Abstract

The fringing-field broadening of a phase-step profile and its dependence on the thickness of a liquid-crystal (LC) cell were studied in a simple, three-electrode LC cell structure consisting of two lateral electrodes biased with a differential voltage and a third, grounded, electrode placed on the opposite substrate. The results were compared both with an approximate analytical model developed earlier for a fringe-field-broadening kernel and with computer simulations. Good agreement between the experiment and the theoretical as well as the simulation results is shown.

© 2005 Optical Society of America

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References

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  1. V. G. Chigrinov, I. N. Kompanets, A. A. Vasiliev, “Behaviour of nematic liquid crystals in inhomogeneous electric fields,” Mol. Cryst. Liq. Cryst. 55, 193–207 (1979).
    [CrossRef]
  2. J. H. Kulick, J. M. Jarem, R. G. Lindquist, S. T. Kowel, M. W. Friends, T. M. Leslie, “Electrostatic and diffraction analysis of a liquid crystal device utilizing fringing fields: applications to three-dimensional displays,” Appl. Opt. 34, 1901–1922 (1995).
    [CrossRef] [PubMed]
  3. B. Verweire, R. Defever, “Limitation of resolution of LCOS-based projection displays by diffraction effects,” presented at 19th International Display Research Conference (EuroDisplay 99), Berlin, Germany,6–9 September 1999.
  4. H. De Smet, J. Van Der Steen, A. Van Calster, “Microdisplays with high pixel count,” Proc. Soc. Inf. Disp. 32, 968–971 (2001).
  5. L. J. Friedman, D. S. Hobbs, S. Lieberman, D. L. Corkum, H. Q. Nguyen, D. P. Resler, R. C. Sharp, T. A. Dorschner, “Spatially resolved phase imaging of programmable liquid crystal grating,” Appl. Opt. 35, 6236–6240 (1996).
    [CrossRef] [PubMed]
  6. V. G. Dominique, A. J. Carney, E. A. Watson, “Measurement and modeling of the angular dispersion in liquid crystal broadband beam steering devices,” Opt. Eng. 35, 3371–3379 (1996).
    [CrossRef]
  7. T. Scharf, M. Bouvier, R. Dändliker, “Multilevel nematic liquid crystal phase grating,”in Eighth International Conference on Nonlinear Optics of Liquid and Photorefractive Crystals, G. V. Klimusheva, A. G. Iljin, eds., Proc. SPIE4418, 31–37 (2001).
    [CrossRef]
  8. M. Bouvier, T. Scharf, “Analysis of nematic liquid crystal binary gratings with high spatial frequency,” Opt. Eng. 39, 2129–2137 (2000).
    [CrossRef]
  9. J. H. Kulick, J. M. Jarem, R. G. Lindquist, S. T. Kowel, M. W. Friends, T. M. Leslie, “Electrostatic and diffraction analysis of a liquid crystal device utilizing fringing fields: applications to three-dimensional displays,” Appl. Opt. 34, 1901–1922 (1995).
    [CrossRef] [PubMed]
  10. M. Oh-e, K. Kondo, “Electro-optical characteristics and switching behavior of the in-plane switching mode,” Appl. Phys. Lett. 67, 3895–3897 (1995).
    [CrossRef]
  11. B. Apter, U. Efron, E. Bahat-Treidel, “On the fringing field effect in liquid crystal beam steering devices,” Appl. Opt. 43, 11–19 (2004).
    [CrossRef] [PubMed]
  12. U. Efron, B. Apter, E. Bahat-Treidel, “Fringing field effect in liquid crystal beam steering devices: an approximate analytical model,” J. Opt. Soc. Am. A 21, 1996–2008 (2004).
    [CrossRef]
  13. M. Oh-e, M. Yoneya, K. Kondo, “Switching of negative and positive dielectro-anisotropic liquid crystals by in-plane electric fields,” J. Appl. Phys. 82, 528–535 (1997).
    [CrossRef]
  14. E. Bahat-Treidel, B. Apter, U. Efron, “Simple method for controlled variation of liquid crystal cell thickness,” Opt. Eng. 43, 3021–3025 (2004).
    [CrossRef]
  15. L. M. Blinov, Electro-Optical and Magneto-Optical Properties of Liquid Crystals (Wiley, New York, 1983), pp. 120–121.
  16. S.-T. Wu, U. Efron, L. D. Hess, “Birefringence measurements of liquid crystals,”Appl. Opt. 23, 3911–3915 (1984).
    [CrossRef] [PubMed]
  17. Autronics-Melcher’s 2dimMOS software. http://www.autronicmelchers.com/index.htm .
  18. F. C. Frank, “On the theory of liquid crystals,” Discuss. Faraday Soc. 25, 19–28 (1958).
    [CrossRef]
  19. C. W. Oseen, “The theory of liquid crystals,” Trans. Faraday Soc. 29, 883–889 (1933).
    [CrossRef]
  20. Because the method of estimating the cell thickness based on the computer simulation results differs from that used in specifying cell thickness based on experimental data, the cell thicknesses that correspond to the same integer multiple, N, of π radians of phase retardation (N = 2, 5, 7, 10) differ by approximately 12–13 % in the two cases, as observed in the caption of Fig. 8. However, whereas the origin of this difference is unclear at this point, its effect on the results is marginal because of its small magnitude.
  21. D. C. Montgomery, G. C. Runger, Applied Statistics and Probability for Engineers, 2nd ed. (Wiley, New York, 1999), Chap. 10.
  22. E. Mansfield, Basic Statistics with Applications (Norton, New York, 1986), Chap. 11.

2004 (3)

2001 (1)

H. De Smet, J. Van Der Steen, A. Van Calster, “Microdisplays with high pixel count,” Proc. Soc. Inf. Disp. 32, 968–971 (2001).

2000 (1)

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

1997 (1)

M. Oh-e, M. Yoneya, K. Kondo, “Switching of negative and positive dielectro-anisotropic liquid crystals by in-plane electric fields,” J. Appl. Phys. 82, 528–535 (1997).
[CrossRef]

1996 (2)

L. J. Friedman, D. S. Hobbs, S. Lieberman, D. L. Corkum, H. Q. Nguyen, D. P. Resler, R. C. Sharp, T. A. Dorschner, “Spatially resolved phase imaging of programmable liquid crystal grating,” Appl. Opt. 35, 6236–6240 (1996).
[CrossRef] [PubMed]

V. G. Dominique, A. J. Carney, E. A. Watson, “Measurement and modeling of the angular dispersion in liquid crystal broadband beam steering devices,” Opt. Eng. 35, 3371–3379 (1996).
[CrossRef]

1995 (3)

1984 (1)

1979 (1)

V. G. Chigrinov, I. N. Kompanets, A. A. Vasiliev, “Behaviour of nematic liquid crystals in inhomogeneous electric fields,” Mol. Cryst. Liq. Cryst. 55, 193–207 (1979).
[CrossRef]

1958 (1)

F. C. Frank, “On the theory of liquid crystals,” Discuss. Faraday Soc. 25, 19–28 (1958).
[CrossRef]

1933 (1)

C. W. Oseen, “The theory of liquid crystals,” Trans. Faraday Soc. 29, 883–889 (1933).
[CrossRef]

Apter, B.

Bahat-Treidel, E.

Blinov, L. M.

L. M. Blinov, Electro-Optical and Magneto-Optical Properties of Liquid Crystals (Wiley, New York, 1983), pp. 120–121.

Bouvier, M.

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

T. Scharf, M. Bouvier, R. Dändliker, “Multilevel nematic liquid crystal phase grating,”in Eighth International Conference on Nonlinear Optics of Liquid and Photorefractive Crystals, G. V. Klimusheva, A. G. Iljin, eds., Proc. SPIE4418, 31–37 (2001).
[CrossRef]

Carney, A. J.

V. G. Dominique, A. J. Carney, E. A. Watson, “Measurement and modeling of the angular dispersion in liquid crystal broadband beam steering devices,” Opt. Eng. 35, 3371–3379 (1996).
[CrossRef]

Chigrinov, V. G.

V. G. Chigrinov, I. N. Kompanets, A. A. Vasiliev, “Behaviour of nematic liquid crystals in inhomogeneous electric fields,” Mol. Cryst. Liq. Cryst. 55, 193–207 (1979).
[CrossRef]

Corkum, D. L.

Dändliker, R.

T. Scharf, M. Bouvier, R. Dändliker, “Multilevel nematic liquid crystal phase grating,”in Eighth International Conference on Nonlinear Optics of Liquid and Photorefractive Crystals, G. V. Klimusheva, A. G. Iljin, eds., Proc. SPIE4418, 31–37 (2001).
[CrossRef]

De Smet, H.

H. De Smet, J. Van Der Steen, A. Van Calster, “Microdisplays with high pixel count,” Proc. Soc. Inf. Disp. 32, 968–971 (2001).

Defever, R.

B. Verweire, R. Defever, “Limitation of resolution of LCOS-based projection displays by diffraction effects,” presented at 19th International Display Research Conference (EuroDisplay 99), Berlin, Germany,6–9 September 1999.

Dominique, V. G.

V. G. Dominique, A. J. Carney, E. A. Watson, “Measurement and modeling of the angular dispersion in liquid crystal broadband beam steering devices,” Opt. Eng. 35, 3371–3379 (1996).
[CrossRef]

Dorschner, T. A.

Efron, U.

Frank, F. C.

F. C. Frank, “On the theory of liquid crystals,” Discuss. Faraday Soc. 25, 19–28 (1958).
[CrossRef]

Friedman, L. J.

Friends, M. W.

Hess, L. D.

Hobbs, D. S.

Jarem, J. M.

Kompanets, I. N.

V. G. Chigrinov, I. N. Kompanets, A. A. Vasiliev, “Behaviour of nematic liquid crystals in inhomogeneous electric fields,” Mol. Cryst. Liq. Cryst. 55, 193–207 (1979).
[CrossRef]

Kondo, K.

M. Oh-e, M. Yoneya, K. Kondo, “Switching of negative and positive dielectro-anisotropic liquid crystals by in-plane electric fields,” J. Appl. Phys. 82, 528–535 (1997).
[CrossRef]

M. Oh-e, K. Kondo, “Electro-optical characteristics and switching behavior of the in-plane switching mode,” Appl. Phys. Lett. 67, 3895–3897 (1995).
[CrossRef]

Kowel, S. T.

Kulick, J. H.

Leslie, T. M.

Lieberman, S.

Lindquist, R. G.

Mansfield, E.

E. Mansfield, Basic Statistics with Applications (Norton, New York, 1986), Chap. 11.

Montgomery, D. C.

D. C. Montgomery, G. C. Runger, Applied Statistics and Probability for Engineers, 2nd ed. (Wiley, New York, 1999), Chap. 10.

Nguyen, H. Q.

Oh-e, M.

M. Oh-e, M. Yoneya, K. Kondo, “Switching of negative and positive dielectro-anisotropic liquid crystals by in-plane electric fields,” J. Appl. Phys. 82, 528–535 (1997).
[CrossRef]

M. Oh-e, K. Kondo, “Electro-optical characteristics and switching behavior of the in-plane switching mode,” Appl. Phys. Lett. 67, 3895–3897 (1995).
[CrossRef]

Oseen, C. W.

C. W. Oseen, “The theory of liquid crystals,” Trans. Faraday Soc. 29, 883–889 (1933).
[CrossRef]

Resler, D. P.

Runger, G. C.

D. C. Montgomery, G. C. Runger, Applied Statistics and Probability for Engineers, 2nd ed. (Wiley, New York, 1999), Chap. 10.

Scharf, T.

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

T. Scharf, M. Bouvier, R. Dändliker, “Multilevel nematic liquid crystal phase grating,”in Eighth International Conference on Nonlinear Optics of Liquid and Photorefractive Crystals, G. V. Klimusheva, A. G. Iljin, eds., Proc. SPIE4418, 31–37 (2001).
[CrossRef]

Sharp, R. C.

Van Calster, A.

H. De Smet, J. Van Der Steen, A. Van Calster, “Microdisplays with high pixel count,” Proc. Soc. Inf. Disp. 32, 968–971 (2001).

Van Der Steen, J.

H. De Smet, J. Van Der Steen, A. Van Calster, “Microdisplays with high pixel count,” Proc. Soc. Inf. Disp. 32, 968–971 (2001).

Vasiliev, A. A.

V. G. Chigrinov, I. N. Kompanets, A. A. Vasiliev, “Behaviour of nematic liquid crystals in inhomogeneous electric fields,” Mol. Cryst. Liq. Cryst. 55, 193–207 (1979).
[CrossRef]

Verweire, B.

B. Verweire, R. Defever, “Limitation of resolution of LCOS-based projection displays by diffraction effects,” presented at 19th International Display Research Conference (EuroDisplay 99), Berlin, Germany,6–9 September 1999.

Watson, E. A.

V. G. Dominique, A. J. Carney, E. A. Watson, “Measurement and modeling of the angular dispersion in liquid crystal broadband beam steering devices,” Opt. Eng. 35, 3371–3379 (1996).
[CrossRef]

Wu, S.-T.

Yoneya, M.

M. Oh-e, M. Yoneya, K. Kondo, “Switching of negative and positive dielectro-anisotropic liquid crystals by in-plane electric fields,” J. Appl. Phys. 82, 528–535 (1997).
[CrossRef]

Appl. Opt. (5)

Appl. Phys. Lett. (1)

M. Oh-e, K. Kondo, “Electro-optical characteristics and switching behavior of the in-plane switching mode,” Appl. Phys. Lett. 67, 3895–3897 (1995).
[CrossRef]

Discuss. Faraday Soc. (1)

F. C. Frank, “On the theory of liquid crystals,” Discuss. Faraday Soc. 25, 19–28 (1958).
[CrossRef]

J. Appl. Phys. (1)

M. Oh-e, M. Yoneya, K. Kondo, “Switching of negative and positive dielectro-anisotropic liquid crystals by in-plane electric fields,” J. Appl. Phys. 82, 528–535 (1997).
[CrossRef]

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

Mol. Cryst. Liq. Cryst. (1)

V. G. Chigrinov, I. N. Kompanets, A. A. Vasiliev, “Behaviour of nematic liquid crystals in inhomogeneous electric fields,” Mol. Cryst. Liq. Cryst. 55, 193–207 (1979).
[CrossRef]

Opt. Eng. (3)

V. G. Dominique, A. J. Carney, E. A. Watson, “Measurement and modeling of the angular dispersion in liquid crystal broadband beam steering devices,” Opt. Eng. 35, 3371–3379 (1996).
[CrossRef]

E. Bahat-Treidel, B. Apter, U. Efron, “Simple method for controlled variation of liquid crystal cell thickness,” Opt. Eng. 43, 3021–3025 (2004).
[CrossRef]

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

Proc. Soc. Inf. Disp. (1)

H. De Smet, J. Van Der Steen, A. Van Calster, “Microdisplays with high pixel count,” Proc. Soc. Inf. Disp. 32, 968–971 (2001).

Trans. Faraday Soc. (1)

C. W. Oseen, “The theory of liquid crystals,” Trans. Faraday Soc. 29, 883–889 (1933).
[CrossRef]

Other (7)

Because the method of estimating the cell thickness based on the computer simulation results differs from that used in specifying cell thickness based on experimental data, the cell thicknesses that correspond to the same integer multiple, N, of π radians of phase retardation (N = 2, 5, 7, 10) differ by approximately 12–13 % in the two cases, as observed in the caption of Fig. 8. However, whereas the origin of this difference is unclear at this point, its effect on the results is marginal because of its small magnitude.

D. C. Montgomery, G. C. Runger, Applied Statistics and Probability for Engineers, 2nd ed. (Wiley, New York, 1999), Chap. 10.

E. Mansfield, Basic Statistics with Applications (Norton, New York, 1986), Chap. 11.

L. M. Blinov, Electro-Optical and Magneto-Optical Properties of Liquid Crystals (Wiley, New York, 1983), pp. 120–121.

B. Verweire, R. Defever, “Limitation of resolution of LCOS-based projection displays by diffraction effects,” presented at 19th International Display Research Conference (EuroDisplay 99), Berlin, Germany,6–9 September 1999.

T. Scharf, M. Bouvier, R. Dändliker, “Multilevel nematic liquid crystal phase grating,”in Eighth International Conference on Nonlinear Optics of Liquid and Photorefractive Crystals, G. V. Klimusheva, A. G. Iljin, eds., Proc. SPIE4418, 31–37 (2001).
[CrossRef]

Autronics-Melcher’s 2dimMOS software. http://www.autronicmelchers.com/index.htm .

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Figures (9)

Fig. 1
Fig. 1

Schematic of the experimental setup.

Fig. 2
Fig. 2

Schematic of the test cell.

Fig. 3
Fig. 3

Experimentally recorded intensities for a 6.1-µm-thick (4π phase retardation) LC cell. The LC voltage is proportional to the time elapsed as indicated on the horizontal axis (5 s/division, representing 1-V increments). The top, wedgelike signal shows the analog voltage as recorded by the scope.

Fig. 4
Fig. 4

(a) Experimental normalized transmission along the three-electrode test cell with cell-gap thicknesses of 3.1, 7.7, 10.8, and 15.4 µm, corresponding to maximum phase retardations of 2π, 5π, 7π, and 10π. (b) Experimental 2π rad retardation profile along the three-electrode test cell with cell-gap thicknesses of 3.1, 7.7, 10.8, and 15.4 µm, corresponding to maximum phase retardations of 2π, 5π, 7π, and 10π. (c) Expanded inset of (b).

Fig. 5
Fig. 5

(a) Calculated broadening kernel for cell-gap thicknesses of 3.1–15.4 µm corresponding to maximum phase retardations of 2π–10π. (b) Theoretical 2π rad retardation profile generated by convolution of the step function with the broadening kernel along the lateral, interelectrode gap of the three-electrode test cell. The cell-gap thicknesses are 3.1–15.4 µm, corresponding to maximum phase retardations of 2π–10π.

Fig. 6
Fig. 6

2dimMOS simulation of 2π rad retardation profile, LC director alignment, and field equipotential lines along the three-electrode test cell with cell-gap thicknesses of 3.5, 8.6, 12.1, and 17.4 µm, corresponding to maximum phase retardations of 2π, 5π, 7π, and 10π.

Fig. 7
Fig. 7

Comparison of experimental and theoretical calculation of 2π rad retardation profiles along the three-electrode test cell with cell-gap thicknesses of 15.4, 10.8, 7.7, and 3.1 µm, corresponding to maximum phase retardations of (a) 10π, (b) 7π, (c) 5π, and (d) 2π.

Fig. 8
Fig. 8

Comparison of experimental and simulations of 2π rad retardation profiles along the three-electrode test cell with cell-gap thicknesses of 15.4, 10.8, 7.7, and 3.1 µm (experimental) and 17.4, 12.1, 8.6, and 3.5 µm (simulation), corresponding to maximum phase retardations of 10π, 7π, 5π, and 2π, respectively.

Fig. 9
Fig. 9

Correlation among the experimental and the computer simulation results and the theoretical model for effective phase-step function broadening as a function of the maximum cell-gap retardation dΔn/λ: experiment versus theory (analytical model), experiment versus computer simulation, and computer simulation versus theory.

Tables (1)

Tables Icon

Table 1 Parameters of the LC Layer Used for Experimental Study and Simulation

Equations (9)

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I = I 0 sin 2 ( 2 β ) sin 2 ( Δ φ / 2 ) ,
Δ φ max ( d ) = 2 π d Δ n max / λ ,
k ( x ) σ 1 exp ( 2 | x | / σ ) ,
σ ( d ) ɛ 3 ɛ ( d ɛ 2 ɛ 2 4 ν ɛ ɛ d min ) ,
d min = λ / Δ n max .
n ( x , z ) = n o n e [ n o 2 cos 2 θ ( x , z ) + n e 2 sin 2 θ ( x , z ) ] 1 / 2 .
Δ ϕ ( x ) = 2 π λ 0 d [ n ( x , z ) n o ] d z .
R Adj 2 = 1 [ ( n 1 ) ( n k ) ( 1 R 2 ) ] ,
r = R 2 = n i = 1 n X i Y i i = 1 n X i i = 1 n Y i [ n i = 1 n X i 2 ( i = 1 n X i ) 2 ] 1 / 2 [ n i = 1 n Y i 2 ( i = 1 n Y i ) 2 ] 1 / 2 .

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