Abstract

Based on the equivalence theorem of a unitary optical system, we proposed an analytical approach to characterize the cell parameters of a twisted nematic liquid-crystal device (TNLCD) with full-field resolution. The spatial distribution of three characteristic parameters of a TNLCD was measured by using a polarizer–sample–analyzer imaging polarimeter so that the untwisted phase retardation, cell thickness, and twisted angle of a TNLCD can be directly calculated through the explicit expressions as a function of the characteristic parameters. The measured results agree well with the given values. This method can be implemented for characterization of a TNLCD in the manufacturing process.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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. I. Moreno, P. Velasquez, C. R. Fernandez-Pousa, M. M. Sanchez-Lopez, and F. Mateos, “Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display,” J. Appl. Phys. 94, 3697–3702 (2003).
    [CrossRef]
  3. J. Arines, V. Durán, Z. Jaroszewicz, J. Ares, E. Tajahuerce, P. Prado, J. Lancis, S. Bará, and V. Climent, “Measurement and compensation of optical aberrations using a single spatial light modulator,” Opt. Express 15, 15287–15292 (2007).
    [CrossRef]
  4. F. Nakano, M. Isogai, and M. Sato, “Simple method of determining liquid crystal tilt-bias angle,” Jpn. J. Appl. Phys. 19, 2013–2014 (1980).
    [CrossRef]
  5. H. L. Ong, “Cell thickness and surface pretilt angle measurements of a planar liquid-crystal cell with obliquely incidence light,” J. Appl. Phys. 71, 140–144 (1992).
    [CrossRef]
  6. C. Chou, Y. C. Huang, and M. Chang, “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]
  7. C. C. Tsai, C. Chou, C. Y. Han, C. H. Hsieh, K. Y. Liao, and Y. F. Chao, “Determination of optical parameters of a twisted-nematic liquid crystal by phase-sensitive optical heterodyne interferometric ellipsometry,” Appl. Opt. 44, 7509–7514 (2005).
    [CrossRef]
  8. H. C. Wei, C. C. Tsai, L. P. Yu, T. E. Lin, C. J. Yu, M. H. Liu, and C. Chou, “Two-dimensional cell parameters of twisted nematic liquid crystal with an amplitude-sensitive heterodyne ellipsometer,” Appl. Opt. 48, 1628–1634 (2009).
    [CrossRef]
  9. M. H. Liu, W. C. Kuo, H. C. Wei, C. C. Tsai, C. J. Yu, B. J. Liang, and C. Chou, “Cell parameter measurement of a twisted nematic liquid crystal device using interferometric polarimeter under normal incidence,” Opt. Express 18, 8759–8766(2010).
    [CrossRef]
  10. T. C. Yu and Y. L. Lo, “A novel heterodyne polarimeter for the multiple-parameter measurements of twisted nematic liquid crystal cell using a genetic algorithm approach,” J. Lightwave Technol. 25, 946–951 (2007).
    [CrossRef]
  11. M. Kawamura, Y. Goto, and S. Sato, “A two-dimensional pretilt angle distribution measurement of twisted nematic liquid crystal cells using Stokes parameters at plural wavelengths,” Jpn. J. Appl. Phys. 43, 709–714 (2004).
    [CrossRef]
  12. 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]
  13. 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]
  14. 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]
  15. 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]
  16. S. T. Tang and H. S. Kwok, “3×3 matrix for unitary optical systems,” J. Opt. Soc. Am. A 18, 2138–2145 (2001).
    [CrossRef]
  17. S. T. Tang and H. S. Kwok, “Characteristic parameters of liquid crystal cells and their measurements,” J. Disp. Technol. 2, 26–31 (2006).
    [CrossRef]
  18. A. Lien, “The general and simplified Jones matrix representations for the high pretilt twisted nematic cell,” J. Appl. Phys. 67, 2853–2856 (1990).
    [CrossRef]
  19. M. C. van de Hulst, Scattering of Light by Small Particles(Wiley, 1957), p. 44.

2010

2009

2007

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

C. C. Tsai, C. Chou, C. Y. Han, C. H. Hsieh, K. Y. Liao, and Y. F. Chao, “Determination of optical parameters of a twisted-nematic liquid crystal by phase-sensitive optical heterodyne interferometric ellipsometry,” Appl. Opt. 44, 7509–7514 (2005).
[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]

2004

M. Kawamura, Y. Goto, and S. Sato, “A two-dimensional pretilt angle distribution measurement of twisted nematic liquid crystal cells using Stokes parameters at plural wavelengths,” Jpn. J. Appl. Phys. 43, 709–714 (2004).
[CrossRef]

2003

I. Moreno, P. Velasquez, C. R. Fernandez-Pousa, M. M. Sanchez-Lopez, and F. Mateos, “Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display,” J. Appl. Phys. 94, 3697–3702 (2003).
[CrossRef]

2001

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]

C. Chou, Y. C. Huang, and M. Chang, “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]

1992

H. L. Ong, “Cell thickness and surface pretilt angle measurements of a planar liquid-crystal cell with obliquely incidence light,” J. Appl. Phys. 71, 140–144 (1992).
[CrossRef]

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,” J. Appl. Phys. 67, 2853–2856 (1990).
[CrossRef]

1980

F. Nakano, M. Isogai, and M. Sato, “Simple method of determining liquid crystal tilt-bias angle,” Jpn. J. Appl. Phys. 19, 2013–2014 (1980).
[CrossRef]

1941

Ares, J.

Arines, J.

Bará, S.

Chang, M.

Chao, Y. F.

Chou, C.

Climent, V.

Durán, V.

J. Arines, V. Durán, Z. Jaroszewicz, J. Ares, E. Tajahuerce, P. Prado, J. Lancis, S. Bará, and V. Climent, “Measurement and compensation of optical aberrations using a single spatial light modulator,” Opt. Express 15, 15287–15292 (2007).
[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]

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]

Fernandez-Pousa, C. R.

I. Moreno, P. Velasquez, C. R. Fernandez-Pousa, M. M. Sanchez-Lopez, and F. Mateos, “Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display,” J. Appl. Phys. 94, 3697–3702 (2003).
[CrossRef]

Goto, Y.

M. Kawamura, Y. Goto, and S. Sato, “A two-dimensional pretilt angle distribution measurement of twisted nematic liquid crystal cells using Stokes parameters at plural wavelengths,” Jpn. J. Appl. Phys. 43, 709–714 (2004).
[CrossRef]

Han, C. Y.

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]

Hsieh, C. H.

Huang, Y. C.

Hurwitz, H.

Isogai, M.

F. Nakano, M. Isogai, and M. Sato, “Simple method of determining liquid crystal tilt-bias angle,” Jpn. J. Appl. Phys. 19, 2013–2014 (1980).
[CrossRef]

Jaroszewicz, Z.

J. Arines, V. Durán, Z. Jaroszewicz, J. Ares, E. Tajahuerce, P. Prado, J. Lancis, S. Bará, and V. Climent, “Measurement and compensation of optical aberrations using a single spatial light modulator,” Opt. Express 15, 15287–15292 (2007).
[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]

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]

Jones, R. C.

Kawamura, M.

M. Kawamura, Y. Goto, and S. Sato, “A two-dimensional pretilt angle distribution measurement of twisted nematic liquid crystal cells using Stokes parameters at plural wavelengths,” Jpn. J. Appl. Phys. 43, 709–714 (2004).
[CrossRef]

Kuo, W. C.

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.

J. Arines, V. Durán, Z. Jaroszewicz, J. Ares, E. Tajahuerce, P. Prado, J. Lancis, S. Bará, and V. Climent, “Measurement and compensation of optical aberrations using a single spatial light modulator,” Opt. Express 15, 15287–15292 (2007).
[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]

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]

Liang, B. J.

Liao, K. Y.

Lien, A.

A. Lien, “The general and simplified Jones matrix representations for the high pretilt twisted nematic cell,” J. Appl. Phys. 67, 2853–2856 (1990).
[CrossRef]

Lin, T. E.

Liu, M. H.

Lo, Y. L.

Lu, K.

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]

Mateos, F.

I. Moreno, P. Velasquez, C. R. Fernandez-Pousa, M. M. Sanchez-Lopez, and F. Mateos, “Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display,” J. Appl. Phys. 94, 3697–3702 (2003).
[CrossRef]

Moreno, I.

I. Moreno, P. Velasquez, C. R. Fernandez-Pousa, M. M. Sanchez-Lopez, and F. Mateos, “Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display,” J. Appl. Phys. 94, 3697–3702 (2003).
[CrossRef]

Nakano, F.

F. Nakano, M. Isogai, and M. Sato, “Simple method of determining liquid crystal tilt-bias angle,” Jpn. J. Appl. Phys. 19, 2013–2014 (1980).
[CrossRef]

Ong, H. L.

H. L. Ong, “Cell thickness and surface pretilt angle measurements of a planar liquid-crystal cell with obliquely incidence light,” J. Appl. Phys. 71, 140–144 (1992).
[CrossRef]

Prado, P.

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]

Sanchez-Lopez, M. M.

I. Moreno, P. Velasquez, C. R. Fernandez-Pousa, M. M. Sanchez-Lopez, and F. Mateos, “Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display,” J. Appl. Phys. 94, 3697–3702 (2003).
[CrossRef]

Sato, M.

F. Nakano, M. Isogai, and M. Sato, “Simple method of determining liquid crystal tilt-bias angle,” Jpn. J. Appl. Phys. 19, 2013–2014 (1980).
[CrossRef]

Sato, S.

M. Kawamura, Y. Goto, and S. Sato, “A two-dimensional pretilt angle distribution measurement of twisted nematic liquid crystal cells using Stokes parameters at plural wavelengths,” Jpn. J. Appl. Phys. 43, 709–714 (2004).
[CrossRef]

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]

Tajahuerce, E.

J. Arines, V. Durán, Z. Jaroszewicz, J. Ares, E. Tajahuerce, P. Prado, J. Lancis, S. Bará, and V. Climent, “Measurement and compensation of optical aberrations using a single spatial light modulator,” Opt. Express 15, 15287–15292 (2007).
[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]

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]

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, C. C.

van de Hulst, M. C.

M. C. van de Hulst, Scattering of Light by Small Particles(Wiley, 1957), p. 44.

Velasquez, P.

I. Moreno, P. Velasquez, C. R. Fernandez-Pousa, M. M. Sanchez-Lopez, and F. Mateos, “Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display,” J. Appl. Phys. 94, 3697–3702 (2003).
[CrossRef]

Wei, H. C.

Yu, C. J.

Yu, L. P.

Yu, T. C.

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.

H. L. Ong, “Cell thickness and surface pretilt angle measurements of a planar liquid-crystal cell with obliquely incidence light,” J. Appl. Phys. 71, 140–144 (1992).
[CrossRef]

I. Moreno, P. Velasquez, C. R. Fernandez-Pousa, M. M. Sanchez-Lopez, and F. Mateos, “Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display,” J. Appl. Phys. 94, 3697–3702 (2003).
[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]

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]

A. Lien, “The general and simplified Jones matrix representations for the high pretilt twisted nematic cell,” J. Appl. Phys. 67, 2853–2856 (1990).
[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. Lightwave Technol.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Jpn. J. Appl. Phys.

F. Nakano, M. Isogai, and M. Sato, “Simple method of determining liquid crystal tilt-bias angle,” Jpn. J. Appl. Phys. 19, 2013–2014 (1980).
[CrossRef]

M. Kawamura, Y. Goto, and S. Sato, “A two-dimensional pretilt angle distribution measurement of twisted nematic liquid crystal cells using Stokes parameters at plural wavelengths,” Jpn. J. Appl. Phys. 43, 709–714 (2004).
[CrossRef]

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]

Opt. Eng.

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

Other

M. C. van de Hulst, Scattering of Light by Small Particles(Wiley, 1957), p. 44.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1.
Fig. 1.

Optical setup of a polarizer–sample–analyzer imaging polarimeter. P, polarizers; TNLCD, twisted nematic liquid crystal device; A, analyzer.

Fig. 2.
Fig. 2.

Spatial distributions of the characteristic parameters of a TNLCD: (a) linear phase retardation Γeq, (b) fast axis orientation ψeq of the equivalently linear phase retarder, (c) polarization rotation angle of the equivalent polarization rotator, and the spatial distributions of the cell parameters of a TNLCD: (d) untwisted phase retardation β, (e) cell thickness d, and (f) twisted angle Ω.

Tables (3)

Tables Icon

Table 1. Output Intensity Ij at Different Azimuth Angles of Polarizer (ξ) and Analyzer (η)

Tables Icon

Table 2. Measured Results of the Characteristic Parameter of a TNLCD

Tables Icon

Table 3. Measured Results of the Cell Parameters of a TNLCD

Equations (45)

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

JTNLCD0=(cosχcosΩ+ΩχsinχsinΩ+iβχsinχcosΩcosχsinΩ+ΩχsinγcosΩ+iβχsinχsinΩcosχsinΩΩχsinχcosΩ+iβχsinχsinΩcosχcosΩ+ΩχsinγsinΩiβχsinχcosΩ).
β=πΔnd/λ0,
χ2=β2+Ω2,
Δn=neff(α)no=[neno/(no2cos2α+ne2sin2α)1/2]no,
JTNLCD=R(D)JTNLCD0R(D)=(T11lcT12lcT21lcT22lc),
R(D)=(cosDsinDsinDcosD),
T11lc=cosχcosΩ+ΩχsinχsinΩ+iβχsinχcos(Ω+2D),
T12lc=cosχsinΩ+ΩχsinχcosΩ+iβχsinχsin(Ω+2D),
T21lc=cosχsinΩΩχsinχcosΩ+iβχsinχsin(Ω+2D),
T22lc=cosχcosΩ+ΩχsinχsinΩiβχsinχcos(Ω+2D).
Jeq=JCB(Φeq)JLB(Γeq,ψeq)=(T11eqT12eqT21eqT22eq),
JLB(Γeq,ψeq)=(cosψeqsinψeqsinψeqcosψeq)(exp(iΓeq2)00exp(iΓeq2))(cosψeqsinψeqsinψeqcosψeq)=(cosΓeq2+isinΓeq2cos2ψeqisinΓeq2sin2ψeqisinΓeq2sin2ψeqcosΓeq2isinΓeq2cos2ψeq),
JCB(Φeq)=(cosΦeqsinΦeqsinΦeqcosΦeq).
T11eq=cosΓeq2cosΦeq+isinΓeq2cos(2ψeqΦeq),
T12eq=cosΓeq2sinΦeq+isinΓeq2sin(2ψeqΦeq),
T21eq=cosΓeq2sinΦeq+isinΓeq2sin(2ψeqΦeq),
T22eq=cosΓeq2cosΦeqisinΓeq2cos(2ψeqΦeq).
cosΓeq2cosΦeq=cosχcosΩ+ΩχsinχsinΩ=Re(T11),
sinΓeq2cos(2ψeqΦeq)=βχsinχcos(Ω+2D)=Im(T11),
cosΓeq2sinΦeq=cosχsinΩ+ΩχsinχcosΩ=Re(T12),
sinΓeq2sin(2ψeqΦeq)=βχsinχsin(Ω+2D)=Im(T12).
sinΓeq2=βχsinχ,
cos(2ψeqΦeq)=cos(Ω+2D),
sin(2ψeqΦeq)=sin(Ω+2D).
Ω=2ψeqΦeq2D.
[Re(T11)]sinΩ+[Re(T12)]cosΩ=Ωχsinχ=cosΓeq2sin(Φeq+Ω).
β=ΩtanΓeq2csc(Ω+Φeq).
d=λ0β/πΔn.
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.
Γeq=cos1{[(m11+m22)2+(m12m21)2]1/21},
ψeq=14tan1[2(m11m12+m21m22)m112m122+m212m222],
Φeq=12tan1(m12m21m11+m22).
Sj=MA(η)MeqMP(ξ)SIN.
Ij=TMjI0,
Mj=1+(m11cos2ξ+m12sin2ξ)cos2η+(m21cos2ξ+m22sin2ξ)sin2η.
m11=I1I3I1+I3,m12=I5I7I5+I7,m21=I2I4I2+I4,m22=I6I8I6+I8.

Metrics