Abstract

Twisted-nematic liquid-crystal devices having high spatial resolution are suitable for spatial light modulators. Phase-modulation characteristics of the devices have been widely studied, but the phase delay calculated using a conventional Jones-matrix model is slightly different from that measured using an interferometer. We propose a modified model whose matrix components are described by angular parameters that are related to the distribution of twist and tilt angles inside the liquid-crystal layer through differential equations. The model is used to simulate phase-modulation characteristics, and the result agrees well with the experimentally measured phase delay.

© 2005 Optical Society of America

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References

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  1. T. H. Barnes, T. Eiju, K. Matsuda, N. Ooyama, “Phase-only modulation using a twisted nematic liquid crystal television,” Appl. Opt. 28, 4845–4852 (1989).
    [CrossRef] [PubMed]
  2. L. G. Neto, D. Roberge, Y. Sheng, “Programmable optical phase-mostly holograms with coupled-mode modulation liquid-crystal television,” Appl. Opt. 34, 1944–1950 (1995).
    [CrossRef] [PubMed]
  3. A. Ogiwara, H. Sakai, J. Ohtsubo, “Application of LCTV to nonlinear speckle correlator,” Opt. Commun. 86, 513–522 (1991).
    [CrossRef]
  4. M. Yamauchi, T. Eiju, “Optimization of twisted nematic liquid crystal panels for spatial light phase modulation,” Opt. Commun. 115, 19–25 (1995).
    [CrossRef]
  5. J. A. Davis, I. Moreno, P. Tsai, “Polarization eigenstates for twisted-nematic liquid-crystal displays,” Appl. Opt. 37, 937–945 (1998).
    [CrossRef]
  6. K. Lu, B. E. A. Saleh, “Theory and design of the liquid crystal TV as an optical spatial phase modulator,” Opt. Eng. 29, 240–246 (1990).
    [CrossRef]
  7. 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]
  8. 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]
  9. J. A. Coy, M. Zaldarriaga, D. F. Grosz, O. E. Martinez, “Characterization of a liquid crystal television as a programmable spatial light modulator,” Opt. Eng. 35, 15–19 (1996).
    [CrossRef]
  10. A. Marquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
    [CrossRef]
  11. A. Marquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. 40, 2558–2564 (2001).
    [CrossRef]
  12. D. W. Berrman, “Optics in smoothly varying anisotropic planar structures: application to liquid-crystal twist cells,” J. Opt. Soc. Am. 63, 1374–1380 (1973).
    [CrossRef]
  13. F. C. Frank, “On the theory of liquid crystals,” Discuss. Faraday Soc. 25, 19–28 (1958).
    [CrossRef]

2002 (1)

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 (1)

A. Marquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. 40, 2558–2564 (2001).
[CrossRef]

2000 (1)

A. Marquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

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 (1)

1996 (1)

J. A. Coy, M. Zaldarriaga, D. F. Grosz, O. E. Martinez, “Characterization of a liquid crystal television as a programmable spatial light modulator,” Opt. Eng. 35, 15–19 (1996).
[CrossRef]

1995 (2)

M. Yamauchi, T. Eiju, “Optimization of twisted nematic liquid crystal panels for spatial light phase modulation,” Opt. Commun. 115, 19–25 (1995).
[CrossRef]

L. G. Neto, D. Roberge, Y. Sheng, “Programmable optical phase-mostly holograms with coupled-mode modulation liquid-crystal television,” Appl. Opt. 34, 1944–1950 (1995).
[CrossRef] [PubMed]

1991 (1)

A. Ogiwara, H. Sakai, J. Ohtsubo, “Application of LCTV to nonlinear speckle correlator,” Opt. Commun. 86, 513–522 (1991).
[CrossRef]

1990 (1)

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

1989 (1)

1973 (1)

1958 (1)

F. C. Frank, “On the theory of liquid crystals,” Discuss. Faraday Soc. 25, 19–28 (1958).
[CrossRef]

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]

Barnes, T. H.

Berrman, D. W.

Campos, J.

A. Marquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. 40, 2558–2564 (2001).
[CrossRef]

A. Marquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

Coy, J. A.

J. A. Coy, M. Zaldarriaga, D. F. Grosz, O. E. Martinez, “Characterization of a liquid crystal television as a programmable spatial light modulator,” Opt. Eng. 35, 15–19 (1996).
[CrossRef]

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]

Davis, J. A.

A. Marquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. 40, 2558–2564 (2001).
[CrossRef]

A. Marquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[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-crystal displays,” Appl. Opt. 37, 937–945 (1998).
[CrossRef]

Eiju, T.

M. Yamauchi, T. Eiju, “Optimization of twisted nematic liquid crystal panels for spatial light phase modulation,” Opt. Commun. 115, 19–25 (1995).
[CrossRef]

T. H. Barnes, T. Eiju, K. Matsuda, N. Ooyama, “Phase-only modulation using a twisted nematic liquid crystal television,” Appl. Opt. 28, 4845–4852 (1989).
[CrossRef] [PubMed]

Frank, F. C.

F. C. Frank, “On the theory of liquid crystals,” Discuss. Faraday Soc. 25, 19–28 (1958).
[CrossRef]

Grosz, D. F.

J. A. Coy, M. Zaldarriaga, D. F. Grosz, O. E. Martinez, “Characterization of a liquid crystal television as a programmable spatial light modulator,” Opt. Eng. 35, 15–19 (1996).
[CrossRef]

Iemmi, C.

A. Marquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. 40, 2558–2564 (2001).
[CrossRef]

A. Marquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

Lu, K.

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

Marquez, A.

A. Marquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. 40, 2558–2564 (2001).
[CrossRef]

A. Marquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

Martinez, O. E.

J. A. Coy, M. Zaldarriaga, D. F. Grosz, O. E. Martinez, “Characterization of a liquid crystal television as a programmable spatial light modulator,” Opt. Eng. 35, 15–19 (1996).
[CrossRef]

Matsuda, K.

Moreno, A.

A. Marquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

Moreno, I.

A. Marquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. 40, 2558–2564 (2001).
[CrossRef]

A. Marquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[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-crystal displays,” Appl. Opt. 37, 937–945 (1998).
[CrossRef]

Neto, L. G.

Ogiwara, A.

A. Ogiwara, H. Sakai, J. Ohtsubo, “Application of LCTV to nonlinear speckle correlator,” Opt. Commun. 86, 513–522 (1991).
[CrossRef]

Ohtsubo, J.

A. Ogiwara, H. Sakai, J. Ohtsubo, “Application of LCTV to nonlinear speckle correlator,” Opt. Commun. 86, 513–522 (1991).
[CrossRef]

Ooyama, N.

Roberge, D.

Robert, A.

A. Marquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

Sakai, H.

A. Ogiwara, H. Sakai, J. Ohtsubo, “Application of LCTV to nonlinear speckle correlator,” Opt. Commun. 86, 513–522 (1991).
[CrossRef]

Saleh, B. E. A.

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

Sheng, Y.

Tsai, P.

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]

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]

M. Yamauchi, T. Eiju, “Optimization of twisted nematic liquid crystal panels for spatial light phase modulation,” Opt. Commun. 115, 19–25 (1995).
[CrossRef]

Yzuel, M. J.

A. Marquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. 40, 2558–2564 (2001).
[CrossRef]

A. Marquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

Zaldarriaga, M.

J. A. Coy, M. Zaldarriaga, D. F. Grosz, O. E. Martinez, “Characterization of a liquid crystal television as a programmable spatial light modulator,” Opt. Eng. 35, 15–19 (1996).
[CrossRef]

Appl. Opt. (3)

Discuss. Faraday Soc. (1)

F. C. Frank, “On the theory of liquid crystals,” Discuss. Faraday Soc. 25, 19–28 (1958).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (2)

A. Ogiwara, H. Sakai, J. Ohtsubo, “Application of LCTV to nonlinear speckle correlator,” Opt. Commun. 86, 513–522 (1991).
[CrossRef]

M. Yamauchi, T. Eiju, “Optimization of twisted nematic liquid crystal panels for spatial light phase modulation,” Opt. Commun. 115, 19–25 (1995).
[CrossRef]

Opt. Eng. (6)

K. Lu, B. E. A. Saleh, “Theory and design of the liquid crystal TV as an optical spatial phase modulator,” Opt. Eng. 29, 240–246 (1990).
[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, 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. Coy, M. Zaldarriaga, D. F. Grosz, O. E. Martinez, “Characterization of a liquid crystal television as a programmable spatial light modulator,” Opt. Eng. 35, 15–19 (1996).
[CrossRef]

A. Marquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

A. Marquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. 40, 2558–2564 (2001).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Twisted-nematic liquid-crystal cell.

Fig. 2
Fig. 2

Definition of azimuth angle ζ and tilt angle η.

Fig. 3
Fig. 3

Distribution of azimuth and tilt angles along the z axis (a), (c), in the off-state and (b), (d), those in the on-state.

Fig. 4
Fig. 4

Distribution of (a) twist angle and (b) birefringence along the z axis defined by Eqs. (5) and (6), respectively. Dotted line, off-state; solid curve, on-state.

Fig. 5
Fig. 5

Distribution of azimuth and tilt angles along the z axis assumed (a), (d), in the conventional model, (b), (e), in the three-layer model, and (c), (f), in the multiple-layer model.

Fig. 6
Fig. 6

Angular parameters of key matrix components.

Fig. 7
Fig. 7

Optical setup for the transmission measurement.

Fig. 8
Fig. 8

Measured values of key matrix components a, b, and c as a function of the gray level.

Fig. 9
Fig. 9

Angular parameters of the key matrix as a function of the gray level.

Fig. 10
Fig. 10

Mach–Zehnder interferometer used to measure the phase delay caused by a TN LCD.

Fig. 11
Fig. 11

Measured and simulated values of the phase delay.

Equations (60)

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ζ ( z ) = α T d z + α T 2 + ψ D             ( - d 2 z d 2 ) ,
η ( z ) = 0             ( - d 2 z d 2 ) .
ζ ( - z ) = α T + 2 ψ D - ζ ( z ) ,
η ( - z ) = η ( z ) .
α ( z ) = ζ ( z ) - ζ ( - z ) = 2 ζ ( z ) - α T - 2 ψ D             ( 0 z d 2 ) ,
β ( z ) = 2 π λ 0 z [ n ( η ) - n o ] d z             ( 0 z d 2 ) ,
1 n 2 ( η ) = cos 2 η n e 2 + sin 2 η n o 2 ,
n ( η ) Δ n cos 2 η + n o             ( for Δ n n o ) ,
β ( z ) 2 π Δ n λ 0 z cos 2 η ( z ) d z .
J = exp [ - i ( ϕ 0 + β T ) ] R ( - ψ D ) R ( - α T ) MR ( ψ D ) ,
ϕ 0 = π d λ ( n e + n o )
β T = β ( d 2 ) .
R ( ξ ) = [ cos ξ sin ξ - sin ξ cos ξ ] .
M C ( α T , β C ) = [ cos γ - i β C γ sin γ α T γ sin γ - α T γ sin γ cos γ + i β C γ sin γ ] ,
γ = α T 2 + β C 2 .
β T = β C = π Δ n d λ cos 2 η C .
j - 1 = exp ( - i β 1 ) R ( - ψ D ) R ( - α 1 ) M C ( α 1 , β 1 ) R ( ψ D ) ,
β 1 = π Δ n d λ cos 2 η 1 .
j 0 = exp ( - i β 0 ) R ( - ψ D - α 1 ) R ( - α 0 ) × M C ( α 0 , β 0 ) R ( ψ D + α 1 ) ,
β 0 = π Δ n d λ cos 2 η 0 .
j 1 = R ( - α 0 - α 1 ) j - 1 R ( α 0 + α 1 ) .
J 3 = j 1 j 0 j - 1 = exp ( - i β T 3 ) R ( - ψ D ) R ( - α T ) × M 3 ( α 0 , β 0 , α 1 , β 1 ) R ( ψ D ) ,
β T 3 = β 0 + 2 β 1 ,
M 3 ( α 0 , β 0 , α 1 , β 1 ) = M C ( α 1 , β 1 ) M C ( α 0 , β 0 ) M C ( α 1 , β 1 ) .
M 3 ( α 0 , β 0 , α 1 , β 1 ) = [ a 3 - i b 3 c 3 - c 3 a 3 + i b 3 ] ,
a 3 = cos γ 0 cos 2 γ 1 - β 0 β 1 γ 0 γ 1 sin γ 0 sin 2 γ 1 ,
b 3 = β 1 γ 1 cos γ 0 sin 2 γ 1 + ( 1 - 2 β 1 2 γ 1 2 sin 2 γ 1 ) × β 0 γ 0 sin γ 0 ,
c 3 = α 0 γ 0 sin γ 0 + 2 ( cos γ 0 cos γ 1 - β 0 β 1 γ 0 γ 1 sin γ 0 sin γ 1 ) α 1 γ 1 sin γ 1 ,
γ k = α k 2 + β k 2 ( k = 0 , 1 ) .
a 3 = cos γ 0 cos 2 β 1 - β 0 γ 0 sin γ 0 sin 2 β 1 ,
b 3 = cos γ 0 sin 2 β 1 + β 0 γ 0 sin γ 0 cos 2 β 1 ,
c 3 = α 0 γ 0 sin γ 0 .
α - k = α k ,
η - k = η k .
β - k = π Δ n λ Δ z cos 2 η - k = β k ,
j k = exp ( - i β k ) R ( - ψ k ) R ( - α k ) M C ( α k , β k ) R ( ψ k ) ,
M C ( p k , q k , s k ) = [ p k - i q k s k - s k p k + i q k ]             ( k = - N , , N ) ,
p k = cos γ k ,
q k = β k γ k sin γ k ,
s k = α k γ k sin γ k ,
γ k = α k 2 + β k 2 ,
p k 2 + q k 2 + s k 2 = 1.
M 2 k + 1 ( a k , b k , c k ) = II j = k - k M C ( p j , q j , s j ) = M C ( p k , q k , s k ) M 2 k - 1 ( a k - 1 , b k - 1 , c k - 1 ) × M C ( p - k , q - k , s - k ) = [ a k - i b k c k - c k a k + i b k ] ( k = 1 , 2 , , N ) ,
a k 2 + b k 2 + c k 2 = 1.
a k = ( p k 2 - q k 2 - s k 2 ) a k - 1 - 2 p k q k b k - 1 - 2 p k s k c k - 1 , b k = 2 p k q k a k - 1 + ( p k 2 - q k 2 + s k 2 ) b k - 1 - 2 q k s k c k - 1 , c k = 2 p k s k a k - 1 - 2 p k s k b k - 1 + ( p k 2 + q k 2 - s k 2 ) c k - 1 .
J 2 N + 1 = II k = N - N j k = exp ( - i β M ) R ( - ψ D ) R ( - α T ) M 2 N + 1 R ( ψ D ) ,
β M = β 0 + 2 k = 1 N β k .
a ( z + 2 Δ z ) - a ( z ) = 2 [ p 2 ( z ) - 1 ] a ( z ) - 2 p ( z ) q ( z ) b ( z ) - 2 p ( z ) s ( z ) c ( z ) , b ( z + 2 Δ z ) - b ( z ) = 2 p ( z ) q ( z ) a ( z ) - 2 q 2 ( z ) b ( z ) - 2 q ( z ) s ( z ) c ( z ) , c ( z + 2 Δ z ) - c ( z ) = 2 p ( z ) s ( z ) a ( z ) - 2 q ( z ) s ( z ) b ( z ) - 2 s 2 ( z ) c ( z ) ,
d a d z = - d β d z b - d α d z c , d b d z = d β d z a , d c d z = d α d z a .
J D = exp ( - i β T ) R ( - ψ D ) R ( - α T ) M D R ( ψ D ) ,
M D ( a , b , c ) = [ a ( d 2 ) - i b ( d 2 ) c ( d 2 ) - c ( d 2 ) a ( d 2 ) + i b ( d 2 ) ] .
tan θ = b 2 + c 2 a ,
tan ϕ = c b .
M D ( θ , ϕ ) = [ cos θ - i sin θ cos ϕ sin θ sin ϕ - sin θ sin ϕ cos θ + i sin θ cos ϕ ] .
d θ d z = sin ϕ d α d z + cos ϕ d β d z , tan θ d ϕ d z = cos ϕ d α d z - sin ϕ d β d z .
θ = α 2 + β 2 , ϕ = tan - 1 α β .
J E = [ f - i g h - i j - h - i j f + i g ] ,
f 2 + g 2 + h 2 + j 2 = 1.
M E = [ a - i b c - c a + i b ] = R ( α T + ψ D ) [ f - i g h - i j - h - i j f + i g ] R ( - ψ D ) .
β C = θ cos ϕ .

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