Abstract

What is believed to be a novel phase-sensitive optical heterodyne interferometric ellipsometer is set up to characterize a twisted-nematic liquid crystal (TN-LC) by the elliptical parameters of the output polarization state. This ellipsometer presents the advantages of both polarized optical heterodyne interferometry and optical photometry, which introduce a polarization modulation that is capable of performing with high-sensitivity on phase detection in real time. The twist angle Φ and the untwisted phase retardation Γ of TN-LC are measured precisely. The experimental results verify that a TN-LC can be treated as identical to an elliptical retarder.

© 2005 Optical Society of America

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References

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  1. C. Chou, Y. C. Huang, M. Chang, “Polarized common path optical heterodyne interferometer for measuring the elliptical birefringence of a quartz wave plate,” Jpn. J. Appl. Phys. 35, 5526–5530 (1996).
    [CrossRef]
  2. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1979), p. 99.
  3. H. K. Teng, C. Chou, C. N. Chang, C. W. Lyu, Y. C. Huang, “Linear birefringence measurement with a differential-phase optical heterodyne polarimeter,” Jpn. J. Appl. Phys. 41, 3140–3144 (2002).
    [CrossRef]
  4. C. H. Lin, C. Chou, K. S. Chang, “Real time interferometric ellipsometry with optical heterodyne and phase lock-in techniques,” Appl. Opt. 29, 5159–5162 (1990).
    [CrossRef] [PubMed]
  5. B. R. Lin, “TN-LC as an elliptical retarder,” Master’s thesis (Institute of Electro-Optical Engineering, National Chao-Tung University, 2004), p. 19.
  6. P. Yeh, C. Gu, Optics of Liquid Crystal Display (Wiley, 1999), pp. 129–130.
  7. Y. C. Huang, M. Chang, C. Chou, “Effect of elliptical birefringence on the measurement of the phase retardation of a quartz wave plate by an optical heterodyne polarimeter,” J. Opt. Soc. Am. A 14, 1367–1372 (1997).
    [CrossRef]
  8. T. Nishiok, T. Kurata, “Novel pretilt angle measure method for twisted-nematic liquid-display cells by apparent retardation measurement,” Jpn. J. Appl. Phys. 40, 6017–6023 (2001).
    [CrossRef]
  9. J. A. Davis, I. Moreno, P. Tsai, “Polarization eigenstates for twisted-nematic liquid-crstal display,” Appl. Opt. 37, 937–945 (1998).
    [CrossRef]
  10. I. Moreno, J. A. Davis, K. G. D’Nelly, D. B. Allison, “Transmission and phase measurement for polarization eigenvectors in twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 37, 3048–3052 (1998).
    [CrossRef]
  11. X. Zhu, Q. Hong, Y. Huang, S. T. Wu, “Eigenmodes of a reflective twisted-nematic liquid-crystal cell,” J. Appl. Phys. 94, 2868–2873 (2003).
    [CrossRef]
  12. I. Scierski, F. Ratajczyk, “The Jones matrix of the real dicroic elliptic object,” Optik 68, 121–125 (1984).
  13. J. A. Davis, J. Nicolas, A. Márquez, “Phasor analysis of eigenvectors generated in liquid-crystal displays,” Appl. Opt. 22, 4579–4584 (2002).
    [CrossRef]
  14. Y. Zhou, Z. He, 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]
  15. J. A. Davis, D. B. Allison, K. G. D’Nelly, M. L. Wilson, I. Moreno, “Ambiguities in measuring the physical parameters for twisted-nematic liquid-crystal spatial light modulators,” Opt. Eng. 38, 705–709 (1999).
    [CrossRef]
  16. H. Kim, Y. H. Lee, “Unique measurement of the parameters of a twisted-nematic liquid-crystal display,” Appl. Opt. 44, 1642–1649 (2005).
    [CrossRef] [PubMed]
  17. N. Konforti, E. Marom, S. T. Wu, “Phase-only modulation with twisted nematic liquid-crystal spatial light modulators,” Opt. Let. 13, 251–253 (1998).
    [CrossRef]
  18. S. Valyukh, A. Slobodyanyuk, V. Sorokin, “Simulation of obliquely incident light propagation through a general twisted nematic liquid crystal cell by the Jones matrix technique,” Semicond. Phys., Quantum Electron. Optoelectron. 3, 258–263 (2000).
  19. R. Giust, J. P. Goedgebuer, “Determination of the twist angle and the retardation properties of twisted nematic liquid crystal television by spectral measurements,” Opt. Eng. 37, 629–634 (1998).
    [CrossRef]

2005 (1)

2003 (1)

X. Zhu, Q. Hong, Y. Huang, S. T. Wu, “Eigenmodes of a reflective twisted-nematic liquid-crystal cell,” J. Appl. Phys. 94, 2868–2873 (2003).
[CrossRef]

2002 (2)

H. K. Teng, C. Chou, C. N. Chang, C. W. Lyu, Y. C. Huang, “Linear birefringence measurement with a differential-phase optical heterodyne polarimeter,” Jpn. J. Appl. Phys. 41, 3140–3144 (2002).
[CrossRef]

J. A. Davis, J. Nicolas, A. Márquez, “Phasor analysis of eigenvectors generated in liquid-crystal displays,” Appl. Opt. 22, 4579–4584 (2002).
[CrossRef]

2001 (1)

T. Nishiok, T. Kurata, “Novel pretilt angle measure method for twisted-nematic liquid-display cells by apparent retardation measurement,” Jpn. J. Appl. Phys. 40, 6017–6023 (2001).
[CrossRef]

2000 (1)

S. Valyukh, A. Slobodyanyuk, V. Sorokin, “Simulation of obliquely incident light propagation through a general twisted nematic liquid crystal cell by the Jones matrix technique,” Semicond. Phys., Quantum Electron. Optoelectron. 3, 258–263 (2000).

1999 (1)

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

1998 (4)

R. Giust, J. P. Goedgebuer, “Determination of the twist angle and the retardation properties of twisted nematic liquid crystal television by spectral measurements,” Opt. Eng. 37, 629–634 (1998).
[CrossRef]

N. Konforti, E. Marom, S. T. Wu, “Phase-only modulation with twisted nematic liquid-crystal spatial light modulators,” Opt. Let. 13, 251–253 (1998).
[CrossRef]

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

I. Moreno, J. A. Davis, K. G. D’Nelly, D. B. Allison, “Transmission and phase measurement for polarization eigenvectors in twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 37, 3048–3052 (1998).
[CrossRef]

1997 (2)

Y. C. Huang, M. Chang, C. Chou, “Effect of elliptical birefringence on the measurement of the phase retardation of a quartz wave plate by an optical heterodyne polarimeter,” J. Opt. Soc. Am. A 14, 1367–1372 (1997).
[CrossRef]

Y. Zhou, Z. He, 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 (1)

C. Chou, Y. C. Huang, M. Chang, “Polarized common path optical heterodyne interferometer for measuring the elliptical birefringence of a quartz wave plate,” Jpn. J. Appl. Phys. 35, 5526–5530 (1996).
[CrossRef]

1990 (1)

1984 (1)

I. Scierski, F. Ratajczyk, “The Jones matrix of the real dicroic elliptic object,” Optik 68, 121–125 (1984).

Allison, D. B.

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

I. Moreno, J. A. Davis, K. G. D’Nelly, D. B. Allison, “Transmission and phase measurement for polarization eigenvectors in twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 37, 3048–3052 (1998).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1979), p. 99.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1979), p. 99.

Chang, C. N.

H. K. Teng, C. Chou, C. N. Chang, C. W. Lyu, Y. C. Huang, “Linear birefringence measurement with a differential-phase optical heterodyne polarimeter,” Jpn. J. Appl. Phys. 41, 3140–3144 (2002).
[CrossRef]

Chang, K. S.

Chang, M.

Y. C. Huang, M. Chang, C. Chou, “Effect of elliptical birefringence on the measurement of the phase retardation of a quartz wave plate by an optical heterodyne polarimeter,” J. Opt. Soc. Am. A 14, 1367–1372 (1997).
[CrossRef]

C. Chou, Y. C. Huang, M. Chang, “Polarized common path optical heterodyne interferometer for measuring the elliptical birefringence of a quartz wave plate,” Jpn. J. Appl. Phys. 35, 5526–5530 (1996).
[CrossRef]

Chou, C.

H. K. Teng, C. Chou, C. N. Chang, C. W. Lyu, Y. C. Huang, “Linear birefringence measurement with a differential-phase optical heterodyne polarimeter,” Jpn. J. Appl. Phys. 41, 3140–3144 (2002).
[CrossRef]

Y. C. Huang, M. Chang, C. Chou, “Effect of elliptical birefringence on the measurement of the phase retardation of a quartz wave plate by an optical heterodyne polarimeter,” J. Opt. Soc. Am. A 14, 1367–1372 (1997).
[CrossRef]

C. Chou, Y. C. Huang, M. Chang, “Polarized common path optical heterodyne interferometer for measuring the elliptical birefringence of a quartz wave plate,” Jpn. J. Appl. Phys. 35, 5526–5530 (1996).
[CrossRef]

C. H. Lin, C. Chou, K. S. Chang, “Real time interferometric ellipsometry with optical heterodyne and phase lock-in techniques,” Appl. Opt. 29, 5159–5162 (1990).
[CrossRef] [PubMed]

D’Nelly, K. G.

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

I. Moreno, J. A. Davis, K. G. D’Nelly, D. B. Allison, “Transmission and phase measurement for polarization eigenvectors in twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 37, 3048–3052 (1998).
[CrossRef]

Davis, J. A.

J. A. Davis, J. Nicolas, A. Márquez, “Phasor analysis of eigenvectors generated in liquid-crystal displays,” Appl. Opt. 22, 4579–4584 (2002).
[CrossRef]

J. A. Davis, D. B. Allison, K. G. D’Nelly, M. L. Wilson, 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, P. Tsai, “Polarization eigenstates for twisted-nematic liquid-crstal display,” Appl. Opt. 37, 937–945 (1998).
[CrossRef]

I. Moreno, J. A. Davis, K. G. D’Nelly, D. B. Allison, “Transmission and phase measurement for polarization eigenvectors in twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 37, 3048–3052 (1998).
[CrossRef]

Giust, R.

R. Giust, J. P. Goedgebuer, “Determination of the twist angle and the retardation properties of twisted nematic liquid crystal television by spectral measurements,” Opt. Eng. 37, 629–634 (1998).
[CrossRef]

Goedgebuer, J. P.

R. Giust, J. P. Goedgebuer, “Determination of the twist angle and the retardation properties of twisted nematic liquid crystal television by spectral measurements,” Opt. Eng. 37, 629–634 (1998).
[CrossRef]

Gu, C.

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

He, Z.

Y. Zhou, Z. He, 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]

Hong, Q.

X. Zhu, Q. Hong, Y. Huang, S. T. Wu, “Eigenmodes of a reflective twisted-nematic liquid-crystal cell,” J. Appl. Phys. 94, 2868–2873 (2003).
[CrossRef]

Huang, Y.

X. Zhu, Q. Hong, Y. Huang, S. T. Wu, “Eigenmodes of a reflective twisted-nematic liquid-crystal cell,” J. Appl. Phys. 94, 2868–2873 (2003).
[CrossRef]

Huang, Y. C.

H. K. Teng, C. Chou, C. N. Chang, C. W. Lyu, Y. C. Huang, “Linear birefringence measurement with a differential-phase optical heterodyne polarimeter,” Jpn. J. Appl. Phys. 41, 3140–3144 (2002).
[CrossRef]

Y. C. Huang, M. Chang, C. Chou, “Effect of elliptical birefringence on the measurement of the phase retardation of a quartz wave plate by an optical heterodyne polarimeter,” J. Opt. Soc. Am. A 14, 1367–1372 (1997).
[CrossRef]

C. Chou, Y. C. Huang, M. Chang, “Polarized common path optical heterodyne interferometer for measuring the elliptical birefringence of a quartz wave plate,” Jpn. J. Appl. Phys. 35, 5526–5530 (1996).
[CrossRef]

Kim, H.

Konforti, N.

N. Konforti, E. Marom, S. T. Wu, “Phase-only modulation with twisted nematic liquid-crystal spatial light modulators,” Opt. Let. 13, 251–253 (1998).
[CrossRef]

Kurata, T.

T. Nishiok, T. Kurata, “Novel pretilt angle measure method for twisted-nematic liquid-display cells by apparent retardation measurement,” Jpn. J. Appl. Phys. 40, 6017–6023 (2001).
[CrossRef]

Lee, Y. H.

Lin, B. R.

B. R. Lin, “TN-LC as an elliptical retarder,” Master’s thesis (Institute of Electro-Optical Engineering, National Chao-Tung University, 2004), p. 19.

Lin, C. H.

Lyu, C. W.

H. K. Teng, C. Chou, C. N. Chang, C. W. Lyu, Y. C. Huang, “Linear birefringence measurement with a differential-phase optical heterodyne polarimeter,” Jpn. J. Appl. Phys. 41, 3140–3144 (2002).
[CrossRef]

Marom, E.

N. Konforti, E. Marom, S. T. Wu, “Phase-only modulation with twisted nematic liquid-crystal spatial light modulators,” Opt. Let. 13, 251–253 (1998).
[CrossRef]

Márquez, A.

J. A. Davis, J. Nicolas, A. Márquez, “Phasor analysis of eigenvectors generated in liquid-crystal displays,” Appl. Opt. 22, 4579–4584 (2002).
[CrossRef]

Moreno, I.

J. A. Davis, D. B. Allison, K. G. D’Nelly, M. L. Wilson, 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, P. Tsai, “Polarization eigenstates for twisted-nematic liquid-crstal display,” Appl. Opt. 37, 937–945 (1998).
[CrossRef]

I. Moreno, J. A. Davis, K. G. D’Nelly, D. B. Allison, “Transmission and phase measurement for polarization eigenvectors in twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 37, 3048–3052 (1998).
[CrossRef]

Nicolas, J.

J. A. Davis, J. Nicolas, A. Márquez, “Phasor analysis of eigenvectors generated in liquid-crystal displays,” Appl. Opt. 22, 4579–4584 (2002).
[CrossRef]

Nishiok, T.

T. Nishiok, T. Kurata, “Novel pretilt angle measure method for twisted-nematic liquid-display cells by apparent retardation measurement,” Jpn. J. Appl. Phys. 40, 6017–6023 (2001).
[CrossRef]

Ratajczyk, F.

I. Scierski, F. Ratajczyk, “The Jones matrix of the real dicroic elliptic object,” Optik 68, 121–125 (1984).

Sato, S.

Y. Zhou, Z. He, 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]

Scierski, I.

I. Scierski, F. Ratajczyk, “The Jones matrix of the real dicroic elliptic object,” Optik 68, 121–125 (1984).

Slobodyanyuk, A.

S. Valyukh, A. Slobodyanyuk, V. Sorokin, “Simulation of obliquely incident light propagation through a general twisted nematic liquid crystal cell by the Jones matrix technique,” Semicond. Phys., Quantum Electron. Optoelectron. 3, 258–263 (2000).

Sorokin, V.

S. Valyukh, A. Slobodyanyuk, V. Sorokin, “Simulation of obliquely incident light propagation through a general twisted nematic liquid crystal cell by the Jones matrix technique,” Semicond. Phys., Quantum Electron. Optoelectron. 3, 258–263 (2000).

Teng, H. K.

H. K. Teng, C. Chou, C. N. Chang, C. W. Lyu, Y. C. Huang, “Linear birefringence measurement with a differential-phase optical heterodyne polarimeter,” Jpn. J. Appl. Phys. 41, 3140–3144 (2002).
[CrossRef]

Tsai, P.

Valyukh, S.

S. Valyukh, A. Slobodyanyuk, V. Sorokin, “Simulation of obliquely incident light propagation through a general twisted nematic liquid crystal cell by the Jones matrix technique,” Semicond. Phys., Quantum Electron. Optoelectron. 3, 258–263 (2000).

Wilson, M. L.

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

Wu, S. T.

X. Zhu, Q. Hong, Y. Huang, S. T. Wu, “Eigenmodes of a reflective twisted-nematic liquid-crystal cell,” J. Appl. Phys. 94, 2868–2873 (2003).
[CrossRef]

N. Konforti, E. Marom, S. T. Wu, “Phase-only modulation with twisted nematic liquid-crystal spatial light modulators,” Opt. Let. 13, 251–253 (1998).
[CrossRef]

Yeh, P.

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

Zhou, Y.

Y. Zhou, Z. He, 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]

Zhu, X.

X. Zhu, Q. Hong, Y. Huang, S. T. Wu, “Eigenmodes of a reflective twisted-nematic liquid-crystal cell,” J. Appl. Phys. 94, 2868–2873 (2003).
[CrossRef]

Appl. Opt. (4)

J. Appl. Phys. (1)

X. Zhu, Q. Hong, Y. Huang, S. T. Wu, “Eigenmodes of a reflective twisted-nematic liquid-crystal cell,” J. Appl. Phys. 94, 2868–2873 (2003).
[CrossRef]

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

Jpn. J. Appl. Phys. (4)

T. Nishiok, T. Kurata, “Novel pretilt angle measure method for twisted-nematic liquid-display cells by apparent retardation measurement,” Jpn. J. Appl. Phys. 40, 6017–6023 (2001).
[CrossRef]

C. Chou, Y. C. Huang, M. Chang, “Polarized common path optical heterodyne interferometer for measuring the elliptical birefringence of a quartz wave plate,” Jpn. J. Appl. Phys. 35, 5526–5530 (1996).
[CrossRef]

Y. Zhou, Z. He, 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]

H. K. Teng, C. Chou, C. N. Chang, C. W. Lyu, Y. C. Huang, “Linear birefringence measurement with a differential-phase optical heterodyne polarimeter,” Jpn. J. Appl. Phys. 41, 3140–3144 (2002).
[CrossRef]

Opt. Eng. (3)

R. Giust, J. P. Goedgebuer, “Determination of the twist angle and the retardation properties of twisted nematic liquid crystal television by spectral measurements,” Opt. Eng. 37, 629–634 (1998).
[CrossRef]

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

I. Moreno, J. A. Davis, K. G. D’Nelly, D. B. Allison, “Transmission and phase measurement for polarization eigenvectors in twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 37, 3048–3052 (1998).
[CrossRef]

Opt. Let. (1)

N. Konforti, E. Marom, S. T. Wu, “Phase-only modulation with twisted nematic liquid-crystal spatial light modulators,” Opt. Let. 13, 251–253 (1998).
[CrossRef]

Optik (1)

I. Scierski, F. Ratajczyk, “The Jones matrix of the real dicroic elliptic object,” Optik 68, 121–125 (1984).

Semicond. Phys., Quantum Electron. Optoelectron. (1)

S. Valyukh, A. Slobodyanyuk, V. Sorokin, “Simulation of obliquely incident light propagation through a general twisted nematic liquid crystal cell by the Jones matrix technique,” Semicond. Phys., Quantum Electron. Optoelectron. 3, 258–263 (2000).

Other (3)

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1979), p. 99.

B. R. Lin, “TN-LC as an elliptical retarder,” Master’s thesis (Institute of Electro-Optical Engineering, National Chao-Tung University, 2004), p. 19.

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

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Figures (7)

Fig. 1
Fig. 1

Experimental setup: HWP, half-wave plate; BS, beam splitter; AOM, acousto-optic modulator; M, mirror; A, analyzer; PBS, polarization beam splitter; D, detector; BPF, bandpass filter; LIA, lock-in amplifier; PC, personal computer.

Fig. 2
Fig. 2

Experimental data of phase retardation versus rotation angle of TN-LC by this setup: (a) without phase-bias correction, (b) with phase-bias correction.

Fig. 3
Fig. 3

Schematic diagram of azimuth angle θ and rubbing angle α of the TN-LC: c indicates the optical axis, i is the direction of rubbing in of the TN-LC, and o is the direction of rubbing out of the TN-LC.

Fig. 4
Fig. 4

Measured phase retardation versus mechanical rotation angle β of the TN-LC.

Fig. 5
Fig. 5

Theoretical calculation of phase retardation versus azimuth angle θ of the TN-LC.

Fig. 6
Fig. 6

Theoretical calculation of phase retardation versus rubbing angle α of the TN-LC.

Fig. 7
Fig. 7

Phase retardation versus rotation angle of the TN-LC under the conditions of (a) the mechanical rotation of the TN-LC as marker (●), (b) the theoretical calculation based on the elliptical phase retarder as marker (--), (c) the theoretical calculation based on a linear TN-LC coincided with (b).

Equations (13)

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I p ( Δ ω t ) = | E p 1 + E p 2 | 2 = I p 1 + I p 2 + 2 I p 1 I p 2 cos ( Δ ω t + δ p ) ,
I s ( Δ ω t ) = | E s 1 + E s 2 | 2 = I s 1 + I s 2 + 2 I s 1 I s 2 cos ( Δ ω t + δ s ) ,
X = E s E p exp [ i ( δ s δ p ) ] = | X | exp ( i δ )
X ( o ) = T 22 X ( i ) + T 21 T 12 X ( i ) + T 11 ,
T ε = [ T 11 T 12 T 21 T 22 ] = [ cos γ 2 + i cos 2 ε sin γ 2 cos 2 ( θ + π 2 ) sin γ 2 sin 2 ε + i cos 2 ε sin γ 2 sin 2 ( θ + π 2 ) sin γ 2 sin 2 ε + i cos 2 ε sin γ 2 sin 2 ( θ + π 2 ) cos γ 2 i cos 2 ε sin γ 2 cos 2 ( θ + π 2 ) ] ,
X ( o ) = T 22 + T 21 T 12 + T 11 .
X ( o ) = | X ( o ) | exp [ i δ ( o ) ] ,
δ ε ( 0 ) = tan 1 ( { 2 sin γ 2 cos 2 ε [ sin γ 2 sin 2 ε × sin 2 ( θ + π 2 ) cos γ 2 cos 2 ( θ + π 2 ) ] } / [ cos 2 γ 2 sin 2 γ 2 sin 2 2 ε 1 2 sin 2 γ 2 cos 2 2 ε × cos 4 ( θ + π 2 ) ] ) .
T LC = [ T 11 T 12 T 21 T 22 ] = [ cos χ cos Φ + Φ χ sin χ sin Φ i Γ / 2 χ sin χ cos ( 2 α + Φ ) cos χ sin Φ + Φ χ sin χ cos Φ i Γ / 2 χ sin χ sin ( 2 α + Φ ) cos χ sin Φ Φ χ sin χ cos Φ i Γ / 2 χ sin χ sin ( 2 α + Φ ) cos χ cos Φ + Φ χ sin χ sin Φ + i Γ / 2 χ sin χ cos ( 2 α + Φ ) ] .
δ LC ( 0 ) = tan 1 ( { 2 ( Γ / 2 χ ) sin χ [ cos χ cos 2 α ( Φ χ ) × sin χ sin 2 α ] } / { cos 2 Φ [ cos 2 χ ( Φ χ ) 2 × sin 2 χ ] + ( Φ χ ) sin 2 χ sin 2 Φ ( Γ / 2 χ ) 2 × sin 2 χ cos ( 4 α + 2 Φ ) } ) ,
χ = Φ 2 + ( Γ 2 ) 2 ,
δ ε ( o ) ( γ , ε , θ ) = δ LC ( o ) ( Γ , Φ , α )
α = θ Φ 2 .

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