Abstract

The 2×2 matrix methods are extended to calculate the optical behaviors of reflective liquid-crystal displays with asymmetric incident and exit angles. Both the unfolding method and the backward-eigenwave method are employed to derive the 2×2 matrix representations. The simulation results for symmetric incident and exit angles from these two methods are identical and agree well with those obtained from the 4×4 matrix method when the air–panel surface reflections are neglected. Further, the derived 2×2 matrix methods are applied to the asymmetric cases with different incident and exit angles. The simulated results on the normally black vertical alignment and normally white mixed-mode twisted nematic reflective displays show reasonably good agreement with the reported experimental data. In addition, a rubbing effect related to contrast values is observed and analyzed in asymmetric reflective cases. We also find that this effect has a significant influence on the contrast ratios once the difference between the incident and exit angles becomes large.

© 2005 Optical Society of America

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References

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  1. S. T. Wu, D. K. Yang, Reflective Liquid Crystal Displays (Wiley, New York, 2001).
  2. M. F. Schiekel, K. Fahrenschon, “Deformation of nematic liquid crystals with vertical orientation in electrical fields,” Appl. Phys. Lett. 19, 391–393 (1971).
    [CrossRef]
  3. S. T. Wu, C. S. Wu, “Mixed-mode twisted nematic liquid crystal cells for reflective displays,” Appl. Phys. Lett. 68, 1455–1457 (1996).
    [CrossRef]
  4. S. Teitler, B. W. Henvis, “Refraction in stratified, anisotropic media,” J. Opt. Soc. Am. 60, 830–834 (1970).
    [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. I. Abdulhalim, L. Benguigui, R. Weil, “Selective reflection by helicoidal liquid crystals. Results of an exact calculation using the 4∗4 characteristic matrix method,” J. Phys. Colloq. 46, 815–825 (1985).
    [CrossRef]
  7. I. Abdulhalim, R. Weil, L. Benguigui, “Dispersion and attenuation of the eigenwaves for light propagation in helicoidal liquid crystals,” Liq. Cryst. 1, 155–167 (1986).
    [CrossRef]
  8. H. Wohler, G. Hass, M. Fritsch, D. A. Mlynski, “Faster 4×4 matrix method for uniaxial inhomogeneous media,” J. Opt. Soc. Am. A 5, 1554–1557 (1988).
    [CrossRef]
  9. K. Eidner, G. Mayer, M. Schmidt, H. Schmiedel, “Optics in stratified media-the use of optical eigenmodes of uniaxial crystal in the 4×4 matrix formalism,” Mol. Cryst. Liq. Cryst. 172, 191–200 (1989).
  10. C. J. Chen, A. Lien, M. I. Nathan, “4×4 and 2×2 matrix formulations for the optics in stratified and biaxial media,” J. Opt. Soc. Am. A 14, 3125–3133 (1997).
    [CrossRef]
  11. I. Abdulhalim, “Analytic propagation matrix method for linear optics of arbitrary biaxial layered media,” J. Opt. A, Pure Appl. Opt. 1, 646–653 (1999).
    [CrossRef]
  12. S. Stallinga, “Berreman 4×4 matrix method for reflective liquid crystal displays,” J. Appl. Phys. 85, 3023–3031 (1999).
    [CrossRef]
  13. C. L. Kuo, C. K. Wei, S. T. Wu, C. S. Wu, “Reflective direct-view display using a mixed-mode twisted nematic cell,” Jpn. J. Appl. Phys., Part 1 36, 1077–1080 (1997).
    [CrossRef]
  14. P. G. De Gennes, J. Prost, The Physics of Liquid Crystals, 2nd ed. (Oxford, London, 1995).
  15. P. Yeh, “Extended Jones matrix method,” J. Opt. Soc. Am. 72, 507–513 (1982).
    [CrossRef]
  16. C. Gu, P. Yeh, “Extended Jones matrix method II,” J. Opt. Soc. Am. A 10, 966–973 (1993).
    [CrossRef]
  17. A. Lien, “Extended Jones matrix representation for the twisted nematic liquid-crystal display at oblique incidence,” Appl. Phys. Lett. 57, 2767–2769 (1990).
    [CrossRef]
  18. A. Lien, C. J. Chen, “A new 2×2 matrix representation for twisted nematic liquid crystal displays at oblique incidence,” Jpn. J. Appl. Phys., Part 1 35, L1200–1203 (1996).
    [CrossRef]
  19. A. Lien, “A detailed derivation of extended Jones matrix representation for twisted nematic liquid crystal displays,” Liq. Cryst. 22, 171–175 (1997).
    [CrossRef]
  20. F. H. Yu, H. S. Kwok, “Comparison of extended Jones matrices for twisted nematic liquid-crystal displays at oblique angles of incidence,” J. Opt. Soc. Am. A 16, 2772–2780 (1999).
    [CrossRef]
  21. I. Abdulhalim, “Exact 2×2 matrix method for the transmission and reflection at the interface between two arbitrarily oriented biaxial crystals,” J. Opt. A, Pure Appl. Opt. 1, 655–661 (1999).
    [CrossRef]
  22. I. Abdulhalim, “2×2 Matrix summation method for multiple reflections and transmissions in a biaxial slab between two anisotropic media,” Opt. Commun. 163, 9–14 (1999).
    [CrossRef]
  23. M. Mansuripur, “Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2×2 matrices,” J. Appl. Phys. 67, 6466–6475 (1990).
    [CrossRef]
  24. E. Cojocaru, “Generalized Abel’s relations for an anisotropic thin film of an arbitrary dielectric tensor,” Appl. Opt. 36, 2825–2829 (1997).
    [CrossRef] [PubMed]
  25. S. Pancharatnam, “Achromatic combination of birefringent plates. II. An achromatic quarter-wave plate,” Proc. Indian Acad. Sci., Sect. A 41, 137–144 (1955).
  26. D. L. Ting, W. C. Chang, C. Y. Liu, J. W. Shiu, C. J. Wen, C. H. Chao, L. S. Chuang, C. C. Chang, “A high brightness and high contrast reflective LCD with micro slant reflector (MSR),” SID Int. Symp. Digest Tech. Papers 30, 954–957 (1999).
    [CrossRef]
  27. N. Sugiura, K. Tashiro, K. Ohmuro, Y. Koike, K. Okamoto, “A novel vertically aligned reflective-color TFT LCD with high contrast ratio,” SID Int. Symp. Digest Tech. Papers 33, 1386–1389 (2002).
    [CrossRef]

2002 (1)

N. Sugiura, K. Tashiro, K. Ohmuro, Y. Koike, K. Okamoto, “A novel vertically aligned reflective-color TFT LCD with high contrast ratio,” SID Int. Symp. Digest Tech. Papers 33, 1386–1389 (2002).
[CrossRef]

1999 (6)

F. H. Yu, H. S. Kwok, “Comparison of extended Jones matrices for twisted nematic liquid-crystal displays at oblique angles of incidence,” J. Opt. Soc. Am. A 16, 2772–2780 (1999).
[CrossRef]

I. Abdulhalim, “Analytic propagation matrix method for linear optics of arbitrary biaxial layered media,” J. Opt. A, Pure Appl. Opt. 1, 646–653 (1999).
[CrossRef]

S. Stallinga, “Berreman 4×4 matrix method for reflective liquid crystal displays,” J. Appl. Phys. 85, 3023–3031 (1999).
[CrossRef]

I. Abdulhalim, “Exact 2×2 matrix method for the transmission and reflection at the interface between two arbitrarily oriented biaxial crystals,” J. Opt. A, Pure Appl. Opt. 1, 655–661 (1999).
[CrossRef]

I. Abdulhalim, “2×2 Matrix summation method for multiple reflections and transmissions in a biaxial slab between two anisotropic media,” Opt. Commun. 163, 9–14 (1999).
[CrossRef]

D. L. Ting, W. C. Chang, C. Y. Liu, J. W. Shiu, C. J. Wen, C. H. Chao, L. S. Chuang, C. C. Chang, “A high brightness and high contrast reflective LCD with micro slant reflector (MSR),” SID Int. Symp. Digest Tech. Papers 30, 954–957 (1999).
[CrossRef]

1997 (4)

A. Lien, “A detailed derivation of extended Jones matrix representation for twisted nematic liquid crystal displays,” Liq. Cryst. 22, 171–175 (1997).
[CrossRef]

C. L. Kuo, C. K. Wei, S. T. Wu, C. S. Wu, “Reflective direct-view display using a mixed-mode twisted nematic cell,” Jpn. J. Appl. Phys., Part 1 36, 1077–1080 (1997).
[CrossRef]

C. J. Chen, A. Lien, M. I. Nathan, “4×4 and 2×2 matrix formulations for the optics in stratified and biaxial media,” J. Opt. Soc. Am. A 14, 3125–3133 (1997).
[CrossRef]

E. Cojocaru, “Generalized Abel’s relations for an anisotropic thin film of an arbitrary dielectric tensor,” Appl. Opt. 36, 2825–2829 (1997).
[CrossRef] [PubMed]

1996 (2)

S. T. Wu, C. S. Wu, “Mixed-mode twisted nematic liquid crystal cells for reflective displays,” Appl. Phys. Lett. 68, 1455–1457 (1996).
[CrossRef]

A. Lien, C. J. Chen, “A new 2×2 matrix representation for twisted nematic liquid crystal displays at oblique incidence,” Jpn. J. Appl. Phys., Part 1 35, L1200–1203 (1996).
[CrossRef]

1993 (1)

1990 (2)

M. Mansuripur, “Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2×2 matrices,” J. Appl. Phys. 67, 6466–6475 (1990).
[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]

1989 (1)

K. Eidner, G. Mayer, M. Schmidt, H. Schmiedel, “Optics in stratified media-the use of optical eigenmodes of uniaxial crystal in the 4×4 matrix formalism,” Mol. Cryst. Liq. Cryst. 172, 191–200 (1989).

1988 (1)

1986 (1)

I. Abdulhalim, R. Weil, L. Benguigui, “Dispersion and attenuation of the eigenwaves for light propagation in helicoidal liquid crystals,” Liq. Cryst. 1, 155–167 (1986).
[CrossRef]

1985 (1)

I. Abdulhalim, L. Benguigui, R. Weil, “Selective reflection by helicoidal liquid crystals. Results of an exact calculation using the 4∗4 characteristic matrix method,” J. Phys. Colloq. 46, 815–825 (1985).
[CrossRef]

1982 (1)

1972 (1)

1971 (1)

M. F. Schiekel, K. Fahrenschon, “Deformation of nematic liquid crystals with vertical orientation in electrical fields,” Appl. Phys. Lett. 19, 391–393 (1971).
[CrossRef]

1970 (1)

1955 (1)

S. Pancharatnam, “Achromatic combination of birefringent plates. II. An achromatic quarter-wave plate,” Proc. Indian Acad. Sci., Sect. A 41, 137–144 (1955).

Abdulhalim, I.

I. Abdulhalim, “2×2 Matrix summation method for multiple reflections and transmissions in a biaxial slab between two anisotropic media,” Opt. Commun. 163, 9–14 (1999).
[CrossRef]

I. Abdulhalim, “Exact 2×2 matrix method for the transmission and reflection at the interface between two arbitrarily oriented biaxial crystals,” J. Opt. A, Pure Appl. Opt. 1, 655–661 (1999).
[CrossRef]

I. Abdulhalim, “Analytic propagation matrix method for linear optics of arbitrary biaxial layered media,” J. Opt. A, Pure Appl. Opt. 1, 646–653 (1999).
[CrossRef]

I. Abdulhalim, R. Weil, L. Benguigui, “Dispersion and attenuation of the eigenwaves for light propagation in helicoidal liquid crystals,” Liq. Cryst. 1, 155–167 (1986).
[CrossRef]

I. Abdulhalim, L. Benguigui, R. Weil, “Selective reflection by helicoidal liquid crystals. Results of an exact calculation using the 4∗4 characteristic matrix method,” J. Phys. Colloq. 46, 815–825 (1985).
[CrossRef]

Benguigui, L.

I. Abdulhalim, R. Weil, L. Benguigui, “Dispersion and attenuation of the eigenwaves for light propagation in helicoidal liquid crystals,” Liq. Cryst. 1, 155–167 (1986).
[CrossRef]

I. Abdulhalim, L. Benguigui, R. Weil, “Selective reflection by helicoidal liquid crystals. Results of an exact calculation using the 4∗4 characteristic matrix method,” J. Phys. Colloq. 46, 815–825 (1985).
[CrossRef]

Berreman, D. W.

Chang, C. C.

D. L. Ting, W. C. Chang, C. Y. Liu, J. W. Shiu, C. J. Wen, C. H. Chao, L. S. Chuang, C. C. Chang, “A high brightness and high contrast reflective LCD with micro slant reflector (MSR),” SID Int. Symp. Digest Tech. Papers 30, 954–957 (1999).
[CrossRef]

Chang, W. C.

D. L. Ting, W. C. Chang, C. Y. Liu, J. W. Shiu, C. J. Wen, C. H. Chao, L. S. Chuang, C. C. Chang, “A high brightness and high contrast reflective LCD with micro slant reflector (MSR),” SID Int. Symp. Digest Tech. Papers 30, 954–957 (1999).
[CrossRef]

Chao, C. H.

D. L. Ting, W. C. Chang, C. Y. Liu, J. W. Shiu, C. J. Wen, C. H. Chao, L. S. Chuang, C. C. Chang, “A high brightness and high contrast reflective LCD with micro slant reflector (MSR),” SID Int. Symp. Digest Tech. Papers 30, 954–957 (1999).
[CrossRef]

Chen, C. J.

C. J. Chen, A. Lien, M. I. Nathan, “4×4 and 2×2 matrix formulations for the optics in stratified and biaxial media,” J. Opt. Soc. Am. A 14, 3125–3133 (1997).
[CrossRef]

A. Lien, C. J. Chen, “A new 2×2 matrix representation for twisted nematic liquid crystal displays at oblique incidence,” Jpn. J. Appl. Phys., Part 1 35, L1200–1203 (1996).
[CrossRef]

Chuang, L. S.

D. L. Ting, W. C. Chang, C. Y. Liu, J. W. Shiu, C. J. Wen, C. H. Chao, L. S. Chuang, C. C. Chang, “A high brightness and high contrast reflective LCD with micro slant reflector (MSR),” SID Int. Symp. Digest Tech. Papers 30, 954–957 (1999).
[CrossRef]

Cojocaru, E.

De Gennes, P. G.

P. G. De Gennes, J. Prost, The Physics of Liquid Crystals, 2nd ed. (Oxford, London, 1995).

Eidner, K.

K. Eidner, G. Mayer, M. Schmidt, H. Schmiedel, “Optics in stratified media-the use of optical eigenmodes of uniaxial crystal in the 4×4 matrix formalism,” Mol. Cryst. Liq. Cryst. 172, 191–200 (1989).

Fahrenschon, K.

M. F. Schiekel, K. Fahrenschon, “Deformation of nematic liquid crystals with vertical orientation in electrical fields,” Appl. Phys. Lett. 19, 391–393 (1971).
[CrossRef]

Fritsch, M.

Gu, C.

Hass, G.

Henvis, B. W.

Koike, Y.

N. Sugiura, K. Tashiro, K. Ohmuro, Y. Koike, K. Okamoto, “A novel vertically aligned reflective-color TFT LCD with high contrast ratio,” SID Int. Symp. Digest Tech. Papers 33, 1386–1389 (2002).
[CrossRef]

Kuo, C. L.

C. L. Kuo, C. K. Wei, S. T. Wu, C. S. Wu, “Reflective direct-view display using a mixed-mode twisted nematic cell,” Jpn. J. Appl. Phys., Part 1 36, 1077–1080 (1997).
[CrossRef]

Kwok, H. S.

Lien, A.

A. Lien, “A detailed derivation of extended Jones matrix representation for twisted nematic liquid crystal displays,” Liq. Cryst. 22, 171–175 (1997).
[CrossRef]

C. J. Chen, A. Lien, M. I. Nathan, “4×4 and 2×2 matrix formulations for the optics in stratified and biaxial media,” J. Opt. Soc. Am. A 14, 3125–3133 (1997).
[CrossRef]

A. Lien, C. J. Chen, “A new 2×2 matrix representation for twisted nematic liquid crystal displays at oblique incidence,” Jpn. J. Appl. Phys., Part 1 35, L1200–1203 (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]

Liu, C. Y.

D. L. Ting, W. C. Chang, C. Y. Liu, J. W. Shiu, C. J. Wen, C. H. Chao, L. S. Chuang, C. C. Chang, “A high brightness and high contrast reflective LCD with micro slant reflector (MSR),” SID Int. Symp. Digest Tech. Papers 30, 954–957 (1999).
[CrossRef]

Mansuripur, M.

M. Mansuripur, “Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2×2 matrices,” J. Appl. Phys. 67, 6466–6475 (1990).
[CrossRef]

Mayer, G.

K. Eidner, G. Mayer, M. Schmidt, H. Schmiedel, “Optics in stratified media-the use of optical eigenmodes of uniaxial crystal in the 4×4 matrix formalism,” Mol. Cryst. Liq. Cryst. 172, 191–200 (1989).

Mlynski, D. A.

Nathan, M. I.

Ohmuro, K.

N. Sugiura, K. Tashiro, K. Ohmuro, Y. Koike, K. Okamoto, “A novel vertically aligned reflective-color TFT LCD with high contrast ratio,” SID Int. Symp. Digest Tech. Papers 33, 1386–1389 (2002).
[CrossRef]

Okamoto, K.

N. Sugiura, K. Tashiro, K. Ohmuro, Y. Koike, K. Okamoto, “A novel vertically aligned reflective-color TFT LCD with high contrast ratio,” SID Int. Symp. Digest Tech. Papers 33, 1386–1389 (2002).
[CrossRef]

Pancharatnam, S.

S. Pancharatnam, “Achromatic combination of birefringent plates. II. An achromatic quarter-wave plate,” Proc. Indian Acad. Sci., Sect. A 41, 137–144 (1955).

Prost, J.

P. G. De Gennes, J. Prost, The Physics of Liquid Crystals, 2nd ed. (Oxford, London, 1995).

Schiekel, M. F.

M. F. Schiekel, K. Fahrenschon, “Deformation of nematic liquid crystals with vertical orientation in electrical fields,” Appl. Phys. Lett. 19, 391–393 (1971).
[CrossRef]

Schmidt, M.

K. Eidner, G. Mayer, M. Schmidt, H. Schmiedel, “Optics in stratified media-the use of optical eigenmodes of uniaxial crystal in the 4×4 matrix formalism,” Mol. Cryst. Liq. Cryst. 172, 191–200 (1989).

Schmiedel, H.

K. Eidner, G. Mayer, M. Schmidt, H. Schmiedel, “Optics in stratified media-the use of optical eigenmodes of uniaxial crystal in the 4×4 matrix formalism,” Mol. Cryst. Liq. Cryst. 172, 191–200 (1989).

Shiu, J. W.

D. L. Ting, W. C. Chang, C. Y. Liu, J. W. Shiu, C. J. Wen, C. H. Chao, L. S. Chuang, C. C. Chang, “A high brightness and high contrast reflective LCD with micro slant reflector (MSR),” SID Int. Symp. Digest Tech. Papers 30, 954–957 (1999).
[CrossRef]

Stallinga, S.

S. Stallinga, “Berreman 4×4 matrix method for reflective liquid crystal displays,” J. Appl. Phys. 85, 3023–3031 (1999).
[CrossRef]

Sugiura, N.

N. Sugiura, K. Tashiro, K. Ohmuro, Y. Koike, K. Okamoto, “A novel vertically aligned reflective-color TFT LCD with high contrast ratio,” SID Int. Symp. Digest Tech. Papers 33, 1386–1389 (2002).
[CrossRef]

Tashiro, K.

N. Sugiura, K. Tashiro, K. Ohmuro, Y. Koike, K. Okamoto, “A novel vertically aligned reflective-color TFT LCD with high contrast ratio,” SID Int. Symp. Digest Tech. Papers 33, 1386–1389 (2002).
[CrossRef]

Teitler, S.

Ting, D. L.

D. L. Ting, W. C. Chang, C. Y. Liu, J. W. Shiu, C. J. Wen, C. H. Chao, L. S. Chuang, C. C. Chang, “A high brightness and high contrast reflective LCD with micro slant reflector (MSR),” SID Int. Symp. Digest Tech. Papers 30, 954–957 (1999).
[CrossRef]

Wei, C. K.

C. L. Kuo, C. K. Wei, S. T. Wu, C. S. Wu, “Reflective direct-view display using a mixed-mode twisted nematic cell,” Jpn. J. Appl. Phys., Part 1 36, 1077–1080 (1997).
[CrossRef]

Weil, R.

I. Abdulhalim, R. Weil, L. Benguigui, “Dispersion and attenuation of the eigenwaves for light propagation in helicoidal liquid crystals,” Liq. Cryst. 1, 155–167 (1986).
[CrossRef]

I. Abdulhalim, L. Benguigui, R. Weil, “Selective reflection by helicoidal liquid crystals. Results of an exact calculation using the 4∗4 characteristic matrix method,” J. Phys. Colloq. 46, 815–825 (1985).
[CrossRef]

Wen, C. J.

D. L. Ting, W. C. Chang, C. Y. Liu, J. W. Shiu, C. J. Wen, C. H. Chao, L. S. Chuang, C. C. Chang, “A high brightness and high contrast reflective LCD with micro slant reflector (MSR),” SID Int. Symp. Digest Tech. Papers 30, 954–957 (1999).
[CrossRef]

Wohler, H.

Wu, C. S.

C. L. Kuo, C. K. Wei, S. T. Wu, C. S. Wu, “Reflective direct-view display using a mixed-mode twisted nematic cell,” Jpn. J. Appl. Phys., Part 1 36, 1077–1080 (1997).
[CrossRef]

S. T. Wu, C. S. Wu, “Mixed-mode twisted nematic liquid crystal cells for reflective displays,” Appl. Phys. Lett. 68, 1455–1457 (1996).
[CrossRef]

Wu, S. T.

C. L. Kuo, C. K. Wei, S. T. Wu, C. S. Wu, “Reflective direct-view display using a mixed-mode twisted nematic cell,” Jpn. J. Appl. Phys., Part 1 36, 1077–1080 (1997).
[CrossRef]

S. T. Wu, C. S. Wu, “Mixed-mode twisted nematic liquid crystal cells for reflective displays,” Appl. Phys. Lett. 68, 1455–1457 (1996).
[CrossRef]

S. T. Wu, D. K. Yang, Reflective Liquid Crystal Displays (Wiley, New York, 2001).

Yang, D. K.

S. T. Wu, D. K. Yang, Reflective Liquid Crystal Displays (Wiley, New York, 2001).

Yeh, P.

Yu, F. H.

Appl. Opt. (1)

Appl. Phys. Lett. (3)

M. F. Schiekel, K. Fahrenschon, “Deformation of nematic liquid crystals with vertical orientation in electrical fields,” Appl. Phys. Lett. 19, 391–393 (1971).
[CrossRef]

S. T. Wu, C. S. Wu, “Mixed-mode twisted nematic liquid crystal cells for reflective displays,” Appl. Phys. Lett. 68, 1455–1457 (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]

J. Appl. Phys. (2)

M. Mansuripur, “Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2×2 matrices,” J. Appl. Phys. 67, 6466–6475 (1990).
[CrossRef]

S. Stallinga, “Berreman 4×4 matrix method for reflective liquid crystal displays,” J. Appl. Phys. 85, 3023–3031 (1999).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (2)

I. Abdulhalim, “Analytic propagation matrix method for linear optics of arbitrary biaxial layered media,” J. Opt. A, Pure Appl. Opt. 1, 646–653 (1999).
[CrossRef]

I. Abdulhalim, “Exact 2×2 matrix method for the transmission and reflection at the interface between two arbitrarily oriented biaxial crystals,” J. Opt. A, Pure Appl. Opt. 1, 655–661 (1999).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (4)

J. Phys. Colloq. (1)

I. Abdulhalim, L. Benguigui, R. Weil, “Selective reflection by helicoidal liquid crystals. Results of an exact calculation using the 4∗4 characteristic matrix method,” J. Phys. Colloq. 46, 815–825 (1985).
[CrossRef]

Jpn. J. Appl. Phys., Part 1 (2)

C. L. Kuo, C. K. Wei, S. T. Wu, C. S. Wu, “Reflective direct-view display using a mixed-mode twisted nematic cell,” Jpn. J. Appl. Phys., Part 1 36, 1077–1080 (1997).
[CrossRef]

A. Lien, C. J. Chen, “A new 2×2 matrix representation for twisted nematic liquid crystal displays at oblique incidence,” Jpn. J. Appl. Phys., Part 1 35, L1200–1203 (1996).
[CrossRef]

Liq. Cryst. (2)

A. Lien, “A detailed derivation of extended Jones matrix representation for twisted nematic liquid crystal displays,” Liq. Cryst. 22, 171–175 (1997).
[CrossRef]

I. Abdulhalim, R. Weil, L. Benguigui, “Dispersion and attenuation of the eigenwaves for light propagation in helicoidal liquid crystals,” Liq. Cryst. 1, 155–167 (1986).
[CrossRef]

Mol. Cryst. Liq. Cryst. (1)

K. Eidner, G. Mayer, M. Schmidt, H. Schmiedel, “Optics in stratified media-the use of optical eigenmodes of uniaxial crystal in the 4×4 matrix formalism,” Mol. Cryst. Liq. Cryst. 172, 191–200 (1989).

Opt. Commun. (1)

I. Abdulhalim, “2×2 Matrix summation method for multiple reflections and transmissions in a biaxial slab between two anisotropic media,” Opt. Commun. 163, 9–14 (1999).
[CrossRef]

Proc. Indian Acad. Sci., Sect. A (1)

S. Pancharatnam, “Achromatic combination of birefringent plates. II. An achromatic quarter-wave plate,” Proc. Indian Acad. Sci., Sect. A 41, 137–144 (1955).

SID Int. Symp. Digest Tech. Papers (2)

D. L. Ting, W. C. Chang, C. Y. Liu, J. W. Shiu, C. J. Wen, C. H. Chao, L. S. Chuang, C. C. Chang, “A high brightness and high contrast reflective LCD with micro slant reflector (MSR),” SID Int. Symp. Digest Tech. Papers 30, 954–957 (1999).
[CrossRef]

N. Sugiura, K. Tashiro, K. Ohmuro, Y. Koike, K. Okamoto, “A novel vertically aligned reflective-color TFT LCD with high contrast ratio,” SID Int. Symp. Digest Tech. Papers 33, 1386–1389 (2002).
[CrossRef]

Other (2)

P. G. De Gennes, J. Prost, The Physics of Liquid Crystals, 2nd ed. (Oxford, London, 1995).

S. T. Wu, D. K. Yang, Reflective Liquid Crystal Displays (Wiley, New York, 2001).

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Figures (12)

Fig. 1
Fig. 1

Schematic diagram of beam path and viewing position in a conventional hand-held reflective LCD device.

Fig. 2
Fig. 2

Schematic diagram of layer division in a direct-view reflective LCD system. The wave-propagation vector k lies on the x z plane, θ inc is the incident angle from the air to the LCD, and θ exit is the exit angle from the LCD to the air.

Fig. 3
Fig. 3

Schematic diagram of layer division in an equivalent double-cell transmissive LCD structure by unfolding method. In this equivalent structure, light enters from the bottom air into the bottom polarizer layer and exits from the top polarizer layer to the top air.

Fig. 4
Fig. 4

Relationship between the image director and the original director in the same coordinate system.  n 1 , n 2 , and n 3 denote the original director, the mirror-image director, and the equivalent mirror-image director, respectively.

Fig. 5
Fig. 5

Schematic diagram of the beam path in the LC cell and the field reflection on the surface of the reflector; the dashed line represents the tangential reflection surface on the bumpy reflector.

Fig. 6
Fig. 6

Schematic diagram of energy-flow-defined reflectance with asymmetric incident and exit angles in R-mode LCD.

Fig. 7
Fig. 7

The top figure shows schematically the forward propagation (bold arrow) and backward propagation (dashed arrow) through a layer. The bottom figure shows that the incident wave from top to bottom can be viewed as the backward part (dashed arrow) with an angle of θ inc .

Fig. 8
Fig. 8

Structure of a broadband quarter-wave film, which comprises one chromatic quarter-wave film and one chromatic half-wave film. The quarter-wave film has an optic axis with an angle of 75° to the transmission axis of the polarizer. The half-wave film has an angle of 15° to the transmission axis of the polarizer.

Fig. 9
Fig. 9

Electro-optical properties of symmetric incidence for (a) normal incidence case with different methods for a VA mode, (b) θ inc = 30 ° and θ exit = 30 ° with different methods for a VA mode, (c) normal-incidence case with different methods for an 80° MTN mode, (d) θ inc = 30 ° and θ exit = 30 ° with different methods for an 80° MTN mode.

Fig. 10
Fig. 10

Rubbing diagrams under asymmetric light incidence for (a) a VA cell, (b) a MTN cell.

Fig. 11
Fig. 11

Viewing-angle-dependent contrast-ratio plot for both VA and MTN modes under different rubbing diagrams.

Fig. 12
Fig. 12

Structure of n th sublayer with thickness d n . Notice that the bottom of this layer is the ( n + 1 st ) layer.

Equations (84)

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J upper = J 1 J 2 J N 1 J N ,
J lower = J N J N 1 J 2 J 1 .
[ E x , N + 1 i E y , N + 1 i ] = J lower [ E x , 1 i E y , 1 i ] .
[ E x , 1 r E y , 1 r ] = J upper [ E x , N + 1 r E y , N + 1 r ] .
E x , N + 1 i = E , N + 1 i cos θ i ,
E y , N + 1 i = E , N + 1 i .
θ i = sin 1 ( sin θ inc n LC ) .
E x , N + 1 r = E , N + 1 r cos θ r ,
E y , N + 1 r = E , N + 1 r ,
θ r = sin 1 ( sin θ exit n LC ) .
[ E x , N + 1 r E y , N + 1 r ] = [ cos θ r cos θ i 0 0 1 ] [ E x , N + 1 i E y , N + 1 i ] .
J TR = [ cos θ r cos θ i 0 0 1 ] .
[ E x , 1 r E y , 1 r ] = J upper J TR J lower [ E x , 1 i E y , 1 i ] ,
J upper J TR J lower = J 1 J 2 J N 1 J N J TR J N J N 1 J 2 J 1 .
J ent = [ 2 cos θ 1 cos θ 1 + n 1 cos θ inc 0 0 2 cos θ inc cos θ inc + n 1 cos θ 1 ] ,
J ext = [ 2 n 1 cos θ exit cos θ 1 + n 1 cos θ exit 0 0 2 n 1 cos θ 1 cos θ exit + n 1 cos θ 1 ] ,
J = J ext J 1 J 2 J N 1 J N J TR J N J N 1 J 2 J 1 J ent .
[ E x , air exit E y , air exit ] = J [ E x , air inc E y , air inc ] ,
R = P out P in = E x , air exit cos θ exit 2 + E y , air exit 2 E x , air inc cos θ inc 2 + E y , air inc 2 .
P in = ( E x , air inc cos θ inc 2 + E y , air inc 2 ) A D ,
P out = ( E x , air exit cos θ exit 2 + E y , air exit 2 ) K L .
A C cos θ 1 = I K cos θ 1 .
η corr = K L A D = I K cos θ exit A C cos θ inc = cos θ 1 cos θ 1 cos θ exit cos θ inc .
R = P out P in = E x , air exit cos θ exit 2 + E y , air exit 2 E x , air inc cos θ inc 2 + E y , air inc 2 cos θ 1 cos θ 1 cos θ exit cos θ inc
[ E x , air inc E y , air inc ] = [ cos θ inc e i φ ] ,
P in = ( E x , air inc cos θ inc 2 + E x , air inc 2 ) A D = 2 A D .
[ E x , air exit E y , air exit ] = [ J 11 J 12 J 21 J 22 ] [ E x , air inc E y , air inc ] .
E x , air exit = J 11 cos θ inc + J 12 e j φ ,
E y , air exit = J 21 cos θ inc + J 22 e j φ .
P out = ( E x , air exit cos θ exit 2 + E x , air exit 2 ) K L .
P out = ( J 11 2 cos 2 θ inc + J 12 2 + J 11 cos θ inc J 12 * e j φ + J 11 * cos θ inc J 12 e j φ cos 2 θ exit + J 21 2 cos 2 θ inc + J 22 2 + J 21 cos θ inc J 22 * e j φ + J 21 * cos θ inc J 22 e j φ ) K L .
P out av = ( J 11 2 cos 2 θ inc + J 12 2 cos 2 θ exit + J 21 2 cos 2 θ inc + J 22 2 ) K L .
R = J 11 2 cos 2 θ inc + J 12 2 + cos 2 θ exit ( J 21 2 cos 2 θ inc + J 22 2 ) 2 cos 2 θ exit cos θ 1 cos θ 1 cos θ exit cos θ inc .
[ E x , N + 1 i E y , N + 1 i ] = J e inc [ E x , 1 i E y , 1 i ] ,
J e inc = ( J e , N ) 1 ( J e , N 1 ) 1 ( J e , 2 ) 1 ( J e , 1 ) 1 ,
[ E x 1 r E y , 1 r ] = J e r [ E x , N + 1 r E y , N + 1 r ] ,
J e r = J e , 1 + J e , 2 + J e , N 1 + J e , N + ,
J e = J ext J e r J TR J e inc J ent .
ϵ = [ ϵ x x ϵ x y ϵ x z ϵ y x ϵ y y ϵ y z ϵ z x ϵ z y ϵ z z ] ,
ϵ x x = n o 2 + ( n e 2 n o 2 ) cos 2 θ cos 2 ϕ ,
ϵ x y = ϵ y x = ( n e 2 n o 2 ) cos 2 θ sin ϕ cos ϕ ,
ϵ x z = ϵ z x = ( n e 2 n o 2 ) sin θ cos θ cos ϕ ,
ϵ y y = n o 2 + ( n e 2 n o 2 ) cos 2 θ sin 2 ϕ ,
ϵ y z = ϵ z y = ( n e 2 n o 2 ) sin θ cos θ sin ϕ ,
ϵ z z = n o 2 + ( n e 2 n o 2 ) sin 2 θ ,
J i = ( S G S 1 ) i , ( i = 1 , 2 , , N ) ,
S = [ 1 c 2 c 1 1 ] ,
G = [ exp ( i k z 1 d i ) 0 0 exp ( i k z 2 d i ) ] ,
k z 1 k 0 = { [ n o 2 ( k x k o ) 2 ] } 1 2 ,
k z 2 k 0 = ϵ x z ϵ z z k x k 0 + n o n e ϵ z z { [ ϵ z z ( 1 n e 2 n o 2 n e 2 cos 2 θ sin 2 ϕ ) ( k x k 0 ) 2 ] } 1 2 ,
c 1 = [ ( k x k 0 ) 2 ϵ z z ] ϵ y x + [ ( k x k 0 ) ( k z 1 k 0 ) + ϵ z x ] ϵ y z [ ( k x k 0 ) 2 + ( k z 1 k 0 ) 2 ϵ y y ] [ ( k x k 0 ) 2 ϵ z z ] ϵ y z ϵ z y ,
c 2 = [ ( k x k 0 ) 2 ϵ z z ] ϵ x y + [ ( k x k 0 ) ( k z 2 k 0 ) + ϵ x z ] ϵ z y [ ( k z 2 k 0 ) 2 ϵ x x ] [ ( k x k 0 ) 2 ϵ z z ] [ ( k x k 0 ) ( k z 2 k 0 ) + ϵ z x ] [ ( k x k 0 ) ( k z 2 k 0 ) + ϵ x z ] .
H ̂ = ( μ 0 ϵ 0 ) 1 2 H .
× E = i k 0 H ̂ ,
× H ̂ = i k 0 ϵ E .
E y z = i k 0 H ̂ x ,
i k x E z + E x z = i k 0 H ̂ y ,
i k x E y = i k 0 H ̂ z ,
H ̂ y z = i k 0 ( ϵ x x E x + ϵ x y E y + ϵ x z E z ) ,
i k x H ̂ z + H ̂ x z = i k 0 ( ϵ y x E x + ϵ y y E y + ϵ y z E z ) ,
i k x H ̂ y = i k 0 ( ϵ z x E x + ϵ z y E y + ϵ z z E z ) .
z [ E x E y H ̂ x H ̂ y ] = i k 0 Q [ E x E y H ̂ x H ̂ y ] ,
Q = [ ϵ z x ϵ z z sin θ k ϵ z y ϵ z z sin θ k 0 1 sin 2 θ k ϵ z z 0 0 1 0 ϵ y x + ϵ y z ϵ z x ϵ z z ϵ y y + ϵ y z ϵ z y ϵ z z + sin 2 θ k 0 ϵ y z ϵ z z sin θ k ϵ x x ϵ x z ϵ z x ϵ z z ϵ x y ϵ x z ϵ z y ϵ z z 0 ϵ x z ϵ z z sin θ k ] .
Q = T [ q 1 q 2 q 3 q 4 ] T 1 ,
[ E x E y H ̂ x H ̂ y ] = T [ U 1 U 2 U 3 U 4 ] ,
T = [ T 11 T 12 T 21 T 22 ] .
z [ U 1 U 2 U 3 U 4 ] = i k 0 [ q 1 q 2 q 3 q 4 ] [ U 1 U 2 U 3 U 4 ] .
[ U 1 U 2 U 3 U 4 ] n , d n = G n [ U 1 U 2 U 3 U 4 ] n , 0 ,
G n = [ exp ( i k z 1 d n ) exp ( i k z 2 d n ) exp ( i k z 3 d n ) exp ( i k z 4 d n ) ] ,
k z 1 = k 0 q 1 ,
k z 2 = k 0 q 2 ,
k z 3 = k 0 q 3 ,
k z 4 = k 0 q 4 .
[ U 1 U 2 ] n , d n = F n [ U 1 U 2 ] n , 0 ,
F n = [ exp ( i k z 1 d n ) 0 0 exp ( i k z 2 d n ) ] .
[ U 3 U 4 ] n , d n = B n [ U 3 U 4 ] n , 0 ,
B n = [ exp ( i k z 3 d n ) 0 0 exp ( i k z 4 d n ) ] .
[ E x E y ] = [ E x E y ] + + [ E x E y ] = T 11 [ U 1 U 2 ] + T 12 [ U 3 U 4 ] .
[ E x E y ] [ E x E y ] + = T 11 [ U 1 U 2 ] .
[ E x E y ] n + = J e , n + [ E x E y ] n + 1 + ,
J e , n + = ( T 11 ) n F n ( T 11 ) n 1
[ E x E y ] = T 12 [ U 3 U 4 ] .
[ E x E y ] n + 1 = ( J e , n ) 1 [ E x E y ] n ,
( J e , n ) 1 = ( T 12 ) n ( B n ) 1 ( T 12 ) n 1

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