Abstract

A liquid-crystal-infiltrated microcavity structure is proposed as a variable-refractive-index material. It has the advantages over previously considered nanostructured materials of having a larger phase-angle change and lower driving voltage. Two-dimensional liquid-crystal director and finite-difference time-domain optical simulations are used to select liquid crystal material parameters and optimize the dimension of the microcavity structured material.

© 2003 Optical Society of America

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References

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  1. H. Matsumoto, K. Hirabayashi, S. Sakata, T. Hayashi, “Tunable wavelength filter using nano-sized droplets of liquid crystal,” IEEE Photon. Technol. Lett. 11, 442–444 (1999).
    [CrossRef]
  2. S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, V. Lehmann, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389–R2392 (2000).
    [CrossRef]
  3. R. J. Ondris-Crawford, “The effect of molecular anchoring and curvature on confined liquid crystal,” Ph.D. dissertation (Kent State University, Kent, Ohio, 1993).
  4. L. M. Blinov, V. G. Chigrinov, Electrooptic Effects in Liquid Crystal Materials (Springer-Verlag, New York, 1996).
  5. E. Hecht, Optics, 3rd ed. (Addison-Wesley Longman, Reading, Mass., 1998), Chap. 10, p. 466.
  6. A. Taflove, S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, Boston, Mass., 2000).
  7. B. Witzigmann, P. Regli, W. Fichtner, “Rigorous electromagnetic simulation of liquid crystal displays,” J. Opt. Soc. Am. A 15, 753–757 (1998).
    [CrossRef]
  8. 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,” Jpn. J. Appl. Phys. 38, 1488–1494 (1999).
    [CrossRef]
  9. E. E. Kriezis, S. J. Elston, “Light wave propagation in liquid crystal display by the 2-D finite-difference time-domain method,” Opt. Commun. 177, 69–77 (2000).
    [CrossRef]
  10. K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
    [CrossRef]
  11. A. Yefet, P. G. Petropoulos, “A staggered fourth-order accuracy explicit finite difference scheme for the time-domain Maxwell’s equations,” J. Comput. Phys. 168, 286–315 (2001).
    [CrossRef]
  12. J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  13. M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), Chap. 8.
  14. C. M. Titus, “Refractive and diffractive liquid crystal beam steering devices,” Ph.D. dissertation (Kent State University, Kent, Ohio, 2000).
  15. R. D. Guenther, Modern Optics (Wiley, New York, 1990), Chap. 9.
  16. J. E. Anderson, P. E. Watson, P. J. Bos, LC3D: Liquid Crystal Display 3-D Director Simulator Software and Technology Guide (Artech House, Boston, Mass., 2001).
  17. P. G. de Gennes, J. Prost, The Physics of Liquid Crystal (Oxford Science, Oxford, UK, 1993).
  18. D. W. Berreman, “Numerical modeling of twisted nematic devices,” Philos. Trans. R. Soc. London Ser. A 309, 203–216 (1983).
    [CrossRef]
  19. H. Mori, E. C. Gartland, J. R. Kelly, P. J. Bos, “Multidimensional director modeling using the Q tensor representation in a liquid crystal cell and its application to the π cell with patterned electrodes,” Jpn. J. Appl. Phys. 38, 135–146 (1999).
    [CrossRef]
  20. R. T. Pogue, R. L. Sutherland, M. G. Schmitt, L. V. Natarajan, S. A. Siwecki, V. P. Tondiglia, T. J. Bunning, “Electrically switchable Bragg gratings from liquid crystal/polymer composites,” Appl. Spectrosc. 54, 12A–28A (2000).
    [CrossRef]

2001

A. Yefet, P. G. Petropoulos, “A staggered fourth-order accuracy explicit finite difference scheme for the time-domain Maxwell’s equations,” J. Comput. Phys. 168, 286–315 (2001).
[CrossRef]

2000

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

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, V. Lehmann, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389–R2392 (2000).
[CrossRef]

R. T. Pogue, R. L. Sutherland, M. G. Schmitt, L. V. Natarajan, S. A. Siwecki, V. P. Tondiglia, T. J. Bunning, “Electrically switchable Bragg gratings from liquid crystal/polymer composites,” Appl. Spectrosc. 54, 12A–28A (2000).
[CrossRef]

1999

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,” Jpn. J. Appl. Phys. 38, 1488–1494 (1999).
[CrossRef]

H. Mori, E. C. Gartland, J. R. Kelly, P. J. Bos, “Multidimensional director modeling using the Q tensor representation in a liquid crystal cell and its application to the π cell with patterned electrodes,” Jpn. J. Appl. Phys. 38, 135–146 (1999).
[CrossRef]

H. Matsumoto, K. Hirabayashi, S. Sakata, T. Hayashi, “Tunable wavelength filter using nano-sized droplets of liquid crystal,” IEEE Photon. Technol. Lett. 11, 442–444 (1999).
[CrossRef]

1998

1994

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

1983

D. W. Berreman, “Numerical modeling of twisted nematic devices,” Philos. Trans. R. Soc. London Ser. A 309, 203–216 (1983).
[CrossRef]

1966

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

Anderson, J. E.

J. E. Anderson, P. E. Watson, P. J. Bos, LC3D: Liquid Crystal Display 3-D Director Simulator Software and Technology Guide (Artech House, Boston, Mass., 2001).

Berenger, J.-P.

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

Berreman, D. W.

D. W. Berreman, “Numerical modeling of twisted nematic devices,” Philos. Trans. R. Soc. London Ser. A 309, 203–216 (1983).
[CrossRef]

Birner, A.

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, V. Lehmann, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389–R2392 (2000).
[CrossRef]

Blinov, L. M.

L. M. Blinov, V. G. Chigrinov, Electrooptic Effects in Liquid Crystal Materials (Springer-Verlag, New York, 1996).

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), Chap. 8.

Bos, P. J.

H. Mori, E. C. Gartland, J. R. Kelly, P. J. Bos, “Multidimensional director modeling using the Q tensor representation in a liquid crystal cell and its application to the π cell with patterned electrodes,” Jpn. J. Appl. Phys. 38, 135–146 (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,” Jpn. J. Appl. Phys. 38, 1488–1494 (1999).
[CrossRef]

J. E. Anderson, P. E. Watson, P. J. Bos, LC3D: Liquid Crystal Display 3-D Director Simulator Software and Technology Guide (Artech House, Boston, Mass., 2001).

Bunning, T. J.

Busch, K.

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, V. Lehmann, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389–R2392 (2000).
[CrossRef]

Chigrinov, V. G.

L. M. Blinov, V. G. Chigrinov, Electrooptic Effects in Liquid Crystal Materials (Springer-Verlag, New York, 1996).

de Gennes, P. G.

P. G. de Gennes, J. Prost, The Physics of Liquid Crystal (Oxford Science, Oxford, UK, 1993).

Elston, S. J.

E. E. Kriezis, S. J. Elston, “Light wave propagation in liquid crystal display by the 2-D finite-difference time-domain method,” Opt. Commun. 177, 69–77 (2000).
[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,” Jpn. J. Appl. Phys. 38, 1488–1494 (1999).
[CrossRef]

H. Mori, E. C. Gartland, J. R. Kelly, P. J. Bos, “Multidimensional director modeling using the Q tensor representation in a liquid crystal cell and its application to the π cell with patterned electrodes,” Jpn. J. Appl. Phys. 38, 135–146 (1999).
[CrossRef]

Gösele, U.

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, V. Lehmann, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389–R2392 (2000).
[CrossRef]

Guenther, R. D.

R. D. Guenther, Modern Optics (Wiley, New York, 1990), Chap. 9.

Hagness, S. C.

A. Taflove, S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, Boston, Mass., 2000).

Hayashi, T.

H. Matsumoto, K. Hirabayashi, S. Sakata, T. Hayashi, “Tunable wavelength filter using nano-sized droplets of liquid crystal,” IEEE Photon. Technol. Lett. 11, 442–444 (1999).
[CrossRef]

Hecht, E.

E. Hecht, Optics, 3rd ed. (Addison-Wesley Longman, Reading, Mass., 1998), Chap. 10, p. 466.

Hirabayashi, K.

H. Matsumoto, K. Hirabayashi, S. Sakata, T. Hayashi, “Tunable wavelength filter using nano-sized droplets of liquid crystal,” IEEE Photon. Technol. Lett. 11, 442–444 (1999).
[CrossRef]

John, S.

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, V. Lehmann, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389–R2392 (2000).
[CrossRef]

Kelly, J. R.

H. Mori, E. C. Gartland, J. R. Kelly, P. J. Bos, “Multidimensional director modeling using the Q tensor representation in a liquid crystal cell and its application to the π cell with patterned electrodes,” Jpn. J. Appl. Phys. 38, 135–146 (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,” Jpn. J. Appl. Phys. 38, 1488–1494 (1999).
[CrossRef]

Kriezis, E. E.

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

Lehmann, V.

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, V. Lehmann, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389–R2392 (2000).
[CrossRef]

Leonard, S. W.

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, V. Lehmann, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389–R2392 (2000).
[CrossRef]

Matsumoto, H.

H. Matsumoto, K. Hirabayashi, S. Sakata, T. Hayashi, “Tunable wavelength filter using nano-sized droplets of liquid crystal,” IEEE Photon. Technol. Lett. 11, 442–444 (1999).
[CrossRef]

Mondia, J. P.

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, V. Lehmann, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389–R2392 (2000).
[CrossRef]

Mori, H.

H. Mori, E. C. Gartland, J. R. Kelly, P. J. Bos, “Multidimensional director modeling using the Q tensor representation in a liquid crystal cell and its application to the π cell with patterned electrodes,” Jpn. J. Appl. Phys. 38, 135–146 (1999).
[CrossRef]

Natarajan, L. V.

Ondris-Crawford, R. J.

R. J. Ondris-Crawford, “The effect of molecular anchoring and curvature on confined liquid crystal,” Ph.D. dissertation (Kent State University, Kent, Ohio, 1993).

Petropoulos, P. G.

A. Yefet, P. G. Petropoulos, “A staggered fourth-order accuracy explicit finite difference scheme for the time-domain Maxwell’s equations,” J. Comput. Phys. 168, 286–315 (2001).
[CrossRef]

Pogue, R. T.

Prost, J.

P. G. de Gennes, J. Prost, The Physics of Liquid Crystal (Oxford Science, Oxford, UK, 1993).

Regli, P.

Sakata, S.

H. Matsumoto, K. Hirabayashi, S. Sakata, T. Hayashi, “Tunable wavelength filter using nano-sized droplets of liquid crystal,” IEEE Photon. Technol. Lett. 11, 442–444 (1999).
[CrossRef]

Schmitt, M. G.

Siwecki, S. A.

Sutherland, R. L.

Taflove, A.

A. Taflove, S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, Boston, Mass., 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,” Jpn. J. Appl. Phys. 38, 1488–1494 (1999).
[CrossRef]

C. M. Titus, “Refractive and diffractive liquid crystal beam steering devices,” Ph.D. dissertation (Kent State University, Kent, Ohio, 2000).

Toader, O.

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, V. Lehmann, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389–R2392 (2000).
[CrossRef]

Tondiglia, V. P.

van Driel, H. M.

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, V. Lehmann, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389–R2392 (2000).
[CrossRef]

Watson, P. E.

J. E. Anderson, P. E. Watson, P. J. Bos, LC3D: Liquid Crystal Display 3-D Director Simulator Software and Technology Guide (Artech House, Boston, Mass., 2001).

Witzigmann, B.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), Chap. 8.

Yee, K.

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

Yefet, A.

A. Yefet, P. G. Petropoulos, “A staggered fourth-order accuracy explicit finite difference scheme for the time-domain Maxwell’s equations,” J. Comput. Phys. 168, 286–315 (2001).
[CrossRef]

Appl. Spectrosc.

IEEE Photon. Technol. Lett.

H. Matsumoto, K. Hirabayashi, S. Sakata, T. Hayashi, “Tunable wavelength filter using nano-sized droplets of liquid crystal,” IEEE Photon. Technol. Lett. 11, 442–444 (1999).
[CrossRef]

IEEE Trans. Antennas Propag.

K. 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. Comput. Phys.

A. Yefet, P. G. Petropoulos, “A staggered fourth-order accuracy explicit finite difference scheme for the time-domain Maxwell’s equations,” J. Comput. Phys. 168, 286–315 (2001).
[CrossRef]

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

J. Opt. Soc. Am. A

Jpn. J. Appl. Phys.

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,” Jpn. J. Appl. Phys. 38, 1488–1494 (1999).
[CrossRef]

H. Mori, E. C. Gartland, J. R. Kelly, P. J. Bos, “Multidimensional director modeling using the Q tensor representation in a liquid crystal cell and its application to the π cell with patterned electrodes,” Jpn. J. Appl. Phys. 38, 135–146 (1999).
[CrossRef]

Opt. Commun.

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

Philos. Trans. R. Soc. London Ser. A

D. W. Berreman, “Numerical modeling of twisted nematic devices,” Philos. Trans. R. Soc. London Ser. A 309, 203–216 (1983).
[CrossRef]

Phys. Rev. B

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, V. Lehmann, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389–R2392 (2000).
[CrossRef]

Other

R. J. Ondris-Crawford, “The effect of molecular anchoring and curvature on confined liquid crystal,” Ph.D. dissertation (Kent State University, Kent, Ohio, 1993).

L. M. Blinov, V. G. Chigrinov, Electrooptic Effects in Liquid Crystal Materials (Springer-Verlag, New York, 1996).

E. Hecht, Optics, 3rd ed. (Addison-Wesley Longman, Reading, Mass., 1998), Chap. 10, p. 466.

A. Taflove, S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, Boston, Mass., 2000).

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), Chap. 8.

C. M. Titus, “Refractive and diffractive liquid crystal beam steering devices,” Ph.D. dissertation (Kent State University, Kent, Ohio, 2000).

R. D. Guenther, Modern Optics (Wiley, New York, 1990), Chap. 9.

J. E. Anderson, P. E. Watson, P. J. Bos, LC3D: Liquid Crystal Display 3-D Director Simulator Software and Technology Guide (Artech House, Boston, Mass., 2001).

P. G. de Gennes, J. Prost, The Physics of Liquid Crystal (Oxford Science, Oxford, UK, 1993).

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Figures (14)

Fig. 1
Fig. 1

(a) Scanning electron microscope image of a microcavity-structure photonic crystal obtained from Ref. 2. (b) PR of the liquid crystal director configuration in a confined cylindrical cavity. (c) ER of the liquid crystal director configuration in a confined cylindrical cavity. Reprinted from Phys. Rev. B (see Ref. 2) by permission.

Fig. 2
Fig. 2

Calculated electrical coherence length with different values of Δ as a function of voltage. One liquid crystal elastic constant approximation of Ki=1.5×10-11N is used. Cell gap is 20.0 µm.

Fig. 3
Fig. 3

Rectangular riblike structured thin film material sandwiched between two substrates. The top substrate does not show in the diagram. The structure periodicity Λ is equal to L1+L2. L1 and L2 are the widths of the single wall and of the channel, respectively.

Fig. 4
Fig. 4

Layout of two-dimensional FDTD computational domain in the xy plane. A plane wave of a Gaussian beam is incident from the bottom of the film, and the beam width equals the width of the calculation domain. The widths of wall and the filling channel are L1 and L2, and their refractive indices are n1 and n2, respectively. The periodicity Λ of the structured film is L1+L2, and the film thickness is d.

Fig. 5
Fig. 5

Geometry of the two-dimensional diffraction problem. The diffraction object lies in the y=y0 plane between x=-d and x=d. The normal to the diffracting object is n=y. Diffracted light is observed at (xfar, yfar).

Fig. 6
Fig. 6

Near-field propagation snapshot of a rectangular riblike structured film of L1=1.0 µm, L2=1.0 µm, and λ=1.55 µm in free space, with nwall=1.50, nfill=1.70, and d=15.0 µm. The unit of the x and y axes is the grid point, which equals λ/20. The field amplitude is represented by the gray-scale bar.

Fig. 7
Fig. 7

Light diffraction pattern at the far field for a rectangular riblike structured film of L1=1.0 µm, L2=1.0 µm, and λ=1.55 µm in free space, with nwall=1.5, nfill=1.7, and d=15.0 µm.

Fig. 8
Fig. 8

Near-field propagation snapshot of a rectangular riblike structured film of L1=0.5 µm, L2=1.0 µm, and λ=1.55 µm in free space, with nwall=1.50, nfill=1.70, and d=15.0 µm. The unit of the x and y axes is the grid point, which equals λ/20. The field amplitude is represented by the gray-scale bar.

Fig. 9
Fig. 9

Light diffraction pattern at the far field for a rectangular riblike structured film of L1=0.50 µm, L2=1 µm, and λ=1.55 µm in free space, with nwall=1.50, nfill=1.70, and d=15.0 µm.

Fig. 10
Fig. 10

Phase angles calculated from a rectangular riblike structured film of L1=0.50 µm, L2=1 µm, and λ=1.55 µm in free space, with nwall=1.50, nfill=1.70, and d=15.0 µm. The dotted line (a) shows the phase angle of light just exiting from the film; the dashed line (b) shows the phase angle of light at the top of the calculation domain; the solid line (effective) shows the effective phase angle of the film.

Fig. 11
Fig. 11

Calculated bulk-wall-confined electrically controlled birefringence liquid-crystal-director configuration in a channel of a rectangular riblike structured film. (a) Voltage=0.0 V, (b) voltage=10.5 V.

Fig. 12
Fig. 12

Effective phase angle of an electrically controlled birefringence liquid crystal rectangular riblike variable-refractive-index film as a function of applied voltage.

Fig. 13
Fig. 13

Calculated bulk-wall-confined hybrid liquid-crystal-director configuration in a channel of a rectangular riblike structured film. (a) Voltage=0.0 V, (b) voltage=10.0 V.

Fig. 14
Fig. 14

Effective phase angle of a hybrid liquid crystal type of rectangular riblike variable-refractive-index film as a function of applied voltage.

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

L=1E4πKiΔ
E=V/d,
sin θm=mλΛ,m=1, 2, 3,,
E(r)t=-1(r)·[×H(r)],
-H(r)t=μ0-1(r)·[×E(r)],
Ψfar(r)=14πSn·[Ψnear(r)·G-G·Ψnear(r)]ds,
Ψfar(xfar, yfar)=exp(-iπ)/48πk-ddexp(-ikR)R×yΨnear(x, y0)+ik(yfar-y0)Ψnear(x, y0)Rdx,
R=[(x-xfar)2+(y0-yfar)2]1/2,k=2π/λ.
Ez(x, t)=E0 exp(-x2/w02)exp[-i(k·r-ωt)],
fg=12K11(·n)2+12K22(n·×n+q0)2+12K33(n××n)2-12D·E.
ninew=niold-Δtγ1[fg]ni,i=x, y, z.
[fg]ni=fgni-ddxfg(dni/dx)-ddyfg(dni/dy).

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