Abstract

In the measurement of a twisted nematic liquid crystal device (TNLCD) by an optical apparatus, the cell parameters of the TNLCD may result in multiple solutions in the measurement that all agree with the measured data; hence manufacturers cannot find a set of correct solutions from among the ambiguous ones. With the help of the optical equivalence theorem of a unitary optical system, the ambiguity of the measured parameters of a TNLCD, including cell parameters and equivalent birefringent parameters, can be simultaneously removed by an analytical approach using a single-wavelength polarimeter. The procedure for unique determination of the cell parameters is performed using a self-consistent condition to select a set of the correct solutions from all the possible solutions. The proposed method can be applied to characterize a generally TNLCD for which the twisted angle is close to 270° and the liquid crystal phase retardation is over 2π.

© 2012 Optical Society of America

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References

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  1. B. E. A. Saleh and K. Lu, “Theory and design of the liquid crystal TV as an optical spatial phase modulator,” Opt. Eng. 29, 240–246 (1990).
    [CrossRef]
  2. J. A. Davis, I. Moreno, and P. Tsai, “Polarization eigenstates for twisted-nematic liquid-crystal displays,” Appl. Opt. 37, 937–945 (1998).
    [CrossRef]
  3. C. Soutar and K. Lu, “Determination of the physical properties of an arbitrary twisted-nematic liquid crystal cell,” Opt. Eng. 33, 2704–2712 (1994).
    [CrossRef]
  4. M. Yamauchi, “Origin and characteristics of ambiguous properties in measuring physical parameters of twisted nematic liquid crystal spatial light modulators,” Opt. Eng. 41, 1134–1141 (2002).
    [CrossRef]
  5. J. A. Davis, D. B. Allison, K. G. D’Nelly, M. L. Wilson, and I. Moreno, “Ambiguities in measuring the physical parameters for twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 38, 705–709 (1999).
    [CrossRef]
  6. H. Kim, and Y. H. Lee, “Unique measurement of the parameters of a twisted-nematic liquid-crystal display,” Appl. Opt. 44, 1642–1648 (2005).
    [CrossRef]
  7. A. Hermerscmidt, S. Quiram, F. Kallmeyer, and E. H. Joachim, “Determination of the Jones matrix of an LC cell and derivation of the physical parameters of the LC molecules,” Proc. SPIE 6587, 65871B (2007).
    [CrossRef]
  8. K. Dev, A. Prakarsa, Y. Jiang, H. Lee, and A. Asundi, “Twisted nematic liquid crystal cell characterization using rotating polarizers including full-field cell gap thickness measurement,” Proc. SPIE 7522, 75224N (2009).
  9. Y. Zhou, Z. He, and S. Sato, “A novel method determining the cell thickness and twist angle of a twisted nematic cell by Stokes parameter measurement,” Jpn. J. Appl. Phys. 36, 2760–2764 (1997).
    [CrossRef]
  10. V. Durán, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, “Cell parameter determination of a twisted-nematic liquid crystal display by single-wavelength polarimetry,” J. Appl. Phys. 97, 043101 (2005).
    [CrossRef]
  11. P. Yeh, and C. Gu, Optics of Liquid Crystal Display (Wiley, 1999), pp. 129–130.
  12. A. Lien, “The general and simplified Jones matrix representations for the high pretilt twisted nematic cell,” Jpn. J. Appl. Phys. 67, 2853 (1990).
    [CrossRef]
  13. V. Durán, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, “Equivalent retarder-rotator approach to on-state twisted nematic liquid crystal displays,” J. Appl. Phys. 99, 113101 (2006).
    [CrossRef]
  14. S. T. Tang, and H. S. Kwok, “3×3 Matrix for unitary optical systems,” J. Opt. Soc. Am. A 18, 2138–2145 (2001).
    [CrossRef]
  15. H. Hurwitz, and R. C. Jones, “A new calculus for the treatment of optical systems. II. Proof of three general equivalence theorems,” J. Opt. Soc. Am. 31, 493–499 (1941).
    [CrossRef]
  16. S. T. Tang, and H. S. Kwok, “Characteristic parameters of liquid crystal cells and their measurements,” J. Disp. Technol. 2, 26–31 (2006).
    [CrossRef]
  17. C.-J. Yu, C. E. Lin, Y. C. Li, L. D. Chou, J. S. Wu, C. C. Lee, and C. Chou, “Dual-frequency heterodyne ellipsometer for characterizing generalized elliptically birefringent media,” Opt. Express 17, 19213–19224 (2009).
    [CrossRef]
  18. C.-J. Yu, Y.-T. Tseng, K.-C. Hsu, and C. Chou, “Full-field characterization of a twisted nematic liquid-crystal device using equivalence theorem of a unitary optical system,” Appl. Opt. 51, 238–244 (2012).
    [CrossRef]
  19. D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. 31, 6676–6683 (1992).
    [CrossRef]
  20. K. Muraki, M. Tsukiji, A. Takayanagi, and N. Umeda, “Simultaneous measurement of linear and circular birefringence with heterodyne interferometer,” Proc. SPIE 2873, 29–32 (1996).
    [CrossRef]
  21. B. Wang, “Measurement of circular and linear birefringence in chiral media and optical materials using the photoelastic modulator,” Proc. SPIE 3535, 294–302 (1999).
    [CrossRef]
  22. C. Hitzenberger, E. Goetzinger, M. Sticker, M. Pircher, and A. Fercher, “Measurement and imaging of birefringence and optic axis orientation by phase resolved polarization sensitive optical coherence tomography,” Opt. Express 9, 780–790 (2001).
    [CrossRef]

2012

2009

C.-J. Yu, C. E. Lin, Y. C. Li, L. D. Chou, J. S. Wu, C. C. Lee, and C. Chou, “Dual-frequency heterodyne ellipsometer for characterizing generalized elliptically birefringent media,” Opt. Express 17, 19213–19224 (2009).
[CrossRef]

K. Dev, A. Prakarsa, Y. Jiang, H. Lee, and A. Asundi, “Twisted nematic liquid crystal cell characterization using rotating polarizers including full-field cell gap thickness measurement,” Proc. SPIE 7522, 75224N (2009).

2007

A. Hermerscmidt, S. Quiram, F. Kallmeyer, and E. H. Joachim, “Determination of the Jones matrix of an LC cell and derivation of the physical parameters of the LC molecules,” Proc. SPIE 6587, 65871B (2007).
[CrossRef]

2006

V. Durán, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, “Equivalent retarder-rotator approach to on-state twisted nematic liquid crystal displays,” J. Appl. Phys. 99, 113101 (2006).
[CrossRef]

S. T. Tang, and H. S. Kwok, “Characteristic parameters of liquid crystal cells and their measurements,” J. Disp. Technol. 2, 26–31 (2006).
[CrossRef]

2005

V. Durán, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, “Cell parameter determination of a twisted-nematic liquid crystal display by single-wavelength polarimetry,” J. Appl. Phys. 97, 043101 (2005).
[CrossRef]

H. Kim, and Y. H. Lee, “Unique measurement of the parameters of a twisted-nematic liquid-crystal display,” Appl. Opt. 44, 1642–1648 (2005).
[CrossRef]

2002

M. Yamauchi, “Origin and characteristics of ambiguous properties in measuring physical parameters of twisted nematic liquid crystal spatial light modulators,” Opt. Eng. 41, 1134–1141 (2002).
[CrossRef]

2001

1999

B. Wang, “Measurement of circular and linear birefringence in chiral media and optical materials using the photoelastic modulator,” Proc. SPIE 3535, 294–302 (1999).
[CrossRef]

J. A. Davis, D. B. Allison, K. G. D’Nelly, M. L. Wilson, and I. Moreno, “Ambiguities in measuring the physical parameters for twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 38, 705–709 (1999).
[CrossRef]

1998

1997

Y. Zhou, Z. He, and S. Sato, “A novel method determining the cell thickness and twist angle of a twisted nematic cell by Stokes parameter measurement,” Jpn. J. Appl. Phys. 36, 2760–2764 (1997).
[CrossRef]

1996

K. Muraki, M. Tsukiji, A. Takayanagi, and N. Umeda, “Simultaneous measurement of linear and circular birefringence with heterodyne interferometer,” Proc. SPIE 2873, 29–32 (1996).
[CrossRef]

1994

C. Soutar and K. Lu, “Determination of the physical properties of an arbitrary twisted-nematic liquid crystal cell,” Opt. Eng. 33, 2704–2712 (1994).
[CrossRef]

1992

1990

B. E. A. Saleh and K. Lu, “Theory and design of the liquid crystal TV as an optical spatial phase modulator,” Opt. Eng. 29, 240–246 (1990).
[CrossRef]

A. Lien, “The general and simplified Jones matrix representations for the high pretilt twisted nematic cell,” Jpn. J. Appl. Phys. 67, 2853 (1990).
[CrossRef]

1941

Allison, D. B.

J. A. Davis, D. B. Allison, K. G. D’Nelly, M. L. Wilson, and I. Moreno, “Ambiguities in measuring the physical parameters for twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 38, 705–709 (1999).
[CrossRef]

Asundi, A.

K. Dev, A. Prakarsa, Y. Jiang, H. Lee, and A. Asundi, “Twisted nematic liquid crystal cell characterization using rotating polarizers including full-field cell gap thickness measurement,” Proc. SPIE 7522, 75224N (2009).

Chou, C.

Chou, L. D.

D’Nelly, K. G.

J. A. Davis, D. B. Allison, K. G. D’Nelly, M. L. Wilson, and I. Moreno, “Ambiguities in measuring the physical parameters for twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 38, 705–709 (1999).
[CrossRef]

Davis, J. A.

J. A. Davis, D. B. Allison, K. G. D’Nelly, M. L. Wilson, and I. Moreno, “Ambiguities in measuring the physical parameters for twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 38, 705–709 (1999).
[CrossRef]

J. A. Davis, I. Moreno, and P. Tsai, “Polarization eigenstates for twisted-nematic liquid-crystal displays,” Appl. Opt. 37, 937–945 (1998).
[CrossRef]

Dev, K.

K. Dev, A. Prakarsa, Y. Jiang, H. Lee, and A. Asundi, “Twisted nematic liquid crystal cell characterization using rotating polarizers including full-field cell gap thickness measurement,” Proc. SPIE 7522, 75224N (2009).

Durán, V.

V. Durán, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, “Equivalent retarder-rotator approach to on-state twisted nematic liquid crystal displays,” J. Appl. Phys. 99, 113101 (2006).
[CrossRef]

V. Durán, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, “Cell parameter determination of a twisted-nematic liquid crystal display by single-wavelength polarimetry,” J. Appl. Phys. 97, 043101 (2005).
[CrossRef]

Fercher, A.

Goetzinger, E.

Goldstein, D. H.

Gu, C.

P. Yeh, and C. Gu, Optics of Liquid Crystal Display (Wiley, 1999), pp. 129–130.

He, Z.

Y. Zhou, Z. He, and S. Sato, “A novel method determining the cell thickness and twist angle of a twisted nematic cell by Stokes parameter measurement,” Jpn. J. Appl. Phys. 36, 2760–2764 (1997).
[CrossRef]

Hermerscmidt, A.

A. Hermerscmidt, S. Quiram, F. Kallmeyer, and E. H. Joachim, “Determination of the Jones matrix of an LC cell and derivation of the physical parameters of the LC molecules,” Proc. SPIE 6587, 65871B (2007).
[CrossRef]

Hitzenberger, C.

Hsu, K.-C.

Hurwitz, H.

Jaroszewicz, Z.

V. Durán, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, “Equivalent retarder-rotator approach to on-state twisted nematic liquid crystal displays,” J. Appl. Phys. 99, 113101 (2006).
[CrossRef]

V. Durán, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, “Cell parameter determination of a twisted-nematic liquid crystal display by single-wavelength polarimetry,” J. Appl. Phys. 97, 043101 (2005).
[CrossRef]

Jiang, Y.

K. Dev, A. Prakarsa, Y. Jiang, H. Lee, and A. Asundi, “Twisted nematic liquid crystal cell characterization using rotating polarizers including full-field cell gap thickness measurement,” Proc. SPIE 7522, 75224N (2009).

Joachim, E. H.

A. Hermerscmidt, S. Quiram, F. Kallmeyer, and E. H. Joachim, “Determination of the Jones matrix of an LC cell and derivation of the physical parameters of the LC molecules,” Proc. SPIE 6587, 65871B (2007).
[CrossRef]

Jones, R. C.

Kallmeyer, F.

A. Hermerscmidt, S. Quiram, F. Kallmeyer, and E. H. Joachim, “Determination of the Jones matrix of an LC cell and derivation of the physical parameters of the LC molecules,” Proc. SPIE 6587, 65871B (2007).
[CrossRef]

Kim, H.

Kwok, H. S.

S. T. Tang, and H. S. Kwok, “Characteristic parameters of liquid crystal cells and their measurements,” J. Disp. Technol. 2, 26–31 (2006).
[CrossRef]

S. T. Tang, and H. S. Kwok, “3×3 Matrix for unitary optical systems,” J. Opt. Soc. Am. A 18, 2138–2145 (2001).
[CrossRef]

Lancis, J.

V. Durán, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, “Equivalent retarder-rotator approach to on-state twisted nematic liquid crystal displays,” J. Appl. Phys. 99, 113101 (2006).
[CrossRef]

V. Durán, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, “Cell parameter determination of a twisted-nematic liquid crystal display by single-wavelength polarimetry,” J. Appl. Phys. 97, 043101 (2005).
[CrossRef]

Lee, C. C.

Lee, H.

K. Dev, A. Prakarsa, Y. Jiang, H. Lee, and A. Asundi, “Twisted nematic liquid crystal cell characterization using rotating polarizers including full-field cell gap thickness measurement,” Proc. SPIE 7522, 75224N (2009).

Lee, Y. H.

Li, Y. C.

Lien, A.

A. Lien, “The general and simplified Jones matrix representations for the high pretilt twisted nematic cell,” Jpn. J. Appl. Phys. 67, 2853 (1990).
[CrossRef]

Lin, C. E.

Lu, K.

C. Soutar and K. Lu, “Determination of the physical properties of an arbitrary twisted-nematic liquid crystal cell,” Opt. Eng. 33, 2704–2712 (1994).
[CrossRef]

B. E. A. Saleh and K. Lu, “Theory and design of the liquid crystal TV as an optical spatial phase modulator,” Opt. Eng. 29, 240–246 (1990).
[CrossRef]

Moreno, I.

J. A. Davis, D. B. Allison, K. G. D’Nelly, M. L. Wilson, and I. Moreno, “Ambiguities in measuring the physical parameters for twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 38, 705–709 (1999).
[CrossRef]

J. A. Davis, I. Moreno, and P. Tsai, “Polarization eigenstates for twisted-nematic liquid-crystal displays,” Appl. Opt. 37, 937–945 (1998).
[CrossRef]

Muraki, K.

K. Muraki, M. Tsukiji, A. Takayanagi, and N. Umeda, “Simultaneous measurement of linear and circular birefringence with heterodyne interferometer,” Proc. SPIE 2873, 29–32 (1996).
[CrossRef]

Pircher, M.

Prakarsa, A.

K. Dev, A. Prakarsa, Y. Jiang, H. Lee, and A. Asundi, “Twisted nematic liquid crystal cell characterization using rotating polarizers including full-field cell gap thickness measurement,” Proc. SPIE 7522, 75224N (2009).

Quiram, S.

A. Hermerscmidt, S. Quiram, F. Kallmeyer, and E. H. Joachim, “Determination of the Jones matrix of an LC cell and derivation of the physical parameters of the LC molecules,” Proc. SPIE 6587, 65871B (2007).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh and K. Lu, “Theory and design of the liquid crystal TV as an optical spatial phase modulator,” Opt. Eng. 29, 240–246 (1990).
[CrossRef]

Sato, S.

Y. Zhou, Z. He, and S. Sato, “A novel method determining the cell thickness and twist angle of a twisted nematic cell by Stokes parameter measurement,” Jpn. J. Appl. Phys. 36, 2760–2764 (1997).
[CrossRef]

Soutar, C.

C. Soutar and K. Lu, “Determination of the physical properties of an arbitrary twisted-nematic liquid crystal cell,” Opt. Eng. 33, 2704–2712 (1994).
[CrossRef]

Sticker, M.

Tajahuerce, E.

V. Durán, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, “Equivalent retarder-rotator approach to on-state twisted nematic liquid crystal displays,” J. Appl. Phys. 99, 113101 (2006).
[CrossRef]

V. Durán, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, “Cell parameter determination of a twisted-nematic liquid crystal display by single-wavelength polarimetry,” J. Appl. Phys. 97, 043101 (2005).
[CrossRef]

Takayanagi, A.

K. Muraki, M. Tsukiji, A. Takayanagi, and N. Umeda, “Simultaneous measurement of linear and circular birefringence with heterodyne interferometer,” Proc. SPIE 2873, 29–32 (1996).
[CrossRef]

Tang, S. T.

S. T. Tang, and H. S. Kwok, “Characteristic parameters of liquid crystal cells and their measurements,” J. Disp. Technol. 2, 26–31 (2006).
[CrossRef]

S. T. Tang, and H. S. Kwok, “3×3 Matrix for unitary optical systems,” J. Opt. Soc. Am. A 18, 2138–2145 (2001).
[CrossRef]

Tsai, P.

Tseng, Y.-T.

Tsukiji, M.

K. Muraki, M. Tsukiji, A. Takayanagi, and N. Umeda, “Simultaneous measurement of linear and circular birefringence with heterodyne interferometer,” Proc. SPIE 2873, 29–32 (1996).
[CrossRef]

Umeda, N.

K. Muraki, M. Tsukiji, A. Takayanagi, and N. Umeda, “Simultaneous measurement of linear and circular birefringence with heterodyne interferometer,” Proc. SPIE 2873, 29–32 (1996).
[CrossRef]

Wang, B.

B. Wang, “Measurement of circular and linear birefringence in chiral media and optical materials using the photoelastic modulator,” Proc. SPIE 3535, 294–302 (1999).
[CrossRef]

Wilson, M. L.

J. A. Davis, D. B. Allison, K. G. D’Nelly, M. L. Wilson, and I. Moreno, “Ambiguities in measuring the physical parameters for twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 38, 705–709 (1999).
[CrossRef]

Wu, J. S.

Yamauchi, M.

M. Yamauchi, “Origin and characteristics of ambiguous properties in measuring physical parameters of twisted nematic liquid crystal spatial light modulators,” Opt. Eng. 41, 1134–1141 (2002).
[CrossRef]

Yeh, P.

P. Yeh, and C. Gu, Optics of Liquid Crystal Display (Wiley, 1999), pp. 129–130.

Yu, C.-J.

Zhou, Y.

Y. Zhou, Z. He, and S. Sato, “A novel method determining the cell thickness and twist angle of a twisted nematic cell by Stokes parameter measurement,” Jpn. J. Appl. Phys. 36, 2760–2764 (1997).
[CrossRef]

Appl. Opt.

J. Appl. Phys.

V. Durán, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, “Cell parameter determination of a twisted-nematic liquid crystal display by single-wavelength polarimetry,” J. Appl. Phys. 97, 043101 (2005).
[CrossRef]

V. Durán, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, “Equivalent retarder-rotator approach to on-state twisted nematic liquid crystal displays,” J. Appl. Phys. 99, 113101 (2006).
[CrossRef]

J. Disp. Technol.

S. T. Tang, and H. S. Kwok, “Characteristic parameters of liquid crystal cells and their measurements,” J. Disp. Technol. 2, 26–31 (2006).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Jpn. J. Appl. Phys.

Y. Zhou, Z. He, and S. Sato, “A novel method determining the cell thickness and twist angle of a twisted nematic cell by Stokes parameter measurement,” Jpn. J. Appl. Phys. 36, 2760–2764 (1997).
[CrossRef]

A. Lien, “The general and simplified Jones matrix representations for the high pretilt twisted nematic cell,” Jpn. J. Appl. Phys. 67, 2853 (1990).
[CrossRef]

Opt. Eng.

C. Soutar and K. Lu, “Determination of the physical properties of an arbitrary twisted-nematic liquid crystal cell,” Opt. Eng. 33, 2704–2712 (1994).
[CrossRef]

M. Yamauchi, “Origin and characteristics of ambiguous properties in measuring physical parameters of twisted nematic liquid crystal spatial light modulators,” Opt. Eng. 41, 1134–1141 (2002).
[CrossRef]

J. A. Davis, D. B. Allison, K. G. D’Nelly, M. L. Wilson, and I. Moreno, “Ambiguities in measuring the physical parameters for twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 38, 705–709 (1999).
[CrossRef]

B. E. A. Saleh and K. Lu, “Theory and design of the liquid crystal TV as an optical spatial phase modulator,” Opt. Eng. 29, 240–246 (1990).
[CrossRef]

Opt. Express

Proc. SPIE

K. Muraki, M. Tsukiji, A. Takayanagi, and N. Umeda, “Simultaneous measurement of linear and circular birefringence with heterodyne interferometer,” Proc. SPIE 2873, 29–32 (1996).
[CrossRef]

B. Wang, “Measurement of circular and linear birefringence in chiral media and optical materials using the photoelastic modulator,” Proc. SPIE 3535, 294–302 (1999).
[CrossRef]

A. Hermerscmidt, S. Quiram, F. Kallmeyer, and E. H. Joachim, “Determination of the Jones matrix of an LC cell and derivation of the physical parameters of the LC molecules,” Proc. SPIE 6587, 65871B (2007).
[CrossRef]

K. Dev, A. Prakarsa, Y. Jiang, H. Lee, and A. Asundi, “Twisted nematic liquid crystal cell characterization using rotating polarizers including full-field cell gap thickness measurement,” Proc. SPIE 7522, 75224N (2009).

Other

P. Yeh, and C. Gu, Optics of Liquid Crystal Display (Wiley, 1999), pp. 129–130.

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Figures (2)

Fig. 1.
Fig. 1.

Optical setup of a single-wavelength polarizer–sample–analyzer polarimeter. L, laser; P, polarizers; S, sample; A, analyzer; and D, photodetector.

Fig. 2.
Fig. 2.

Calculated result of substituting (a) first ambiguity combination, (b) second ambiguity combination, (c) third ambiguity combination, and (d) fourth ambiguity combination into the self-consistent condition, Eq. (52). The solid line represents the calculated values of the right-hand side of Eq. (52), and the discrete symbols indicate the calculated values of the left-hand side of Eq. (52) at the mth ambiguity combination of the equivalent birefringent parameters (Γm, ψm). The meaning of each of the discrete symbols is given by the following: open circle (χ+mn, Ωm0), open square (χ+mn, Ωm+), open triangle (χ+mn, Ωm), filled circle (χmn, Ωm0), filled square (χmn, Ωm+), and filled triangle (χmn, Ωm).

Tables (5)

Tables Icon

Table 1. Detected Intensity Ij at Different Azimuthal Angles of the Polarizer (P) and Analyzer (A)

Tables Icon

Table 2. Possible Solutions of the Equivalent Birefringent Parameters of the Linear Phase Retarder Calculated from Eqs. (44) and (45)

Tables Icon

Table 3. Lists of the Possible Combinations of Equivalent Birefringent Parameters and the Cell Parameters

Tables Icon

Table 4. Cell Parameters and the Equivalent Birefringent Parameters of an STNLCD

Tables Icon

Table 5. Calculated Results of the Equivalent Birefringent Parameters of an STNLCD with the Same Physical Properties of Birefringence and Thickness as Described in Section 5 but Having Four Different Rotation Senses of Rubbing Direction and Twisted Angle

Equations (63)

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

JLC0=(a0+ib0c0+id0c0+id0a0ib0),
a0=11+u2sinΩsin(1+u2Ω)+cosΩcos(1+u2Ω),
b0=u1+u2cosΩsin(1+u2Ω),
c0=11+u2cosΩsin(1+u2Ω)sinΩcos(1+u2Ω),
d0=u1+u2sinΩsin(1+u2Ω),
u=πlλ0Ω[ne(α)no],
β=π[ne(α)no]l/λ0
ne(α)=ne1+[(ne/no)21]sin2α
a0=ΩχsinΩsinχ+cosΩcosχ,
b0=βχcosΩsinχ,
c0=ΩχcosΩsinχsinΩcosχ,
d0=βχsinΩsinχ,
χ=1+u2Ω=Ω2+β2.
JLC=R(D)JLC0(β,Ω)R(D)=(cosDsinDsinDcosD)(a0+ib0c0+id0c0+id0a0ib0)(cosDsinDsinDcosD)=(alc+iblcclc+idlcclc+idlcalciblc),
alc=cosχcosΩ+ΩχsinχsinΩ,
blc=βχsinχcos(Ω+2D),
clc=cosχsinΩ+ΩχsinχcosΩ,
dlc=βχsinχsin(Ω+2D).
Jeq=JCB(Φeq)JLB(Γeq,ψeq)=(aeq+ibeqceq+ideqceq+ideqaeqibeq),
JCB(Φeq)=(cosΦeqsinΦeqsinΦeqcosΦeq)
JLB=(cosΓeq2+isinΓeq2cos2ψeqisinΓeq2sin2ψeqisinΓeq2sin2ψeqcosΓeq2isinΓeq2cos2ψeq),
aeq=cosΓeq2cosΦeq,
beq=sinΓeq2cos(2ψeqΦeq),
ceq=cosΓeq2sinΦeq,
deq=sinΓeq2sin(2ψeqΦeq).
Ω=2ψeqΦeq2D,
β=ΩtanΓeq2csc(Φeq+Ω)=(2ψeqΦeq2D)tanΓeq2csc(2ψeq2D).
cosχ=cosΓeq2cos(2ψeq2D).
Meq=(10000m11m12m130m21m22m230m31m32m33),
m11=cos2Γeq2cos2Φeq+sin2Γeq2cos(4ψeq2Φeq),
m12=cos2Γeq2sin2Φeq+sin2Γeq2sin(4ψeq2Φeq),
m13=sinΓeqsin(2Φeq2ψeq),
m21=cos2Γeq2sin2Φeq+sin2Γeq2sin(4ψeq2Φeq),
m22=cos2Γeq2cos2Φeqsin2Γeq2cos(4ψeq2Φeq),
m23=sinΓeqcos(2Φeq2ψeq),
m31=sinΓeqsin2ψeq,
m32=sinΓeqcos2ψeq,
m33=cosΓeq.
Sj=MA(A)MeqMP(P)SL(0°),
SL(0°)=(1100)TI0,
Ij=TSMjI0,
Mj=1+(m11cos2A+m21sin2A)cos2P+(m12cos2A+m22sin2A)sin2P,
m11=I1I3I1+I3,m12=I5I7I5+I7,m21=I2I4I2+I4,andm22=I6I8I6+I8.
ΓM=cos1{1[(m11m22)2+(m12+m21)2]1/2},
ψM=14tan1[2(m11m12+m21m22)m112m122+m212m222],
ΦM=12tan1(m12m21m11+m22).
1st ambiguity combination:(Γ1,ψ1)=(ΓM1,ψM1),2nd ambiguity combination:(Γ2,ψ2)=(ΓM1,ψM2),3rd ambiguity combination:(Γ3,ψ3)=(ΓM2,ψM1),4th ambiguity combination:(Γ4,ψ4)=(ΓM2,ψM2).
Ωmp=2πq+Ωm0,
χ+mn=2nπ+χm0,orχmn=2(n+1)πχm0,
Ωm0=2ψm+ΦM2D,
χm0=cos1[cosΓm2cos(2ψm2D)].
Ωχsinχ=cosΓeq2sin(2ψeq2D).
tan(dlc/blc)=tan(deq/beq).
Ω+2D=2ψeqΦeq,
Ω=2ψm+ΦM2D±2nπ=Ωm0±2nπ,n=0,1,2,3,,
acosΩ=cosχcos2Ω+ΩχsinχsinΩcosΩ=cosΓeq2cosΦeqcosΩ,
csinΩ=cosχsin2Ω+ΩχsinχsinΩcosΩ=cosΓeq2sinΦeqsinΩ.
cosχ=cosΓeq2cos(2ψeq2D).
χm0=cos1[cosΓm2cos(2ψm2D)],
χ+mn=2nπ+χm0,andχmn=2(n+1)πχm0,wheren=0,1,2,3.
asinΩ=cosχcosΩsinΩ+Ωχsinχsin2Ω=cosΓeq2cosΦeqsinΩ,
ccosΩ=cosχcosΩsinΩ+Ωχsinχcos2Ω=cosΓeq2sinΦeqcosΩ,
Ωχsinχ=cosΓeq2sin(2ψeq2D).

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