. V. Vastenov, eds., Proc. SPIE4403, 257–261 (2001).
  27. P. Blattner, “Light fields emerging from periodic optical microstructures,” Ph.D. thesis (University of Neuchâtel, 1999).

2005 (1)

T. John, J. Heuer, R. Stannarius, “Influence of excitation wave forms and frequencies on the fundamental time symmetry of the system dynamics, studied in nematic electroconvection,” Phys. Rev. E 71, 056307 (2005).
[CrossRef]

2004 (1)

T. John, R. Stannarius, “Preparation of subharmonic patterns in nematic electroconvection,” Phys. Rev. E 70, 025202 (2004).
[CrossRef]

2003 (1)

T. John, U. Behn, R. Stannarius, “Laser diffraction by periodic dynamic patterns in anisotropic fluids,” Eur. Phys. J. B 35, 267–278 (2003).
[CrossRef]

2002 (3)

S. Trainoff, D. Cannell, “Physical optics treatment of the shadowgraph,” Phys. Fluids 14, 1340–1363 (2002).
[CrossRef]

T. Scharf, C. Bohley, “Light propagation through alignment-patterned liquid crystal gratings,” Mol. Cryst. Liq. Cryst. Sci. Technol., Sect. A 375, 491–500 (2002).
[CrossRef]

E. Kriezis, “A comparative study of light scattering from liquid crystal droplets,” Microwave Opt. Technol. Lett. 35, 437–441 (2002).
[CrossRef]

2000 (1)

M. Bouvier, T. Scharf, “Analysis of nematic liquid crystal binary gratings with high spatial frequency,” Opt. Eng. 39, 2129–2137 (2000).
[CrossRef]

1999 (2)

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]

C. M. Titus, P. J. Bos, J. R. Kelly, E. C. Gartland, “Comparison of analytical calculations to finite-difference time-domain simulations of one-dimensional spatially varying anisotropic liquid crystal structures,” J. Appl. Phys. 38, 1188–1190 (1999).
[CrossRef]

1998 (3)

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

H. Amm, M. Grigutsch, R. Stannarius, “Optical characterization of electroconvection in nematics,” Mol. Cryst. Liq. Cryst. Sci. Technol., Sect. A 320, 11–27 (1998).
[CrossRef]

H. Amm, M. Grigutsch, R. Stannarius, “Spatiotemporal analysis of electroconvection in nematics,” Z. Naturforsch., A: Phys. Sci. 53a, 117–126 (1998).

1997 (1)

1994 (1)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 184–208 (1994).
[CrossRef]

1989 (1)

S. Rasenat, G. Hartung, B. L. Winkler, I. Rehberg, “The shadowgraph method in convection experiments,” Exp. Fluids 7, 412–420 (1989).
[CrossRef]

1988 (1)

1981 (1)

G. Mur, “Absorbing boundary conditions for the finite-difference approximation of time-domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat. 23, 377–382 (1981).
[CrossRef]

1972 (1)

T. O. Carroll, “Liquid crystal diffraction grating,” J. Appl. Phys. 43, 767–770 (1972).
[CrossRef]

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]

1963 (1)

R. Williams, “Domains in liquid crystals,” J. Chem. Phys. 39, 384–388 (1963).
[CrossRef]

Amm, H.

H. Amm, M. Grigutsch, R. Stannarius, “Optical characterization of electroconvection in nematics,” Mol. Cryst. Liq. Cryst. Sci. Technol., Sect. A 320, 11–27 (1998).
[CrossRef]

H. Amm, M. Grigutsch, R. Stannarius, “Spatiotemporal analysis of electroconvection in nematics,” Z. Naturforsch., A: Phys. Sci. 53a, 117–126 (1998).

Behn, U.

T. John, U. Behn, R. Stannarius, “Laser diffraction by periodic dynamic patterns in anisotropic fluids,” Eur. Phys. J. B 35, 267–278 (2003).
[CrossRef]

Berenger, J. P.

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 184–208 (1994).
[CrossRef]

Blattner, P.

R. Dändliker, P. Blattner, C. Rockstuhl, H. P. Herzig, “Phase singularities generated by optical microstructures: Theory and experimental results,” in Singular Optics (Optical Vortices) Fundamentals and Applications, M. S. Soskin and M. V. Vastenov, eds., Proc. SPIE4403, 257–261 (2001).

P. Blattner, “Light fields emerging from periodic optical microstructures,” Ph.D. thesis (University of Neuchâtel, 1999).

Bohley, C.

T. Scharf, C. Bohley, “Light propagation through alignment-patterned liquid crystal gratings,” Mol. Cryst. Liq. Cryst. Sci. Technol., Sect. A 375, 491–500 (2002).
[CrossRef]

C. Bohley, Polarization Optics of Periodic Media (UFO-Verlag, 2004).

Bos, P. J.

C. M. Titus, P. J. Bos, J. R. Kelly, E. C. Gartland, “Comparison of analytical calculations to finite-difference time-domain simulations of one-dimensional spatially varying anisotropic liquid crystal structures,” J. Appl. Phys. 38, 1188–1190 (1999).
[CrossRef]

Bouvier, M.

M. Bouvier, T. Scharf, “Analysis of nematic liquid crystal binary gratings with high spatial frequency,” Opt. Eng. 39, 2129–2137 (2000).
[CrossRef]

Cannell, D.

S. Trainoff, D. Cannell, “Physical optics treatment of the shadowgraph,” Phys. Fluids 14, 1340–1363 (2002).
[CrossRef]

Carroll, T. O.

T. O. Carroll, “Liquid crystal diffraction grating,” J. Appl. Phys. 43, 767–770 (1972).
[CrossRef]

Dändliker, R.

R. Dändliker, P. Blattner, C. Rockstuhl, H. P. Herzig, “Phase singularities generated by optical microstructures: Theory and experimental results,” in Singular Optics (Optical Vortices) Fundamentals and Applications, M. S. Soskin and M. V. Vastenov, eds., Proc. SPIE4403, 257–261 (2001).

de Gennes, P. G.

P. G. de Gennes, The Physics of Liquid Crystals (Clarendon, 1993).

Elston, S. J.

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]

Fichtner, W.

Gartland, E. C.

C. M. Titus, P. J. Bos, J. R. Kelly, E. C. Gartland, “Comparison of analytical calculations to finite-difference time-domain simulations of one-dimensional spatially varying anisotropic liquid crystal structures,” J. Appl. Phys. 38, 1188–1190 (1999).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Grigutsch, M.

H. Amm, M. Grigutsch, R. Stannarius, “Spatiotemporal analysis of electroconvection in nematics,” Z. Naturforsch., A: Phys. Sci. 53a, 117–126 (1998).

H. Amm, M. Grigutsch, R. Stannarius, “Optical characterization of electroconvection in nematics,” Mol. Cryst. Liq. Cryst. Sci. Technol., Sect. A 320, 11–27 (1998).
[CrossRef]

Hagness, S. C.

A. Taflove, S. C. Hagness, Computational Electrodynamics (Artech House, 2000).

Hartung, G.

S. Rasenat, G. Hartung, B. L. Winkler, I. Rehberg, “The shadowgraph method in convection experiments,” Exp. Fluids 7, 412–420 (1989).
[CrossRef]

Herzig, H. P.

R. Dändliker, P. Blattner, C. Rockstuhl, H. P. Herzig, “Phase singularities generated by optical microstructures: Theory and experimental results,” in Singular Optics (Optical Vortices) Fundamentals and Applications, M. S. Soskin and M. V. Vastenov, eds., Proc. SPIE4403, 257–261 (2001).

Heuer, J.

T. John, J. Heuer, R. Stannarius, “Influence of excitation wave forms and frequencies on the fundamental time symmetry of the system dynamics, studied in nematic electroconvection,” Phys. Rev. E 71, 056307 (2005).
[CrossRef]

John, T.

T. John, J. Heuer, R. Stannarius, “Influence of excitation wave forms and frequencies on the fundamental time symmetry of the system dynamics, studied in nematic electroconvection,” Phys. Rev. E 71, 056307 (2005).
[CrossRef]

T. John, R. Stannarius, “Preparation of subharmonic patterns in nematic electroconvection,” Phys. Rev. E 70, 025202 (2004).
[CrossRef]

T. John, U. Behn, R. Stannarius, “Laser diffraction by periodic dynamic patterns in anisotropic fluids,” Eur. Phys. J. B 35, 267–278 (2003).
[CrossRef]

Kelly, J. R.

C. M. Titus, P. J. Bos, J. R. Kelly, E. C. Gartland, “Comparison of analytical calculations to finite-difference time-domain simulations of one-dimensional spatially varying anisotropic liquid crystal structures,” J. Appl. Phys. 38, 1188–1190 (1999).
[CrossRef]

Kosmopoulos, J. A.

Kramer, L.

L. Kramer, W. Pesch, “Electrohydrodynamic instabilities,” in Pattern Formation in Liquid Crystals, A. Buka and L. Kramer, eds. (Springer, 1996), pp. 221–255.
[CrossRef]

Kriezis, E.

E. Kriezis, “A comparative study of light scattering from liquid crystal droplets,” Microwave Opt. Technol. Lett. 35, 437–441 (2002).
[CrossRef]

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]

Mur, G.

G. Mur, “Absorbing boundary conditions for the finite-difference approximation of time-domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat. 23, 377–382 (1981).
[CrossRef]

Pesch, W.

L. Kramer, W. Pesch, “Electrohydrodynamic instabilities,” in Pattern Formation in Liquid Crystals, A. Buka and L. Kramer, eds. (Springer, 1996), pp. 221–255.
[CrossRef]

Rasenat, S.

S. Rasenat, G. Hartung, B. L. Winkler, I. Rehberg, “The shadowgraph method in convection experiments,” Exp. Fluids 7, 412–420 (1989).
[CrossRef]

Regli, P.

Rehberg, I.

S. Rasenat, G. Hartung, B. L. Winkler, I. Rehberg, “The shadowgraph method in convection experiments,” Exp. Fluids 7, 412–420 (1989).
[CrossRef]

Rockstuhl, C.

R. Dändliker, P. Blattner, C. Rockstuhl, H. P. Herzig, “Phase singularities generated by optical microstructures: Theory and experimental results,” in Singular Optics (Optical Vortices) Fundamentals and Applications, M. S. Soskin and M. V. Vastenov, eds., Proc. SPIE4403, 257–261 (2001).

Scharf, T.

T. Scharf, C. Bohley, “Light propagation through alignment-patterned liquid crystal gratings,” Mol. Cryst. Liq. Cryst. Sci. Technol., Sect. A 375, 491–500 (2002).
[CrossRef]

M. Bouvier, T. Scharf, “Analysis of nematic liquid crystal binary gratings with high spatial frequency,” Opt. Eng. 39, 2129–2137 (2000).
[CrossRef]

Stannarius, R.

T. John, J. Heuer, R. Stannarius, “Influence of excitation wave forms and frequencies on the fundamental time symmetry of the system dynamics, studied in nematic electroconvection,” Phys. Rev. E 71, 056307 (2005).
[CrossRef]

T. John, R. Stannarius, “Preparation of subharmonic patterns in nematic electroconvection,” Phys. Rev. E 70, 025202 (2004).
[CrossRef]

T. John, U. Behn, R. Stannarius, “Laser diffraction by periodic dynamic patterns in anisotropic fluids,” Eur. Phys. J. B 35, 267–278 (2003).
[CrossRef]

H. Amm, M. Grigutsch, R. Stannarius, “Optical characterization of electroconvection in nematics,” Mol. Cryst. Liq. Cryst. Sci. Technol., Sect. A 320, 11–27 (1998).
[CrossRef]

H. Amm, M. Grigutsch, R. Stannarius, “Spatiotemporal analysis of electroconvection in nematics,” Z. Naturforsch., A: Phys. Sci. 53a, 117–126 (1998).

Taflove, A.

A. Taflove, S. C. Hagness, Computational Electrodynamics (Artech House, 2000).

Titus, C. M.

C. M. Titus, P. J. Bos, J. R. Kelly, E. C. Gartland, “Comparison of analytical calculations to finite-difference time-domain simulations of one-dimensional spatially varying anisotropic liquid crystal structures,” J. Appl. Phys. 38, 1188–1190 (1999).
[CrossRef]

Trainoff, S.

S. Trainoff, D. Cannell, “Physical optics treatment of the shadowgraph,” Phys. Fluids 14, 1340–1363 (2002).
[CrossRef]

Williams, R.

R. Williams, “Domains in liquid crystals,” J. Chem. Phys. 39, 384–388 (1963).
[CrossRef]

Winkler, B. L.

S. Rasenat, G. Hartung, B. L. Winkler, I. Rehberg, “The shadowgraph method in convection experiments,” Exp. Fluids 7, 412–420 (1989).
[CrossRef]

Witzigmann, B.

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]

Zenginoglou, H. M.

Appl. Opt. (1)

Eur. Phys. J. B (1)

T. John, U. Behn, R. Stannarius, “Laser diffraction by periodic dynamic patterns in anisotropic fluids,” Eur. Phys. J. B 35, 267–278 (2003).
[CrossRef]

Exp. Fluids (1)

S. Rasenat, G. Hartung, B. L. Winkler, I. Rehberg, “The shadowgraph method in convection experiments,” Exp. Fluids 7, 412–420 (1989).
[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]

IEEE Trans. Electromagn. Compat. (1)

G. Mur, “Absorbing boundary conditions for the finite-difference approximation of time-domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat. 23, 377–382 (1981).
[CrossRef]

J. Appl. Phys. (2)

C. M. Titus, P. J. Bos, J. R. Kelly, E. C. Gartland, “Comparison of analytical calculations to finite-difference time-domain simulations of one-dimensional spatially varying anisotropic liquid crystal structures,” J. Appl. Phys. 38, 1188–1190 (1999).
[CrossRef]

T. O. Carroll, “Liquid crystal diffraction grating,” J. Appl. Phys. 43, 767–770 (1972).
[CrossRef]

J. Chem. Phys. (1)

R. Williams, “Domains in liquid crystals,” J. Chem. Phys. 39, 384–388 (1963).
[CrossRef]

J. Comput. Phys. (1)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 184–208 (1994).
[CrossRef]

J. Opt. Soc. Am. A (2)

Microwave Opt. Technol. Lett. (1)

E. Kriezis, “A comparative study of light scattering from liquid crystal droplets,” Microwave Opt. Technol. Lett. 35, 437–441 (2002).
[CrossRef]

Mol. Cryst. Liq. Cryst. Sci. Technol., Sect. A (2)

T. Scharf, C. Bohley, “Light propagation through alignment-patterned liquid crystal gratings,” Mol. Cryst. Liq. Cryst. Sci. Technol., Sect. A 375, 491–500 (2002).
[CrossRef]

H. Amm, M. Grigutsch, R. Stannarius, “Optical characterization of electroconvection in nematics,” Mol. Cryst. Liq. Cryst. Sci. Technol., Sect. A 320, 11–27 (1998).
[CrossRef]

Opt. Commun. (1)

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. Eng. (1)

M. Bouvier, T. Scharf, “Analysis of nematic liquid crystal binary gratings with high spatial frequency,” Opt. Eng. 39, 2129–2137 (2000).
[CrossRef]

Phys. Fluids (1)

S. Trainoff, D. Cannell, “Physical optics treatment of the shadowgraph,” Phys. Fluids 14, 1340–1363 (2002).
[CrossRef]

Phys. Rev. E (2)

T. John, R. Stannarius, “Preparation of subharmonic patterns in nematic electroconvection,” Phys. Rev. E 70, 025202 (2004).
[CrossRef]

T. John, J. Heuer, R. Stannarius, “Influence of excitation wave forms and frequencies on the fundamental time symmetry of the system dynamics, studied in nematic electroconvection,” Phys. Rev. E 71, 056307 (2005).
[CrossRef]

Z. Naturforsch., A: Phys. Sci. (1)

H. Amm, M. Grigutsch, R. Stannarius, “Spatiotemporal analysis of electroconvection in nematics,” Z. Naturforsch., A: Phys. Sci. 53a, 117–126 (1998).

Other (7)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

R. Dändliker, P. Blattner, C. Rockstuhl, H. P. Herzig, “Phase singularities generated by optical microstructures: Theory and experimental results,” in Singular Optics (Optical Vortices) Fundamentals and Applications, M. S. Soskin and M. V. Vastenov, eds., Proc. SPIE4403, 257–261 (2001).

P. Blattner, “Light fields emerging from periodic optical microstructures,” Ph.D. thesis (University of Neuchâtel, 1999).

P. G. de Gennes, The Physics of Liquid Crystals (Clarendon, 1993).

L. Kramer, W. Pesch, “Electrohydrodynamic instabilities,” in Pattern Formation in Liquid Crystals, A. Buka and L. Kramer, eds. (Springer, 1996), pp. 221–255.
[CrossRef]

C. Bohley, Polarization Optics of Periodic Media (UFO-Verlag, 2004).

A. Taflove, S. C. Hagness, Computational Electrodynamics (Artech House, 2000).

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Figures (13)

Fig. 1
Fig. 1

Scheme of the director configuration in a nematic cell with convection rolls for φ max = 30 ° . The arrows symbolize the convection flow field.

Fig. 2
Fig. 2

Director field and momentary pictures of the calculated E x field in an EHC cell with period p = 50 μ m , thickness d = 50 μ m and for different director deflections at different heights z. Note the biased aspect ratio, which is chosen for clarification of the field structure.

Fig. 3
Fig. 3

E x field phases at the exit ( z = d = 50 μ m ) of the EHC cell with a pattern period of p = 50 μ m and director deflection angles φ max = ( left ) 5°, (right) 10°. The phases are calculated by the FDTD method (solid curve) and with the analytical approach [Eq. (2)] in John et al.[22] (dashed curve). Each curve extends over one period of the director pattern. The convection rolls are arranged as in Fig. 2.

Fig. 4
Fig. 4

Light intensity in the exit plane of an EHC cell with p = 50 μ m , d = 50 μ m , φ max = 20 ° calculated by the FDTD method (solid curve) and with the ray-trace approach of John et al.[22] (dashed curve).

Fig. 5
Fig. 5

Phase of the electric field at the exit of an EHC cell with p = 20.0 μ m , d = 20.2 μ m , φ max = 10 ° , calculated by the FDTD method (solid curve) and with the analytical approach of John et al.[22] (dashed curve).

Fig. 6
Fig. 6

Top: Phase pattern of the electric field at the exit of an EHC cell with d = 25 μ m ; image size is 150 μ m , U = 5.29 V just above the onset threshold U c , ε = ( U 2 U c 2 1 ) = 0.10 (left), and 6.57 V , ε = 0.69 (right), f = 90 Hz < f c . Dark areas correspond to a small phase, high contrast to large phase regions. Bottom: The cross sections of the images above, taken in the center normal to the rolls, d = 25 μ m ; ε = ( left ) 0.10, (right) 0.69; f = 90 Hz . Note the change of the periodicity of the pattern from p 2 to p.

Fig. 7
Fig. 7

Dependence of the phase difference 2 Δ ϕ on the control parameter ε at a frequency of 40 Hz , cell thickness d = 25 μ m .

Fig. 8
Fig. 8

Momentary picture of the E x field in the glass layer directly behind the exit of an EHC cell with a thickness of 20 μ m and for different director deflections. The TM-polarized light enters the structure from below (see Fig. 2). Note the biased aspect ratio, which is chosen for clarification of the field structure.

Fig. 9
Fig. 9

(Top) electric field phase and (bottom) intensity at the exit of an EHC cell with thickness d = 20.2 μ m ; period p = ( left ) 8.3 μ m , (right) 4.5 μ m ; for different director deflection angles.

Fig. 10
Fig. 10

(Left) I 2 I 0 , (right) I 1 I 0 for a cell with d = 20.2 μ m and for different φ max : solid curve, analytical approach of Eq. (3) and in John et al.[22] [Eq. (37) therein]; circles, FDTD for p s = 8.3 μ m ; crosses, FDTD for p d = 4.5 μ m .

Fig. 11
Fig. 11

Time trajectories for φ max in the (left) subharmonic and (right) dielectric regime. T 0 , s is the temporal excitation period for the subharmonic regime with T 0 , s = 20 ms and T 0 , d is the temporal excitation period for the dielectric regime with T 0 , d = 12.5 ms at sawtooth excitation.

Fig. 12
Fig. 12

(Top) calculated and (bottom) measured spatiotemporal patterns for the subharmonic resp. dielectric regime with p s = 9.6 μ m , T 0 , s = 25 ms resp. p d = 5.5 μ m , T 0 , d = 10 ms for the measured patterns, which are taken from John et al.[15]

Fig. 13
Fig. 13

Left: Momentary picture of the calculated E x field with two opposite phase singularities (in the white circle) in the glass plate 5 μ m behind the middle of the exit of a 20 μ m cell with 4.5 μ m period and φ max = 20 ° . Right: E x field with two opposite phase singularities (each in a white circle) in the glass plate 3 μ m behind the exit plane of d = 12 μ m cell with same parameters. Light propagation is from bottom to the top of the picture.

Equations (4)

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φ ( x , z ) = φ max sin ( 2 π x p ) sin ( π z d ) .
ϕ = 2 π n e d ( n e 2 n o 2 ) φ max 2 sin ( 4 π x p ) 8 λ n o 2 .
I 2 I 0 = 1 4 [ π n e d ( n e 2 n o 2 ) 4 λ n o 2 ] 2 φ max 4 .
E ( t ) = { ( 2 t T 0 1 ) E 0 , for 0 t < T 0 E ( t mod T 0 ) , elsewhere } .

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