Abstract

Detailed formulas were derived by using Yeh’s formulation of the extended 2×2 Jones matrix and applied to twisted nematic liquid-crystal displays at oblique angles of incidence. Numerical comparisons of this extended Jones matrix method with the exact 4×4 Berreman matrix and with the extended Jones matrix of Lien are presented. The various extended Jones matrix formulations differ in their approach to boundary-condition matching between the model birefringent layers. We show that Yeh’s version is a more accurate approximation to the full 4×4 matrix. This extended Jones matrix method is fast, direct, and simple. It is also physically more intuitive.

© 1999 Optical Society of America

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References

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  1. H. Wohler, “Numerical methods for parameter optimization of liquid crystal display,” in Proceedings of the Society for Information Display International Symposium, Santa Ana, California (Society for Information Display, San Jose, Calif., 1991), pp. 582–585.
  2. M. Schadt, W. Helfrich, “Voltage-dependent optical activity of twisted nematic liquid crystal,” Appl. Phys. Lett. 18, 127–128 (1971).
    [CrossRef]
  3. T. J. Scheffer, J. Nehring, “A new highly multiplexable liquid crystal display,” Appl. Phys. Lett. 45, 1021–1023 (1984).
    [CrossRef]
  4. G. H. Gooch, H. A. Tarry, “The optical properties of twisted nematic liquid crystal structures with twist angles <90°,” J. Phys. D 8, 1575–1584 (1975).
    [CrossRef]
  5. D. W. Berreman, “Optics in stratified and anisotropic media: 4×4 matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972);“Optics in smoothly varying anisotropic planar structure: application to liquid crystal twist cells,” J. Opt. Soc. Am. 63, 1374–1380 (1973).
    [CrossRef]
  6. H. Wöhler, G. Hass, M. Fritsch, D. A. Mlynski, “Faster 4×4 matrix method for inhomogeneous uniaxial media,” J. Opt. Soc. Am. A 5, 1554–1557 (1988).
    [CrossRef]
  7. K. H. Yang, “Elimination of the Fabry–Perot effect in the 4×4 matrix method for inhomogeneous uniaxial media,” J. Appl. Phys. 66, 1550–1554 (1990).
    [CrossRef]
  8. P. Yeh, “Extended Jones matrix method,” J. Opt. Soc. Am. 72, 507–513 (1982).
    [CrossRef]
  9. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), pp. 201–253.
  10. C. Gu, P. Yeh, “Extended Jones matrix method. II,” J. Opt. Soc. Am. A 10, 966–973 (1993).
    [CrossRef]
  11. A. Lien, “Extended Jones matrix representation for the twisted nematic liquid-crystal display at oblique incidence,” Appl. Phys. Lett. 57, 2767–2769 (1990).
    [CrossRef]
  12. A. Lien, C. J. Chen, “A new 2×2 matrix representation for twisted nematic liquid crystal displays at oblique incidence,” Jpn. J. Appl. Phys. 35, L1200–L1203 (1996).
    [CrossRef]
  13. Kanghua Lu, B. E. A. Saleh, “Reducing Berreman’s 4×4 formulation of liquid crystal display optics to 2×2 Jones vector equations.” Opt. Lett. 17, 1557–1559 (1993).
    [CrossRef]
  14. C. J. Chen, A. Lien, M. I. Nathan, “A general method to solve the deformation profile of chiral nematic LCDs with asymmetric pretilt,” in Society for Information Display International Symposium, Orlando, Florida (Society for Information Display, San Jose, Calif., 1995), pp. 548–551.
  15. H. J. Deuling, “Deformation pattern of twisted nematic liquid crystal layers in an electric field,” Mol. Cryst. Liq. Cryst. 27, 123–131 (1974).
    [CrossRef]
  16. B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), pp. 193–237.
  17. T. J. Scheffer, J. Nehring, “Accurate determination of liquid crystal tilt bias angles,” J. Appl. Phys. 48, 1783–1792 (1977).
    [CrossRef]
  18. K. Y. Han, T. Miyashita, T. Uchida, “Accurate determination and measurement error of the pretilt angle in the liquid crystal cell,” Jpn. J. Appl. Phys. Part 2 32, L277–L279 (1993).
    [CrossRef]
  19. K. Y. Han, T. Miyashita, T. Uchida, “Accurate measurement of pretilt angle in the liquid crystal cell by an improved crystal rotation method,” Mol. Cryst. Liq. Cryst. 241, 147–157 (1994).
    [CrossRef]
  20. K. Shrota, M. Yaginuma, K. Ishikawa, H. Takezoe, A. Fukuda, “Modified crystal rotation method for measuring high pretilt angle in the liquid crystal cells,” Jpn. J. Appl. Phys. Part 1 34, 4905–4906 (1995).
    [CrossRef]
  21. L. M. Blinov, V. G. Chigrinov, Electrooptic Effects in Liquid Crystal Materials (Springer-Verlag, New York, 1994).
  22. H. S. Kwok, “Parameter space representation of liquid crystal display operating mode,” J. Appl. Phys. 80, 3687–3693 (1996).
    [CrossRef]
  23. S. T. Tang, F. H. Yu, J. Chen, M. Wong, H. C. Huang, H. S. Kwok, “Reflective nematic liquid crystal displays. I. Retardation compensation,” J. Appl. Phys. 81, 5924–5929 (1997).
    [CrossRef]
  24. F. H. Yu, J. Chen, S. T. Tang, H. S. Kwok, “Reflective nematic liquid crystal displays. II. Elimination of retardation film and rear polarizer,” J. Appl. Phys. 82, 5287–5295 (1997).
    [CrossRef]
  25. J. Chen, F. H. Yu, H. C. Huang, H. S. Kwok, “Reflective supertwisted liquid crystal displays,” Jpn. J. Appl. Phys., Part 1 37, 217–223 (1998).
    [CrossRef]
  26. J. Lekner, “Optical properties of a uniaxial layer,” Pure Appl. Opt. 3, 821–837 (1994).
    [CrossRef]

1998

J. Chen, F. H. Yu, H. C. Huang, H. S. Kwok, “Reflective supertwisted liquid crystal displays,” Jpn. J. Appl. Phys., Part 1 37, 217–223 (1998).
[CrossRef]

1997

S. T. Tang, F. H. Yu, J. Chen, M. Wong, H. C. Huang, H. S. Kwok, “Reflective nematic liquid crystal displays. I. Retardation compensation,” J. Appl. Phys. 81, 5924–5929 (1997).
[CrossRef]

F. H. Yu, J. Chen, S. T. Tang, H. S. Kwok, “Reflective nematic liquid crystal displays. II. Elimination of retardation film and rear polarizer,” J. Appl. Phys. 82, 5287–5295 (1997).
[CrossRef]

1996

A. Lien, C. J. Chen, “A new 2×2 matrix representation for twisted nematic liquid crystal displays at oblique incidence,” Jpn. J. Appl. Phys. 35, L1200–L1203 (1996).
[CrossRef]

H. S. Kwok, “Parameter space representation of liquid crystal display operating mode,” J. Appl. Phys. 80, 3687–3693 (1996).
[CrossRef]

1995

K. Shrota, M. Yaginuma, K. Ishikawa, H. Takezoe, A. Fukuda, “Modified crystal rotation method for measuring high pretilt angle in the liquid crystal cells,” Jpn. J. Appl. Phys. Part 1 34, 4905–4906 (1995).
[CrossRef]

1994

K. Y. Han, T. Miyashita, T. Uchida, “Accurate measurement of pretilt angle in the liquid crystal cell by an improved crystal rotation method,” Mol. Cryst. Liq. Cryst. 241, 147–157 (1994).
[CrossRef]

J. Lekner, “Optical properties of a uniaxial layer,” Pure Appl. Opt. 3, 821–837 (1994).
[CrossRef]

1993

1990

A. Lien, “Extended Jones matrix representation for the twisted nematic liquid-crystal display at oblique incidence,” Appl. Phys. Lett. 57, 2767–2769 (1990).
[CrossRef]

K. H. Yang, “Elimination of the Fabry–Perot effect in the 4×4 matrix method for inhomogeneous uniaxial media,” J. Appl. Phys. 66, 1550–1554 (1990).
[CrossRef]

1988

1984

T. J. Scheffer, J. Nehring, “A new highly multiplexable liquid crystal display,” Appl. Phys. Lett. 45, 1021–1023 (1984).
[CrossRef]

1982

1977

T. J. Scheffer, J. Nehring, “Accurate determination of liquid crystal tilt bias angles,” J. Appl. Phys. 48, 1783–1792 (1977).
[CrossRef]

1975

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

1974

H. J. Deuling, “Deformation pattern of twisted nematic liquid crystal layers in an electric field,” Mol. Cryst. Liq. Cryst. 27, 123–131 (1974).
[CrossRef]

1972

1971

M. Schadt, W. Helfrich, “Voltage-dependent optical activity of twisted nematic liquid crystal,” Appl. Phys. Lett. 18, 127–128 (1971).
[CrossRef]

Berreman, D. W.

Blinov, L. M.

L. M. Blinov, V. G. Chigrinov, Electrooptic Effects in Liquid Crystal Materials (Springer-Verlag, New York, 1994).

Chen, C. J.

A. Lien, C. J. Chen, “A new 2×2 matrix representation for twisted nematic liquid crystal displays at oblique incidence,” Jpn. J. Appl. Phys. 35, L1200–L1203 (1996).
[CrossRef]

C. J. Chen, A. Lien, M. I. Nathan, “A general method to solve the deformation profile of chiral nematic LCDs with asymmetric pretilt,” in Society for Information Display International Symposium, Orlando, Florida (Society for Information Display, San Jose, Calif., 1995), pp. 548–551.

Chen, J.

J. Chen, F. H. Yu, H. C. Huang, H. S. Kwok, “Reflective supertwisted liquid crystal displays,” Jpn. J. Appl. Phys., Part 1 37, 217–223 (1998).
[CrossRef]

F. H. Yu, J. Chen, S. T. Tang, H. S. Kwok, “Reflective nematic liquid crystal displays. II. Elimination of retardation film and rear polarizer,” J. Appl. Phys. 82, 5287–5295 (1997).
[CrossRef]

S. T. Tang, F. H. Yu, J. Chen, M. Wong, H. C. Huang, H. S. Kwok, “Reflective nematic liquid crystal displays. I. Retardation compensation,” J. Appl. Phys. 81, 5924–5929 (1997).
[CrossRef]

Chigrinov, V. G.

L. M. Blinov, V. G. Chigrinov, Electrooptic Effects in Liquid Crystal Materials (Springer-Verlag, New York, 1994).

Deuling, H. J.

H. J. Deuling, “Deformation pattern of twisted nematic liquid crystal layers in an electric field,” Mol. Cryst. Liq. Cryst. 27, 123–131 (1974).
[CrossRef]

Fritsch, M.

Fukuda, A.

K. Shrota, M. Yaginuma, K. Ishikawa, H. Takezoe, A. Fukuda, “Modified crystal rotation method for measuring high pretilt angle in the liquid crystal cells,” Jpn. J. Appl. Phys. Part 1 34, 4905–4906 (1995).
[CrossRef]

Gooch, G. H.

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

Gu, C.

Han, K. Y.

K. Y. Han, T. Miyashita, T. Uchida, “Accurate measurement of pretilt angle in the liquid crystal cell by an improved crystal rotation method,” Mol. Cryst. Liq. Cryst. 241, 147–157 (1994).
[CrossRef]

K. Y. Han, T. Miyashita, T. Uchida, “Accurate determination and measurement error of the pretilt angle in the liquid crystal cell,” Jpn. J. Appl. Phys. Part 2 32, L277–L279 (1993).
[CrossRef]

Hass, G.

Helfrich, W.

M. Schadt, W. Helfrich, “Voltage-dependent optical activity of twisted nematic liquid crystal,” Appl. Phys. Lett. 18, 127–128 (1971).
[CrossRef]

Huang, H. C.

J. Chen, F. H. Yu, H. C. Huang, H. S. Kwok, “Reflective supertwisted liquid crystal displays,” Jpn. J. Appl. Phys., Part 1 37, 217–223 (1998).
[CrossRef]

S. T. Tang, F. H. Yu, J. Chen, M. Wong, H. C. Huang, H. S. Kwok, “Reflective nematic liquid crystal displays. I. Retardation compensation,” J. Appl. Phys. 81, 5924–5929 (1997).
[CrossRef]

Ishikawa, K.

K. Shrota, M. Yaginuma, K. Ishikawa, H. Takezoe, A. Fukuda, “Modified crystal rotation method for measuring high pretilt angle in the liquid crystal cells,” Jpn. J. Appl. Phys. Part 1 34, 4905–4906 (1995).
[CrossRef]

Kwok, H. S.

J. Chen, F. H. Yu, H. C. Huang, H. S. Kwok, “Reflective supertwisted liquid crystal displays,” Jpn. J. Appl. Phys., Part 1 37, 217–223 (1998).
[CrossRef]

S. T. Tang, F. H. Yu, J. Chen, M. Wong, H. C. Huang, H. S. Kwok, “Reflective nematic liquid crystal displays. I. Retardation compensation,” J. Appl. Phys. 81, 5924–5929 (1997).
[CrossRef]

F. H. Yu, J. Chen, S. T. Tang, H. S. Kwok, “Reflective nematic liquid crystal displays. II. Elimination of retardation film and rear polarizer,” J. Appl. Phys. 82, 5287–5295 (1997).
[CrossRef]

H. S. Kwok, “Parameter space representation of liquid crystal display operating mode,” J. Appl. Phys. 80, 3687–3693 (1996).
[CrossRef]

Lekner, J.

J. Lekner, “Optical properties of a uniaxial layer,” Pure Appl. Opt. 3, 821–837 (1994).
[CrossRef]

Lien, A.

A. Lien, C. J. Chen, “A new 2×2 matrix representation for twisted nematic liquid crystal displays at oblique incidence,” Jpn. J. Appl. Phys. 35, L1200–L1203 (1996).
[CrossRef]

A. Lien, “Extended Jones matrix representation for the twisted nematic liquid-crystal display at oblique incidence,” Appl. Phys. Lett. 57, 2767–2769 (1990).
[CrossRef]

C. J. Chen, A. Lien, M. I. Nathan, “A general method to solve the deformation profile of chiral nematic LCDs with asymmetric pretilt,” in Society for Information Display International Symposium, Orlando, Florida (Society for Information Display, San Jose, Calif., 1995), pp. 548–551.

Lu, Kanghua

Miyashita, T.

K. Y. Han, T. Miyashita, T. Uchida, “Accurate measurement of pretilt angle in the liquid crystal cell by an improved crystal rotation method,” Mol. Cryst. Liq. Cryst. 241, 147–157 (1994).
[CrossRef]

K. Y. Han, T. Miyashita, T. Uchida, “Accurate determination and measurement error of the pretilt angle in the liquid crystal cell,” Jpn. J. Appl. Phys. Part 2 32, L277–L279 (1993).
[CrossRef]

Mlynski, D. A.

Nathan, M. I.

C. J. Chen, A. Lien, M. I. Nathan, “A general method to solve the deformation profile of chiral nematic LCDs with asymmetric pretilt,” in Society for Information Display International Symposium, Orlando, Florida (Society for Information Display, San Jose, Calif., 1995), pp. 548–551.

Nehring, J.

T. J. Scheffer, J. Nehring, “A new highly multiplexable liquid crystal display,” Appl. Phys. Lett. 45, 1021–1023 (1984).
[CrossRef]

T. J. Scheffer, J. Nehring, “Accurate determination of liquid crystal tilt bias angles,” J. Appl. Phys. 48, 1783–1792 (1977).
[CrossRef]

Saleh, B. E. A.

Schadt, M.

M. Schadt, W. Helfrich, “Voltage-dependent optical activity of twisted nematic liquid crystal,” Appl. Phys. Lett. 18, 127–128 (1971).
[CrossRef]

Scheffer, T. J.

T. J. Scheffer, J. Nehring, “A new highly multiplexable liquid crystal display,” Appl. Phys. Lett. 45, 1021–1023 (1984).
[CrossRef]

T. J. Scheffer, J. Nehring, “Accurate determination of liquid crystal tilt bias angles,” J. Appl. Phys. 48, 1783–1792 (1977).
[CrossRef]

Shrota, K.

K. Shrota, M. Yaginuma, K. Ishikawa, H. Takezoe, A. Fukuda, “Modified crystal rotation method for measuring high pretilt angle in the liquid crystal cells,” Jpn. J. Appl. Phys. Part 1 34, 4905–4906 (1995).
[CrossRef]

Takezoe, H.

K. Shrota, M. Yaginuma, K. Ishikawa, H. Takezoe, A. Fukuda, “Modified crystal rotation method for measuring high pretilt angle in the liquid crystal cells,” Jpn. J. Appl. Phys. Part 1 34, 4905–4906 (1995).
[CrossRef]

Tang, S. T.

S. T. Tang, F. H. Yu, J. Chen, M. Wong, H. C. Huang, H. S. Kwok, “Reflective nematic liquid crystal displays. I. Retardation compensation,” J. Appl. Phys. 81, 5924–5929 (1997).
[CrossRef]

F. H. Yu, J. Chen, S. T. Tang, H. S. Kwok, “Reflective nematic liquid crystal displays. II. Elimination of retardation film and rear polarizer,” J. Appl. Phys. 82, 5287–5295 (1997).
[CrossRef]

Tarry, H. A.

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

Teich, M. C.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), pp. 193–237.

Uchida, T.

K. Y. Han, T. Miyashita, T. Uchida, “Accurate measurement of pretilt angle in the liquid crystal cell by an improved crystal rotation method,” Mol. Cryst. Liq. Cryst. 241, 147–157 (1994).
[CrossRef]

K. Y. Han, T. Miyashita, T. Uchida, “Accurate determination and measurement error of the pretilt angle in the liquid crystal cell,” Jpn. J. Appl. Phys. Part 2 32, L277–L279 (1993).
[CrossRef]

Wohler, H.

H. Wohler, “Numerical methods for parameter optimization of liquid crystal display,” in Proceedings of the Society for Information Display International Symposium, Santa Ana, California (Society for Information Display, San Jose, Calif., 1991), pp. 582–585.

Wöhler, H.

Wong, M.

S. T. Tang, F. H. Yu, J. Chen, M. Wong, H. C. Huang, H. S. Kwok, “Reflective nematic liquid crystal displays. I. Retardation compensation,” J. Appl. Phys. 81, 5924–5929 (1997).
[CrossRef]

Yaginuma, M.

K. Shrota, M. Yaginuma, K. Ishikawa, H. Takezoe, A. Fukuda, “Modified crystal rotation method for measuring high pretilt angle in the liquid crystal cells,” Jpn. J. Appl. Phys. Part 1 34, 4905–4906 (1995).
[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. 66, 1550–1554 (1990).
[CrossRef]

Yeh, P.

Yu, F. H.

J. Chen, F. H. Yu, H. C. Huang, H. S. Kwok, “Reflective supertwisted liquid crystal displays,” Jpn. J. Appl. Phys., Part 1 37, 217–223 (1998).
[CrossRef]

F. H. Yu, J. Chen, S. T. Tang, H. S. Kwok, “Reflective nematic liquid crystal displays. II. Elimination of retardation film and rear polarizer,” J. Appl. Phys. 82, 5287–5295 (1997).
[CrossRef]

S. T. Tang, F. H. Yu, J. Chen, M. Wong, H. C. Huang, H. S. Kwok, “Reflective nematic liquid crystal displays. I. Retardation compensation,” J. Appl. Phys. 81, 5924–5929 (1997).
[CrossRef]

Appl. Phys. Lett.

M. Schadt, W. Helfrich, “Voltage-dependent optical activity of twisted nematic liquid crystal,” Appl. Phys. Lett. 18, 127–128 (1971).
[CrossRef]

T. J. Scheffer, J. Nehring, “A new highly multiplexable liquid crystal display,” Appl. Phys. Lett. 45, 1021–1023 (1984).
[CrossRef]

A. Lien, “Extended Jones matrix representation for the twisted nematic liquid-crystal display at oblique incidence,” Appl. Phys. Lett. 57, 2767–2769 (1990).
[CrossRef]

J. Appl. Phys.

T. J. Scheffer, J. Nehring, “Accurate determination of liquid crystal tilt bias angles,” J. Appl. Phys. 48, 1783–1792 (1977).
[CrossRef]

K. H. Yang, “Elimination of the Fabry–Perot effect in the 4×4 matrix method for inhomogeneous uniaxial media,” J. Appl. Phys. 66, 1550–1554 (1990).
[CrossRef]

H. S. Kwok, “Parameter space representation of liquid crystal display operating mode,” J. Appl. Phys. 80, 3687–3693 (1996).
[CrossRef]

S. T. Tang, F. H. Yu, J. Chen, M. Wong, H. C. Huang, H. S. Kwok, “Reflective nematic liquid crystal displays. I. Retardation compensation,” J. Appl. Phys. 81, 5924–5929 (1997).
[CrossRef]

F. H. Yu, J. Chen, S. T. Tang, H. S. Kwok, “Reflective nematic liquid crystal displays. II. Elimination of retardation film and rear polarizer,” J. Appl. Phys. 82, 5287–5295 (1997).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. D

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

Jpn. J. Appl. Phys.

A. Lien, C. J. Chen, “A new 2×2 matrix representation for twisted nematic liquid crystal displays at oblique incidence,” Jpn. J. Appl. Phys. 35, L1200–L1203 (1996).
[CrossRef]

Jpn. J. Appl. Phys. Part 1

K. Shrota, M. Yaginuma, K. Ishikawa, H. Takezoe, A. Fukuda, “Modified crystal rotation method for measuring high pretilt angle in the liquid crystal cells,” Jpn. J. Appl. Phys. Part 1 34, 4905–4906 (1995).
[CrossRef]

Jpn. J. Appl. Phys. Part 2

K. Y. Han, T. Miyashita, T. Uchida, “Accurate determination and measurement error of the pretilt angle in the liquid crystal cell,” Jpn. J. Appl. Phys. Part 2 32, L277–L279 (1993).
[CrossRef]

Jpn. J. Appl. Phys., Part 1

J. Chen, F. H. Yu, H. C. Huang, H. S. Kwok, “Reflective supertwisted liquid crystal displays,” Jpn. J. Appl. Phys., Part 1 37, 217–223 (1998).
[CrossRef]

Mol. Cryst. Liq. Cryst.

K. Y. Han, T. Miyashita, T. Uchida, “Accurate measurement of pretilt angle in the liquid crystal cell by an improved crystal rotation method,” Mol. Cryst. Liq. Cryst. 241, 147–157 (1994).
[CrossRef]

H. J. Deuling, “Deformation pattern of twisted nematic liquid crystal layers in an electric field,” Mol. Cryst. Liq. Cryst. 27, 123–131 (1974).
[CrossRef]

Opt. Lett.

Pure Appl. Opt.

J. Lekner, “Optical properties of a uniaxial layer,” Pure Appl. Opt. 3, 821–837 (1994).
[CrossRef]

Other

L. M. Blinov, V. G. Chigrinov, Electrooptic Effects in Liquid Crystal Materials (Springer-Verlag, New York, 1994).

C. J. Chen, A. Lien, M. I. Nathan, “A general method to solve the deformation profile of chiral nematic LCDs with asymmetric pretilt,” in Society for Information Display International Symposium, Orlando, Florida (Society for Information Display, San Jose, Calif., 1995), pp. 548–551.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), pp. 193–237.

H. Wohler, “Numerical methods for parameter optimization of liquid crystal display,” in Proceedings of the Society for Information Display International Symposium, Santa Ana, California (Society for Information Display, San Jose, Calif., 1991), pp. 582–585.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), pp. 201–253.

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Figures (10)

Fig. 1
Fig. 1

Geometry of the input light.

Fig. 2
Fig. 2

Definition of various angles in the general twist cell.

Fig. 3
Fig. 3

Comparison of optical transmission as a function of incidence angle θair calculated by three methods for a 90° TN cell filled with E-70. The incident light polar angle is ϕ=0°. In this and the following figures, 2 by 2 refers to the 2×2 Jones matrix in Ref. 11, m 2 by 2 refers to the 2×2 Jones matrix in Ref. 12, New 2 by 2 refers to the Jones matrix derived in this paper, and 4 by 4 refers to the full 4×4 Berreman matrix.

Fig. 4
Fig. 4

Absolute errors of the 2×2 methods as compared with that of the 4×4 method for the case of Fig. 3.

Fig. 5
Fig. 5

Comparison of optical transmission as a function of incidence angle θair calculated by three methods for a 90° TN cell filled with E-70. The incident light polar angle is ϕ=-45°.

Fig. 6
Fig. 6

Absolute errors of the 2×2 methods as compared with that of the 4×4 method for the case of Fig. 5.

Fig. 7
Fig. 7

Comparison of optical transmission as a function of incidence angle θair calculated by three methods for a 90° TN cell filled with E-70. The incident light polar angle is ϕ=45°.

Fig. 8
Fig. 8

Absolute errors of the 2×2 methods as compared with that of the 4×4 method for the case of Fig. 7.

Fig. 9
Fig. 9

Three sets of curves comparing optical transmission as a function of applied voltage calculated by three methods for viewing angles of (ϕ, θi)=(0°, 0°), (0°, 22.5°), (0°, 45°), (-180°, 22.5°), and (-180°, 45°).

Fig. 10
Fig. 10

Comparison of absolute errors for the various cases depicted in Fig. 9.

Tables (1)

Tables Icon

Table 1 Parameters Used in the Calculations

Equations (52)

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

EeEo=R(p, e)R(s, e)R(p, o)R(s, o) EpEs.
EpEx=R(e, p)R(o, p)R(e, s)R(o, s)× exp(-ikσezh)00exp(-ikσozh)× R(p, e)R(s, e)R(p, o)R(s, o) EpEs=JEpEs,
J=R2GR1.
R(p, e)=R(e, p)=p  e,
R(s, e)=R(e, s)=s  e,
R(p, o)=R(o, p)=p  o,
R(s, o)=R(o, s)=s  o;
R2=R1.
J=JNJN-1  Jn  J3J2J1,
Jn=[R2GR1]n,
ENpENs=JE1pE1s.
Ki=[sin θo0cos θo].
s=[0-10],
p=Ki×s|Ki×s|=[cos θo0-sin θo].
Ko=[sin θo0cos θo],
Ke=[no sin θo0ne(θe)cos θe]
cos θe=σez/ne(θe),
Dn=[cos ϕn cos θnsin ϕn cos θnsin θn],
o=Dn×Ko|Dn×Ko|=sin ϕn cos θn cos θoAosin θn sin θo-cos ϕn cos θn cos θoAo-sin ϕn cos θn sin θoAo,
e=Ke×o|Ke×o|=ne(θe)(cos θe)(cos ϕn cos θn cos θo-sin θn sin θo)Ae(sin ϕn cos θn)[no sin2 θo+ne(θe)cos θe cos θo]Ae-no(sin θo)(cos ϕn cos θn cos θo-sin θn sin θo)Ae.
Ao=[sin2 ϕn cos2 θn+(cos ϕn cos θn cos θo-sin θn sin θo)2]1/2,
Ae={[no2 sin2 θo+ne2(θe)cos2 θe]Δ2+(sin2 ϕn cos2 θn)×[no sin2 θo+ne(θe)cos θe cos θo]2}1/2,
Δ=cos ϕn cos θn cos θo-sin θn sin θo.
σez=(ne2-no2)sin θn cos θn cos ϕn sin θono2+(ne2-no2)sin2 θn+noneno2+(ne2-no2)sin2 θn×no2+(ne2-no2)sin2 θn-1-ne2-no2ne2 cos2 θn sin2 ϕnsin2 θo1/2,
σoz=(no2-sin2 θo)1/2.
R(p, e)=[no sin2 θo+ne(θe)cos θe cos θo](cos ϕn cos θn cos θo-sin θn sin θo)Ae,
R(s, e)=-(sin ϕn cos θn)[no sin2 θo+ne(θe)cos θe cos θo]Ae,
R(p, o)=sin ϕn cos θnAo,
R(s, o)=cos ϕn cos θn cos θo-sin θn sin θoAo.
R1=cos ϕn-sin ϕnsin ϕncos ϕn,
σez=none[no2+(ne2-no2)sin2 θn]1/2,
σoz=no
Ao=[sin2 ϕn cos2 θn+(cos ϕn cos θn cos θo-sin θn sin θo)2]1/2,
Ae=noAo,
R1=cos ϕn cos θn cos θo-sin θn sin θoAo-sin ϕn cos θnAosin ϕn cos θnAocos ϕn cos θn cos θo-sin θn sin θoAo.
Ao=cos θn.
R1=1001,
σez=none[no2+(ne2-no2)sin2 θn]1/2,
σoz=no.
J=exp-iki=1Nσezh00exp(-ikσozd),
σez=(ne2-no2)sin θn cos θnno2+(ne2-no2)sin2 θn sin θo+noneno2+(ne2-no2)sin2 θn×[no2+(ne2-no2)sin2 θn-sin2 θo]1/2,
σoz=(no2-sin2 θo)1/2.
J=exp(-ikσezd)00exp(-ikσozd),
I=12 sin2πλ (σez-σoz)d.
E1pE1s=Tpgl00Tsgl Tppg00Tspg cos2(ψ1+ϕ)sin(ψ1+ϕ)cos(ψ1+ϕ)sin(ψ1+ϕ)cos(ψ1+ϕ)sin2(ψ1+ϕ) Tpap00Tsap Eair pEair s,
Tpap=[sin(2θair)sin(2θp)]1/2sin(θair+θp)cos(θair-θp),
Tsap=[sin(2θair)sin(2θp)]1/2sin(θair+θp)
θp=sin-1[nair(sin θair)/np],
Eair pEair s=Tppa00Tspa cos2(ψ2+ϕ)sin(ψ2+ϕ)cos(ψ2+ϕ)sin(ψ2+ϕ)cos(ψ2+ϕ)sin2(ψ2+ϕ)× Tpgp00Tsgp Tplg00Tslg EpENs,
Tppa=[sin(2θp)sin(2θair)]1/2sin(θp+θair)cos(θp-θair),
Tspa=[sin(2θp)sin(2θair)]1/2sin(θp+θair).
I=Eair p2+Eair s2Eair p2+Eair s2.

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