Abstract

General theories for light propagation are applied to specific liquid crystal displays. For the case of a homogeneous molecular distribution as well as that of a uniform twist we elaborate the classical Poincaré sphere method and a modified Jones calculus. The evolution of the polarization ellipse in the different configurations is studied and is depicted in an original way, which gives a better understanding of the color effects in liquid-crystal displays. Practical quantitative calorimetric conclusions are drawn in order to facilitate the experimental study of thickness variations in liquid crystal displays.

© 1994 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1964), Chap. 14.
  2. R. Guenther, Modern Optics (Wiley, New York, 1990), Chap. 13, pp. 534–545.
  3. W. A. Shurcliff, Polarized Light, Production and Use (Harvard U. Press, Cambridge, Mass., 1962), Chaps. 1 and 2.
  4. R. C. Jones, “A new calculus for the treatment of optical systems. I. Description and discussion of the calculus,”J. Opt. Soc. Am. 31, 488–493 (1941).
    [CrossRef]
  5. D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4 matrix formulation,”J. Opt. Soc. Am. 62, 502–510 (1972).
    [CrossRef]
  6. D. W. Berreman, “Optics in smoothly varying anisotropic planar structures: application to liquid-crystal twist cells,”J. Opt. Soc. Am. 63, 1374–1380 (1973).
    [CrossRef]
  7. J. Bigelow, R. Kashnow, “Poincaré sphere analysis of liquid crystal optics,” Appl. Opt. 16, 2090–2096 (1977).
    [CrossRef] [PubMed]
  8. R. M. A. Azzam, N. M. Bashara, “Simplified approach to the propagation of polarized light in anisotropic media: application to liquid crystals,”J. Opt. Soc. Am. 62, 1252–1257 (1972).
    [CrossRef]
  9. C. Gooch, H. Tarry, “Optical characteristics of twisted nematic liquid-crystal films,” Electron. Lett. 10, 2–4 (1974).
    [CrossRef]
  10. C. Gooch, H. Tarry, “The optical properties of twisted nematic liquid crystal structures with twist angles ≤ 90°,”J. Phys. D 8, 1575–1584 (1975).
    [CrossRef]
  11. E. Raynes, R. Smith, “The design of liquid crystal materials for supertwist displays,” in Proceedings of the Seventh International Display Research Conference, Eurodisplay ’87 (The Institute of Physics and The Society for Information Display, London, 1987), p. 100.19.
  12. A. De Vos, C. Reynaerts, F. Cuypers, “Optical characteristics of ferroelectric liquid crystal cells,” Ferroelectrics 113, 467–487 (1991).
    [CrossRef]
  13. A. De Vos, C. Reynaerts, “Optical transmission of ferroelectric liquid-crystal displays,” J. Appl. Phys. 65, 2616–2619 (1989).
    [CrossRef]
  14. F. Cuypers, Ph.D. dissertation (Faculty of Engineering, Rijksuniversiteit Gent, Gent, Belgium, 1989).
  15. C. Mauguin, “Sur les cristaux liquides de Lehmann,” Bull. Soc. Fr. Mineral. 34, 71 (1911).
  16. M. Schadt, F. Leenhouts, “Electro-optical performance of a new, black-white and highly multiplexable liquid crystal display,” Appl. Phys. Lett. 50, 236–238 (1987).
    [CrossRef]
  17. T. Benzschawel, “Colorimetry of displays,” in SID Seminar Lecture Notes (Society for Information Display, New Orleans, La., 1987), p. 8.1.
  18. G. J. Chamberlin, D. G. Chamberlin, Colour, Its Measurement, Computation and Application (Heyden, London, 1980), Chap. 4, pp. 54–55.

1991

A. De Vos, C. Reynaerts, F. Cuypers, “Optical characteristics of ferroelectric liquid crystal cells,” Ferroelectrics 113, 467–487 (1991).
[CrossRef]

1989

A. De Vos, C. Reynaerts, “Optical transmission of ferroelectric liquid-crystal displays,” J. Appl. Phys. 65, 2616–2619 (1989).
[CrossRef]

1987

M. Schadt, F. Leenhouts, “Electro-optical performance of a new, black-white and highly multiplexable liquid crystal display,” Appl. Phys. Lett. 50, 236–238 (1987).
[CrossRef]

1977

1975

C. Gooch, H. Tarry, “The optical properties of twisted nematic liquid crystal structures with twist angles ≤ 90°,”J. Phys. D 8, 1575–1584 (1975).
[CrossRef]

1974

C. Gooch, H. Tarry, “Optical characteristics of twisted nematic liquid-crystal films,” Electron. Lett. 10, 2–4 (1974).
[CrossRef]

1973

1972

1941

1911

C. Mauguin, “Sur les cristaux liquides de Lehmann,” Bull. Soc. Fr. Mineral. 34, 71 (1911).

Azzam, R. M. A.

Bashara, N. M.

Benzschawel, T.

T. Benzschawel, “Colorimetry of displays,” in SID Seminar Lecture Notes (Society for Information Display, New Orleans, La., 1987), p. 8.1.

Berreman, D. W.

Bigelow, J.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1964), Chap. 14.

Chamberlin, D. G.

G. J. Chamberlin, D. G. Chamberlin, Colour, Its Measurement, Computation and Application (Heyden, London, 1980), Chap. 4, pp. 54–55.

Chamberlin, G. J.

G. J. Chamberlin, D. G. Chamberlin, Colour, Its Measurement, Computation and Application (Heyden, London, 1980), Chap. 4, pp. 54–55.

Cuypers, F.

A. De Vos, C. Reynaerts, F. Cuypers, “Optical characteristics of ferroelectric liquid crystal cells,” Ferroelectrics 113, 467–487 (1991).
[CrossRef]

F. Cuypers, Ph.D. dissertation (Faculty of Engineering, Rijksuniversiteit Gent, Gent, Belgium, 1989).

De Vos, A.

A. De Vos, C. Reynaerts, F. Cuypers, “Optical characteristics of ferroelectric liquid crystal cells,” Ferroelectrics 113, 467–487 (1991).
[CrossRef]

A. De Vos, C. Reynaerts, “Optical transmission of ferroelectric liquid-crystal displays,” J. Appl. Phys. 65, 2616–2619 (1989).
[CrossRef]

Gooch, C.

C. Gooch, H. Tarry, “The optical properties of twisted nematic liquid crystal structures with twist angles ≤ 90°,”J. Phys. D 8, 1575–1584 (1975).
[CrossRef]

C. Gooch, H. Tarry, “Optical characteristics of twisted nematic liquid-crystal films,” Electron. Lett. 10, 2–4 (1974).
[CrossRef]

Guenther, R.

R. Guenther, Modern Optics (Wiley, New York, 1990), Chap. 13, pp. 534–545.

Jones, R. C.

Kashnow, R.

Leenhouts, F.

M. Schadt, F. Leenhouts, “Electro-optical performance of a new, black-white and highly multiplexable liquid crystal display,” Appl. Phys. Lett. 50, 236–238 (1987).
[CrossRef]

Mauguin, C.

C. Mauguin, “Sur les cristaux liquides de Lehmann,” Bull. Soc. Fr. Mineral. 34, 71 (1911).

Raynes, E.

E. Raynes, R. Smith, “The design of liquid crystal materials for supertwist displays,” in Proceedings of the Seventh International Display Research Conference, Eurodisplay ’87 (The Institute of Physics and The Society for Information Display, London, 1987), p. 100.19.

Reynaerts, C.

A. De Vos, C. Reynaerts, F. Cuypers, “Optical characteristics of ferroelectric liquid crystal cells,” Ferroelectrics 113, 467–487 (1991).
[CrossRef]

A. De Vos, C. Reynaerts, “Optical transmission of ferroelectric liquid-crystal displays,” J. Appl. Phys. 65, 2616–2619 (1989).
[CrossRef]

Schadt, M.

M. Schadt, F. Leenhouts, “Electro-optical performance of a new, black-white and highly multiplexable liquid crystal display,” Appl. Phys. Lett. 50, 236–238 (1987).
[CrossRef]

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light, Production and Use (Harvard U. Press, Cambridge, Mass., 1962), Chaps. 1 and 2.

Smith, R.

E. Raynes, R. Smith, “The design of liquid crystal materials for supertwist displays,” in Proceedings of the Seventh International Display Research Conference, Eurodisplay ’87 (The Institute of Physics and The Society for Information Display, London, 1987), p. 100.19.

Tarry, H.

C. Gooch, H. Tarry, “The optical properties of twisted nematic liquid crystal structures with twist angles ≤ 90°,”J. Phys. D 8, 1575–1584 (1975).
[CrossRef]

C. Gooch, H. Tarry, “Optical characteristics of twisted nematic liquid-crystal films,” Electron. Lett. 10, 2–4 (1974).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1964), Chap. 14.

Appl. Opt.

Appl. Phys. Lett.

M. Schadt, F. Leenhouts, “Electro-optical performance of a new, black-white and highly multiplexable liquid crystal display,” Appl. Phys. Lett. 50, 236–238 (1987).
[CrossRef]

Bull. Soc. Fr. Mineral.

C. Mauguin, “Sur les cristaux liquides de Lehmann,” Bull. Soc. Fr. Mineral. 34, 71 (1911).

Electron. Lett.

C. Gooch, H. Tarry, “Optical characteristics of twisted nematic liquid-crystal films,” Electron. Lett. 10, 2–4 (1974).
[CrossRef]

Ferroelectrics

A. De Vos, C. Reynaerts, F. Cuypers, “Optical characteristics of ferroelectric liquid crystal cells,” Ferroelectrics 113, 467–487 (1991).
[CrossRef]

J. Appl. Phys.

A. De Vos, C. Reynaerts, “Optical transmission of ferroelectric liquid-crystal displays,” J. Appl. Phys. 65, 2616–2619 (1989).
[CrossRef]

J. Opt. Soc. Am.

J. Phys. D

C. Gooch, H. Tarry, “The optical properties of twisted nematic liquid crystal structures with twist angles ≤ 90°,”J. Phys. D 8, 1575–1584 (1975).
[CrossRef]

Other

E. Raynes, R. Smith, “The design of liquid crystal materials for supertwist displays,” in Proceedings of the Seventh International Display Research Conference, Eurodisplay ’87 (The Institute of Physics and The Society for Information Display, London, 1987), p. 100.19.

F. Cuypers, Ph.D. dissertation (Faculty of Engineering, Rijksuniversiteit Gent, Gent, Belgium, 1989).

T. Benzschawel, “Colorimetry of displays,” in SID Seminar Lecture Notes (Society for Information Display, New Orleans, La., 1987), p. 8.1.

G. J. Chamberlin, D. G. Chamberlin, Colour, Its Measurement, Computation and Application (Heyden, London, 1980), Chap. 4, pp. 54–55.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1964), Chap. 14.

R. Guenther, Modern Optics (Wiley, New York, 1990), Chap. 13, pp. 534–545.

W. A. Shurcliff, Polarized Light, Production and Use (Harvard U. Press, Cambridge, Mass., 1962), Chaps. 1 and 2.

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Figures (15)

Fig. 1
Fig. 1

All polarization states of a monochromatic wave with constant intensity can be represented on a sphere: the Poincaré sphere. Linear polarization is indicated on the equator. For each representation point, half of the azimuth gives the orientation (θ) of the polarization ellipse and half of the elevation corresponds to the angle of ellipticity (). H, V, horizontal and vertical linear polarization, respectively; R, L, right- and left-circular light, respectively.

Fig. 2
Fig. 2

Polarization ellipse, specified by two angles, θ and , which denote the orientation and the ellipticity, respectively.

Fig. 3
Fig. 3

Important angles for the transmission calculations. The x axis is the fast axis of the liquid crystal at the side of the polarizer. The polarizer’s position is defined by θ0, and the opening angle between analyzer and polarizer is γ. Φ denotes the total twist of the LC structure. The analyzer’s position with respect to the molecules on its side of the display is defined by ζ. Finally, the orientation of the transmitted polarization ellipse is indicated by the angles ρ and ψ.

Fig. 4
Fig. 4

Evolution of the polarization in the homogeneous case, indicated on the Poincaré sphere. The representation point X describes a circle around the fast axis, which is horizontally oriented.

Fig. 5
Fig. 5

In the case of a TN display, the representation point describes a cycloidal curve on the Poincaré sphere. When the thickness of the display is well chosen with respect to the wavelength of the traveling light, vertical linear polarization appears on the analyzer’s side of the display. C is a circle that is the intersection of the sphere and a cone; see text.

Fig. 6
Fig. 6

Evolution of the polarization in the TN case, for both a first minimum (curve 1) and a second minimum (curve 2) display. The maximal ellipticity is reached at the highest points on the cycloidal curves, corresponding to = π/6 and = π/12 for curve 1 and curve 2, respectively.

Fig. 7
Fig. 7

Evolution of the polarization ellipse in the case of a first minimum TN cell, depicted in an original way. Each horizontal section of the three-dimensional surface gives the polarization ellipse at the corresponding depth in the LC layer. Clearly, the ellipticity reaches its maximum in the middle of the display thickness.

Fig. 8
Fig. 8

Surface described by the polarization ellipse in the second minimum TN case, which is more complicated than for a first minimum display. At the half-way point in the display, an intermediate linear polarization appears.

Fig. 9
Fig. 9

Poincaré representation of the polarization in the STN display, describing an elegant curve that crosses the equator four times.

Fig. 10
Fig. 10

Surface described by the polarization ellipse in the STN display, demonstrating the high torsion and showing the three linear polarizations within the LC layer itself. Clearly, a small deviation of the display thickness or a small variation of the light wavelength causes an important ellipticity at the boundary. This phenomenon will lead to a high sensitivity of the display color to thickness variations.

Fig. 11
Fig. 11

For the analytical treatment of the polarization state, a stereographic projection of the Poincaré sphere is used. The south pole (left-circular polarization) is the projection center.

Fig. 12
Fig. 12

Color variations in the homogeneous display are illustrated. The symbols in the center of the diagram represent the bright mode (with crossed polarizers). The influence of thickness variations on the display color is more important for the dark mode, as illustrated by the outer points. In each case the parameter is the ratio d/d0 of the actual display thickness and the ideal one. C denotes the standard source.

Fig. 13
Fig. 13

Color variations in the first minimum TN display are very small in the operation mode, as can be concluded from the coincidence of the representation points in the middle of the diagram. For the dark mode the evolution of the outer points illustrates the so-called Gooch–Tarry relation. C denotes the standard source.

Fig. 14
Fig. 14

In the case of a STN display, even the bright mode is very sensitive to thickness variations. C denotes the standard source.

Fig. 15
Fig. 15

Color variations in the STN display, dark mode. Since the human eye’s sensitivity to color differences is high in the violet corner (lower left), thickness variations can easily be observed in this mode. C denotes the standard source.

Tables (1)

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Table 1 Transmission Formulas for Various Types of LCD

Equations (11)

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α = 2 π / p , Φ = α d , g = π Δ n λ , u = g / α = π Δ n Φ d λ , β = α 1 + u 2 .
T ( λ ) = 1 / 2 ( 1 + cos 2 cos 2 ψ ) = 1 / 2 [ 1 + cos 2 cos 2 ( ζ - ρ ) ] .
1 + u 2 = 2 m
1 + u 2 = p ,
χ ( χ 0 , z ) = ( β - i α tan β z ) χ 0 + i g tan β z ( i g tan β z ) χ 0 + ( β + i α tan β z ) exp ( i 2 α z ) ,
θ = 1 / 2 Arg ( χ ) ,
tan = 1 - χ 1 + χ .
χ ( - 1 / χ 0 ¯ , z ) = - 1 χ ¯ ( χ 0 , z ) .
β α = 1 + u 2 = 2 m 2 n + 1 ,
β / α = 1 + u 2 = p / q ,
cos 2 cos 2 ρ = ( g 2 β 2 + α 2 β 2 cos 2 β d ) cos 2 θ 0 + α β sin 2 β d sin 2 θ 0 , cos 2 sin 2 ρ = cos 2 β d sin 2 θ 0 - α β sin 2 β d cos 2 θ 0 .

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