Abstract

Using the extended Jones calculus for oblique incidence, we have succeeded in representing light propagation through birefringent media on the Poincaré sphere for arbitrary propagation directions. The situation is far more complicated than for the case of perpendicular transmission. To achieve a simple representation method we had to define appropriate conventions. One can use the method to evaluate the efficiency of existing viewing-angle compensation foils. The Poincaré sphere turns out to be a valid geometrical method for the design of new compensation foils that optimize the contrast ratio. The different parameters of a homogeneous compensation foil that is applied to a liquid-crystal display to improve the viewing-angle properties can be systematically determined.

© 1999 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  7. P. Yeh, “Extended Jones matrix method,” J. Opt. Soc. Am. 72, 507–513 (1982).
    [CrossRef]
  8. A. Lien, “Extended Jones matrix representation for the twisted nematic liquid-crystal display at oblique incidence,” Appl. Phys. Lett. 57, 2767–2769 (1990).
    [CrossRef]
  9. A. Lien, “A detailed derivation of extended Jones matrix representation for twisted nematic liquid crystal displays,” Liq. Cryst. 22, 171–175 (1997).
    [CrossRef]
  10. W. A. Shurcliff, Polarized Light, Production and Use (Harvard U. Press, Cambridge, Mass., 1962).
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    [CrossRef] [PubMed]
  12. A. De Meyere, “Light propagation and color variations in liquid crystal displays,” J. Opt. Soc. Am. A 11, 731–739 (1994).
    [CrossRef]
  13. O. Itoh, K. Kondo, N. Kikuchi, H. Madokoro, “Liquid crystal display apparatus having driving voltage to enable light of many wavelengths to take a same sign along one axis of a Poincaré sphere.” U.S. patent5,485,295 (16January1996).
  14. K. H. Kang, J. M. Moon, T. Yoon, J. C. Kim, G. Lee, E. Lee, “Design of a reflective multicolor STN-LCD on the Poincaré sphere,” in Eurodisplay ’96 Birmingham (Society for Information Display, San Jose, Calif., 1996), pp. 383–386.
  15. A. De Meyere, B. Maximus, C. Colpaert, “Optical compensation films for LCD’s: design on the Poincaré sphere,” in Eurodisplay ’96 Birmingham (Society for Information Display, San Jose, Calif., 1996), pp. 355–358.
  16. B. K. Winker, I. W. J. Gunning, D. B. Taber, L. G. Hale, “Optical compensator for improved gray scale performance in liquid crystal display,” U.S. patent5,504,603 (2April1996).
  17. G. Koch, B. K. Winker, I. W. J. Gunning, “Slay/twist compensator for improved gray scale performance in twisted nematic liquid crystal displays,” international patent application WO 96/10768 (11April1996).
  18. B. K. Winker, L. G. Hale, D. B. Taber, I. W. J. Gunning, “Viewing angle enhancement for vertically aligned cholesteric liquid crystal displays,” international patent application WO 96/10774 (11April1996).
  19. P. van de Witte, S. Stallinga, J. van Haaren, “Novel compensation foils for active matrix TN displays,” in Symposium Digest Boston ’97 (Society for Information Display, San Jose, Calif., 1997), pp. 687–690.
  20. G. X. M. Jones, “TN-LCD wide-viewing-angle optimization,” in Symposium Digest Anaheim ’98 (Society for Information Display, San Jose, Calif., 1998), pp. 475–478.
    [CrossRef]
  21. H. Pauwels, K. Vermeirsch, J. Fornier, A. De Meyere, “Analytical expression for off-axis light transmission through homogeneously oriented LC,” Mol. Cryst. Liq. Cryst. 9, 147–156 (1998).
  22. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964).
  23. H. F. Jones, Groups, Representations and Physics (Adam Hilger, Bristol, UK, 1990).
    [CrossRef]

1998 (1)

H. Pauwels, K. Vermeirsch, J. Fornier, A. De Meyere, “Analytical expression for off-axis light transmission through homogeneously oriented LC,” Mol. Cryst. Liq. Cryst. 9, 147–156 (1998).

1997 (1)

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

1995 (1)

E. De Ley, A. De Meyere, J. Fornier, B. Maximus, P. De Meerleer, C. Colpaert, H. Pauwels, “A simple numerical transmission model for general LCD-configurations,” Mol. Cryst. Liq. Cryst. 262, 577–590 (1995); paper also presented at the Fifteenth International Liquid Crystal Conference, Budapest, Hungary, 3–8 July 1994.

1994 (1)

1990 (1)

A. Lien, “Extended Jones matrix representation for the twisted nematic liquid-crystal display at oblique incidence,” Appl. Phys. Lett. 57, 2767–2769 (1990).
[CrossRef]

1982 (1)

1979 (1)

1977 (1)

1974 (1)

1972 (2)

1941 (1)

Azzam, R. M. A.

Bashara, N. M.

Berreman, D. W.

Bigelow, J.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964).

Colpaert, C.

E. De Ley, A. De Meyere, J. Fornier, B. Maximus, P. De Meerleer, C. Colpaert, H. Pauwels, “A simple numerical transmission model for general LCD-configurations,” Mol. Cryst. Liq. Cryst. 262, 577–590 (1995); paper also presented at the Fifteenth International Liquid Crystal Conference, Budapest, Hungary, 3–8 July 1994.

A. De Meyere, B. Maximus, C. Colpaert, “Optical compensation films for LCD’s: design on the Poincaré sphere,” in Eurodisplay ’96 Birmingham (Society for Information Display, San Jose, Calif., 1996), pp. 355–358.

De Ley, E.

E. De Ley, A. De Meyere, J. Fornier, B. Maximus, P. De Meerleer, C. Colpaert, H. Pauwels, “A simple numerical transmission model for general LCD-configurations,” Mol. Cryst. Liq. Cryst. 262, 577–590 (1995); paper also presented at the Fifteenth International Liquid Crystal Conference, Budapest, Hungary, 3–8 July 1994.

De Meerleer, P.

E. De Ley, A. De Meyere, J. Fornier, B. Maximus, P. De Meerleer, C. Colpaert, H. Pauwels, “A simple numerical transmission model for general LCD-configurations,” Mol. Cryst. Liq. Cryst. 262, 577–590 (1995); paper also presented at the Fifteenth International Liquid Crystal Conference, Budapest, Hungary, 3–8 July 1994.

De Meyere, A.

H. Pauwels, K. Vermeirsch, J. Fornier, A. De Meyere, “Analytical expression for off-axis light transmission through homogeneously oriented LC,” Mol. Cryst. Liq. Cryst. 9, 147–156 (1998).

E. De Ley, A. De Meyere, J. Fornier, B. Maximus, P. De Meerleer, C. Colpaert, H. Pauwels, “A simple numerical transmission model for general LCD-configurations,” Mol. Cryst. Liq. Cryst. 262, 577–590 (1995); paper also presented at the Fifteenth International Liquid Crystal Conference, Budapest, Hungary, 3–8 July 1994.

A. De Meyere, “Light propagation and color variations in liquid crystal displays,” J. Opt. Soc. Am. A 11, 731–739 (1994).
[CrossRef]

A. De Meyere, B. Maximus, C. Colpaert, “Optical compensation films for LCD’s: design on the Poincaré sphere,” in Eurodisplay ’96 Birmingham (Society for Information Display, San Jose, Calif., 1996), pp. 355–358.

Fornier, J.

H. Pauwels, K. Vermeirsch, J. Fornier, A. De Meyere, “Analytical expression for off-axis light transmission through homogeneously oriented LC,” Mol. Cryst. Liq. Cryst. 9, 147–156 (1998).

E. De Ley, A. De Meyere, J. Fornier, B. Maximus, P. De Meerleer, C. Colpaert, H. Pauwels, “A simple numerical transmission model for general LCD-configurations,” Mol. Cryst. Liq. Cryst. 262, 577–590 (1995); paper also presented at the Fifteenth International Liquid Crystal Conference, Budapest, Hungary, 3–8 July 1994.

Gunning, I. W. J.

G. Koch, B. K. Winker, I. W. J. Gunning, “Slay/twist compensator for improved gray scale performance in twisted nematic liquid crystal displays,” international patent application WO 96/10768 (11April1996).

B. K. Winker, L. G. Hale, D. B. Taber, I. W. J. Gunning, “Viewing angle enhancement for vertically aligned cholesteric liquid crystal displays,” international patent application WO 96/10774 (11April1996).

B. K. Winker, I. W. J. Gunning, D. B. Taber, L. G. Hale, “Optical compensator for improved gray scale performance in liquid crystal display,” U.S. patent5,504,603 (2April1996).

Hale, L. G.

B. K. Winker, I. W. J. Gunning, D. B. Taber, L. G. Hale, “Optical compensator for improved gray scale performance in liquid crystal display,” U.S. patent5,504,603 (2April1996).

B. K. Winker, L. G. Hale, D. B. Taber, I. W. J. Gunning, “Viewing angle enhancement for vertically aligned cholesteric liquid crystal displays,” international patent application WO 96/10774 (11April1996).

Itoh, O.

O. Itoh, K. Kondo, N. Kikuchi, H. Madokoro, “Liquid crystal display apparatus having driving voltage to enable light of many wavelengths to take a same sign along one axis of a Poincaré sphere.” U.S. patent5,485,295 (16January1996).

Jones, G. X. M.

G. X. M. Jones, “TN-LCD wide-viewing-angle optimization,” in Symposium Digest Anaheim ’98 (Society for Information Display, San Jose, Calif., 1998), pp. 475–478.
[CrossRef]

Jones, H. F.

H. F. Jones, Groups, Representations and Physics (Adam Hilger, Bristol, UK, 1990).
[CrossRef]

Jones, R. C.

Kang, K. H.

K. H. Kang, J. M. Moon, T. Yoon, J. C. Kim, G. Lee, E. Lee, “Design of a reflective multicolor STN-LCD on the Poincaré sphere,” in Eurodisplay ’96 Birmingham (Society for Information Display, San Jose, Calif., 1996), pp. 383–386.

Kashnow, R.

Kikuchi, N.

O. Itoh, K. Kondo, N. Kikuchi, H. Madokoro, “Liquid crystal display apparatus having driving voltage to enable light of many wavelengths to take a same sign along one axis of a Poincaré sphere.” U.S. patent5,485,295 (16January1996).

Kim, J. C.

K. H. Kang, J. M. Moon, T. Yoon, J. C. Kim, G. Lee, E. Lee, “Design of a reflective multicolor STN-LCD on the Poincaré sphere,” in Eurodisplay ’96 Birmingham (Society for Information Display, San Jose, Calif., 1996), pp. 383–386.

Koch, G.

G. Koch, B. K. Winker, I. W. J. Gunning, “Slay/twist compensator for improved gray scale performance in twisted nematic liquid crystal displays,” international patent application WO 96/10768 (11April1996).

Kondo, K.

O. Itoh, K. Kondo, N. Kikuchi, H. Madokoro, “Liquid crystal display apparatus having driving voltage to enable light of many wavelengths to take a same sign along one axis of a Poincaré sphere.” U.S. patent5,485,295 (16January1996).

Lee, E.

K. H. Kang, J. M. Moon, T. Yoon, J. C. Kim, G. Lee, E. Lee, “Design of a reflective multicolor STN-LCD on the Poincaré sphere,” in Eurodisplay ’96 Birmingham (Society for Information Display, San Jose, Calif., 1996), pp. 383–386.

Lee, G.

K. H. Kang, J. M. Moon, T. Yoon, J. C. Kim, G. Lee, E. Lee, “Design of a reflective multicolor STN-LCD on the Poincaré sphere,” in Eurodisplay ’96 Birmingham (Society for Information Display, San Jose, Calif., 1996), pp. 383–386.

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]

A. Lien, “Extended Jones matrix representation for the twisted nematic liquid-crystal display at oblique incidence,” Appl. Phys. Lett. 57, 2767–2769 (1990).
[CrossRef]

Madokoro, H.

O. Itoh, K. Kondo, N. Kikuchi, H. Madokoro, “Liquid crystal display apparatus having driving voltage to enable light of many wavelengths to take a same sign along one axis of a Poincaré sphere.” U.S. patent5,485,295 (16January1996).

Maximus, B.

E. De Ley, A. De Meyere, J. Fornier, B. Maximus, P. De Meerleer, C. Colpaert, H. Pauwels, “A simple numerical transmission model for general LCD-configurations,” Mol. Cryst. Liq. Cryst. 262, 577–590 (1995); paper also presented at the Fifteenth International Liquid Crystal Conference, Budapest, Hungary, 3–8 July 1994.

A. De Meyere, B. Maximus, C. Colpaert, “Optical compensation films for LCD’s: design on the Poincaré sphere,” in Eurodisplay ’96 Birmingham (Society for Information Display, San Jose, Calif., 1996), pp. 355–358.

Moon, J. M.

K. H. Kang, J. M. Moon, T. Yoon, J. C. Kim, G. Lee, E. Lee, “Design of a reflective multicolor STN-LCD on the Poincaré sphere,” in Eurodisplay ’96 Birmingham (Society for Information Display, San Jose, Calif., 1996), pp. 383–386.

Pauwels, H.

H. Pauwels, K. Vermeirsch, J. Fornier, A. De Meyere, “Analytical expression for off-axis light transmission through homogeneously oriented LC,” Mol. Cryst. Liq. Cryst. 9, 147–156 (1998).

E. De Ley, A. De Meyere, J. Fornier, B. Maximus, P. De Meerleer, C. Colpaert, H. Pauwels, “A simple numerical transmission model for general LCD-configurations,” Mol. Cryst. Liq. Cryst. 262, 577–590 (1995); paper also presented at the Fifteenth International Liquid Crystal Conference, Budapest, Hungary, 3–8 July 1994.

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light, Production and Use (Harvard U. Press, Cambridge, Mass., 1962).

Stallinga, S.

P. van de Witte, S. Stallinga, J. van Haaren, “Novel compensation foils for active matrix TN displays,” in Symposium Digest Boston ’97 (Society for Information Display, San Jose, Calif., 1997), pp. 687–690.

Taber, D. B.

B. K. Winker, L. G. Hale, D. B. Taber, I. W. J. Gunning, “Viewing angle enhancement for vertically aligned cholesteric liquid crystal displays,” international patent application WO 96/10774 (11April1996).

B. K. Winker, I. W. J. Gunning, D. B. Taber, L. G. Hale, “Optical compensator for improved gray scale performance in liquid crystal display,” U.S. patent5,504,603 (2April1996).

van de Witte, P.

P. van de Witte, S. Stallinga, J. van Haaren, “Novel compensation foils for active matrix TN displays,” in Symposium Digest Boston ’97 (Society for Information Display, San Jose, Calif., 1997), pp. 687–690.

van Haaren, J.

P. van de Witte, S. Stallinga, J. van Haaren, “Novel compensation foils for active matrix TN displays,” in Symposium Digest Boston ’97 (Society for Information Display, San Jose, Calif., 1997), pp. 687–690.

Vermeirsch, K.

H. Pauwels, K. Vermeirsch, J. Fornier, A. De Meyere, “Analytical expression for off-axis light transmission through homogeneously oriented LC,” Mol. Cryst. Liq. Cryst. 9, 147–156 (1998).

Winker, B. K.

G. Koch, B. K. Winker, I. W. J. Gunning, “Slay/twist compensator for improved gray scale performance in twisted nematic liquid crystal displays,” international patent application WO 96/10768 (11April1996).

B. K. Winker, L. G. Hale, D. B. Taber, I. W. J. Gunning, “Viewing angle enhancement for vertically aligned cholesteric liquid crystal displays,” international patent application WO 96/10774 (11April1996).

B. K. Winker, I. W. J. Gunning, D. B. Taber, L. G. Hale, “Optical compensator for improved gray scale performance in liquid crystal display,” U.S. patent5,504,603 (2April1996).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964).

Yeh, P.

Yoon, T.

K. H. Kang, J. M. Moon, T. Yoon, J. C. Kim, G. Lee, E. Lee, “Design of a reflective multicolor STN-LCD on the Poincaré sphere,” in Eurodisplay ’96 Birmingham (Society for Information Display, San Jose, Calif., 1996), pp. 383–386.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

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. Opt. Soc. Am. (6)

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

Liq. Cryst. (1)

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

Mol. Cryst. Liq. Cryst. (2)

E. De Ley, A. De Meyere, J. Fornier, B. Maximus, P. De Meerleer, C. Colpaert, H. Pauwels, “A simple numerical transmission model for general LCD-configurations,” Mol. Cryst. Liq. Cryst. 262, 577–590 (1995); paper also presented at the Fifteenth International Liquid Crystal Conference, Budapest, Hungary, 3–8 July 1994.

H. Pauwels, K. Vermeirsch, J. Fornier, A. De Meyere, “Analytical expression for off-axis light transmission through homogeneously oriented LC,” Mol. Cryst. Liq. Cryst. 9, 147–156 (1998).

Other (11)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964).

H. F. Jones, Groups, Representations and Physics (Adam Hilger, Bristol, UK, 1990).
[CrossRef]

W. A. Shurcliff, Polarized Light, Production and Use (Harvard U. Press, Cambridge, Mass., 1962).

O. Itoh, K. Kondo, N. Kikuchi, H. Madokoro, “Liquid crystal display apparatus having driving voltage to enable light of many wavelengths to take a same sign along one axis of a Poincaré sphere.” U.S. patent5,485,295 (16January1996).

K. H. Kang, J. M. Moon, T. Yoon, J. C. Kim, G. Lee, E. Lee, “Design of a reflective multicolor STN-LCD on the Poincaré sphere,” in Eurodisplay ’96 Birmingham (Society for Information Display, San Jose, Calif., 1996), pp. 383–386.

A. De Meyere, B. Maximus, C. Colpaert, “Optical compensation films for LCD’s: design on the Poincaré sphere,” in Eurodisplay ’96 Birmingham (Society for Information Display, San Jose, Calif., 1996), pp. 355–358.

B. K. Winker, I. W. J. Gunning, D. B. Taber, L. G. Hale, “Optical compensator for improved gray scale performance in liquid crystal display,” U.S. patent5,504,603 (2April1996).

G. Koch, B. K. Winker, I. W. J. Gunning, “Slay/twist compensator for improved gray scale performance in twisted nematic liquid crystal displays,” international patent application WO 96/10768 (11April1996).

B. K. Winker, L. G. Hale, D. B. Taber, I. W. J. Gunning, “Viewing angle enhancement for vertically aligned cholesteric liquid crystal displays,” international patent application WO 96/10774 (11April1996).

P. van de Witte, S. Stallinga, J. van Haaren, “Novel compensation foils for active matrix TN displays,” in Symposium Digest Boston ’97 (Society for Information Display, San Jose, Calif., 1997), pp. 687–690.

G. X. M. Jones, “TN-LCD wide-viewing-angle optimization,” in Symposium Digest Anaheim ’98 (Society for Information Display, San Jose, Calif., 1998), pp. 475–478.
[CrossRef]

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Figures (15)

Fig. 1
Fig. 1

(a) Definition of θ o , θ c , and Φ c . (b) Definition of ψ.

Fig. 2
Fig. 2

Representation of three arbitrary viewing directions and the corresponding s directions.

Fig. 3
Fig. 3

(a) Definition of the arc distance 2β between two representation points. (b) Representation points of polarizations that lead to equal transmission intensities through a polarizer with representation point A.

Fig. 4
Fig. 4

Different viewing directions: representation example.

Fig. 5
Fig. 5

Light leakage through crossed polarizers.

Fig. 6
Fig. 6

Effect of a homogeneous birefringent slab.

Fig. 7
Fig. 7

Characterization of viewing direction k o and the optical axis c of a compensation foil.

Fig. 8
Fig. 8

Polarization states of an ideal TN LCD upon the Poincaré sphere.

Fig. 9
Fig. 9

Value of 2ψ F as a function of the azimuthal angle.

Fig. 10
Fig. 10

F curves as functions of the azimuthal and the inclination angles; compensation foil: Φ c = 0°, θ c = 0°.

Fig. 11
Fig. 11

Polarization states of the film-compensated TN LCD upon the Poincaré sphere.

Fig. 12
Fig. 12

(a) Conoscopic plot of the contrast ratio (CR) of the uncompensated ideal TN LCD. (b) Conoscopic plot of the contrast ratio of the compensated ideal TN LCD.

Fig. 13
Fig. 13

(a) Poincaré sphere representation of the transmission of a real TN LCD. (b) Poincaré sphere representation of the transmission of the TN LCD plus one compensating layer.

Fig. 14
Fig. 14

(a) Conoscopic diagram of the contrast ratio of the uncompensated TN-LCD. (b) Conoscopic diagram of the contrast ratio of the developed FF TN-LCD.

Fig. 15
Fig. 15

Relation between θ, ∊ and α, δ.

Tables (1)

Tables Icon

Table 1 Different Viewing Directions: Calculation of Transmission for Oblique Incidence

Equations (19)

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

o=c×ko|c×ko|, e=ko×o|ko×o|.
c=1x cos Φc+1y sin Φcsin θc+1z cos θc.
cos ψ=cos θo sin Φc sin θc-sin θo cos θc1-sin θo sin Φc sin θc+cos θo cos θc21/2, sin ψ=-cos Φc sin θc1-sin θo sin Φc sin θc + cos θo cos θc21/2.
EsEp=T0R-ψPRψTiEsEp.
T=1/21+cos2β,
T=1/21+cos 2ψA-ψP=cos2ψA-ψP,
Γ=kez-kozd=2πλdcos θo Δn sin2 ψF.
T=1/41+cos 2β=1/41+cos 2ψA-ψFcos 2ψF-ψP+sin 2ψA-ψFsin 2ψF-ψPcosπ-Γ=1/82+cos 22ψF-ψA-ψP+cos 2ψA-ψP+cos 22ψF-ψA-ψP-cos 2ψA-ψPsin2Γ/2-cos2Γ/2=1/41+sin2Γ/2cos 22ψF-ψA-ψP+cos2Γ/2cos 2ψA-ψP=1/2cos2Γ/2cos2ψA-ψP+sin2Γ/2cos22ψF-ψA-ψP.
EsEp=ts00tpcos αsin α expjδ=ts2+tp2tsts2+tp2cos αtpts2+tp2sin α expjδ=ts2+tp2cos αsin α expjδ.
=ExEy=a1 expjδ1a2 expjδ2.
=a12+a22 expjδ1a1a12+a22a1a12+a22expjδ1-δ2=a expjδ1cosαsinαexpjδ.
x=a cosαcosωt, y=a sinαcosωt+δ
x2 sin2α-2xy cosδsinαcosα+y2 cos2α=a2 sin2δsin2αcos2α.
2 tanθ1-tan2θ=2 cosδsinαcosαcos2α-sin2α, tan 2θ=tan 2α cosδ.
tan2θ=tan2αcosδ, sin2=sin2αsinδ.
cos2α=cos2θcos2, tanδ=tan2sin2θ.
s0=|Ex|2+|Ey|2=a2, s1=|Ex|2-|Ey|2=a2 cos 2 cos 2θ, s2=|Eπ/4|2-|E-π/4|2=1/2|Ex+Ey|2-1/2|Ex-Ey|2=a2 cos 2 sin 2θ, s3=|Er|2-|El|2=1/2|Ex-jEy|2-1/2|Ex+jEy|2=a2 sin 2.
=cos α0 expjζ0sin α0 exp-jδ0-ζ0-sin α0 expjδ0-ζ0cos α0 exp-jζ0cos α1sin α1 expjδ1=cos α0 cos α1 expjζ0+sin α0 sin α1 expjδ1-δ0+ζ0-sin α0 cos α1 expjδ0-ζ0+cos α0 sin α1 expjδ1-ζ0,
T=|cos α0 cos α1 expjζ0+sin α0 sin α1 expjδ1-δ0+ζ0|2=cos2 α0 cos2 α1+sin2 α0 sin2 α1+2 cos α0 cos α1 sin α0 sin α1 cosδ1-δ0=1/21+cos 2α0 cos 2α1+sin 2α0 sin 2α1 cosδ1-δ0=1/21+cos 2β.

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