Abstract

The Poincaré sphere representation, a geometrical method for solving problems involving the propagation of polarized light through birefringent and optically active media, is applied to several electrooptic liquid crystal problems. The emphasis is on the twisted nematic case, for which the quiescent state solution was given by Mauguin in 1911. The Poincaré construction shows that the normal modes for the undeformed twisted nematic layer are slightly elliptically polarized and suggests convenient experiments for measuring the ellipticity. For the field-activated state, a construction is indicated as an alternative to matrix-multiplication methods.

© 1977 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. Poincaré, Theorie Mathematique de la Lumiere (Paris, 1892), Vol. 2, p. 275.
  2. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).
  3. H. G. Jerrard, J. Opt. Soc. Am. 44, 634 (1954) and references therein.
    [CrossRef]
  4. N. H. Hartshorne, A. Stuart, Crystals and the Polarizing Microscope (Edward Arnold, London, 1970).
  5. C. Mauguin, Bull. Soc. Fran. Mineral 34, 71 (1911).
  6. C. Mauguin, Ref. 5, p. 6.
  7. G. N. Ramachandran, S. Ramaseshan, J. Opt. Soc. Am. 42, 49 (1952).
    [CrossRef]
  8. M. Schadt, W. Helfrich, Appl. Phys. Lett. 18, 127 (1971).
    [CrossRef]
  9. A theorem of which this is a special case is proved in Ref. 7.
  10. C. Z. Van Doorn, Phys. Lett. A 42, 537 (1973).
    [CrossRef]
  11. D. W. Berreman, J. Opt. Soc. Am. 63, 1374 (1973).
    [CrossRef]
  12. F. M. Leslie, Mol. Cryst. Liq. Cryst. 12, 57 (1970).
    [CrossRef]
  13. J. Grinberg, A. D. Jacobson, J. Opt. Soc. Am. 66, 1003 (1976).
    [CrossRef]
  14. J. Grinberg et al., IEEE Trans. Electron Devices ED-22, 775 (1975).
    [CrossRef]
  15. A. Sussman, IEEE Trans. Parts, Hybrids, Packag. PHP-8, 4, 24 (Dec.1972).
  16. E. P. Raynes, Electron Lett. 9, 101 (1973).
    [CrossRef]
  17. I. A. Shanks, Electron. Lett. 10, 90 (1974).
    [CrossRef]
  18. T. J. Scheffer, J. Appl. Phys. 44, 4799 (1973).
    [CrossRef]

1976 (1)

1975 (1)

J. Grinberg et al., IEEE Trans. Electron Devices ED-22, 775 (1975).
[CrossRef]

1974 (1)

I. A. Shanks, Electron. Lett. 10, 90 (1974).
[CrossRef]

1973 (4)

T. J. Scheffer, J. Appl. Phys. 44, 4799 (1973).
[CrossRef]

E. P. Raynes, Electron Lett. 9, 101 (1973).
[CrossRef]

C. Z. Van Doorn, Phys. Lett. A 42, 537 (1973).
[CrossRef]

D. W. Berreman, J. Opt. Soc. Am. 63, 1374 (1973).
[CrossRef]

1972 (1)

A. Sussman, IEEE Trans. Parts, Hybrids, Packag. PHP-8, 4, 24 (Dec.1972).

1971 (1)

M. Schadt, W. Helfrich, Appl. Phys. Lett. 18, 127 (1971).
[CrossRef]

1970 (1)

F. M. Leslie, Mol. Cryst. Liq. Cryst. 12, 57 (1970).
[CrossRef]

1954 (1)

1952 (1)

1911 (1)

C. Mauguin, Bull. Soc. Fran. Mineral 34, 71 (1911).

Berreman, D. W.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).

Grinberg, J.

J. Grinberg, A. D. Jacobson, J. Opt. Soc. Am. 66, 1003 (1976).
[CrossRef]

J. Grinberg et al., IEEE Trans. Electron Devices ED-22, 775 (1975).
[CrossRef]

Hartshorne, N. H.

N. H. Hartshorne, A. Stuart, Crystals and the Polarizing Microscope (Edward Arnold, London, 1970).

Helfrich, W.

M. Schadt, W. Helfrich, Appl. Phys. Lett. 18, 127 (1971).
[CrossRef]

Jacobson, A. D.

Jerrard, H. G.

Leslie, F. M.

F. M. Leslie, Mol. Cryst. Liq. Cryst. 12, 57 (1970).
[CrossRef]

Mauguin, C.

C. Mauguin, Bull. Soc. Fran. Mineral 34, 71 (1911).

C. Mauguin, Ref. 5, p. 6.

Poincaré, H.

H. Poincaré, Theorie Mathematique de la Lumiere (Paris, 1892), Vol. 2, p. 275.

Ramachandran, G. N.

Ramaseshan, S.

Raynes, E. P.

E. P. Raynes, Electron Lett. 9, 101 (1973).
[CrossRef]

Schadt, M.

M. Schadt, W. Helfrich, Appl. Phys. Lett. 18, 127 (1971).
[CrossRef]

Scheffer, T. J.

T. J. Scheffer, J. Appl. Phys. 44, 4799 (1973).
[CrossRef]

Shanks, I. A.

I. A. Shanks, Electron. Lett. 10, 90 (1974).
[CrossRef]

Stuart, A.

N. H. Hartshorne, A. Stuart, Crystals and the Polarizing Microscope (Edward Arnold, London, 1970).

Sussman, A.

A. Sussman, IEEE Trans. Parts, Hybrids, Packag. PHP-8, 4, 24 (Dec.1972).

Van Doorn, C. Z.

C. Z. Van Doorn, Phys. Lett. A 42, 537 (1973).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).

Appl. Phys. Lett. (1)

M. Schadt, W. Helfrich, Appl. Phys. Lett. 18, 127 (1971).
[CrossRef]

Bull. Soc. Fran. Mineral (1)

C. Mauguin, Bull. Soc. Fran. Mineral 34, 71 (1911).

Electron Lett. (1)

E. P. Raynes, Electron Lett. 9, 101 (1973).
[CrossRef]

Electron. Lett. (1)

I. A. Shanks, Electron. Lett. 10, 90 (1974).
[CrossRef]

IEEE Trans. Electron Devices (1)

J. Grinberg et al., IEEE Trans. Electron Devices ED-22, 775 (1975).
[CrossRef]

IEEE Trans. Parts, Hybrids, Packag. (1)

A. Sussman, IEEE Trans. Parts, Hybrids, Packag. PHP-8, 4, 24 (Dec.1972).

J. Appl. Phys. (1)

T. J. Scheffer, J. Appl. Phys. 44, 4799 (1973).
[CrossRef]

J. Opt. Soc. Am. (4)

Mol. Cryst. Liq. Cryst. (1)

F. M. Leslie, Mol. Cryst. Liq. Cryst. 12, 57 (1970).
[CrossRef]

Phys. Lett. A (1)

C. Z. Van Doorn, Phys. Lett. A 42, 537 (1973).
[CrossRef]

Other (5)

N. H. Hartshorne, A. Stuart, Crystals and the Polarizing Microscope (Edward Arnold, London, 1970).

H. Poincaré, Theorie Mathematique de la Lumiere (Paris, 1892), Vol. 2, p. 275.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).

C. Mauguin, Ref. 5, p. 6.

A theorem of which this is a special case is proved in Ref. 7.

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 (17)

Fig. 1
Fig. 1

Parameters defining the azimuth α and ellipticity β of a polarization state.

Fig. 2
Fig. 2

Coordinates of a point P(2α, 2β) on the Poincaré sphere. The poles CL and CR denote left- and right-handed circularly polarized light; the equator is the locus of all linearly polarized states.

Fig. 3
Fig. 3

Linearly polarized light is incident on a uniaxial-birefringent crystal with an azimuth ψ with respect to the crystal fast axis.

Fig. 4
Fig. 4

Poincaré construction for the case shown in Fig. 3. The final state Q is found by rotating the initial state P through the angle θ about the xy axis.

Fig. 5
Fig. 5

Pure optical rotation on the Poincaré sphere.

Fig. 6
Fig. 6

Construction (due to Ramachandran and Ramaseshan7) for combined birefringence and optical rotation. Rotate P through △ about the OJ axis; △ and γ are functions of birefringence and optical rotation.

Fig. 7
Fig. 7

Coordinate system for the description of a twisted nematic configuration.

Fig. 8
Fig. 8

Poincaré construction and equatorial projection for the transmission of linearly polarized light P through a stack of N birefringent platelets, the axes of each of which are rotated through δα with respect to adjacent ones.5

Fig. 9
Fig. 9

Poincaré construction for a continuously twisted nematic layer.5 In the limit of large N, the construction of Fig. 8 leads to this picture of two pairs of cones which roll along the equatorial plane. A and B denote the slow LC axes at the entrance and exit boundaries. RH and LH denote right-handed and left-handed twist. The cone halfangle ω gives the normal modes, such as at O.

Fig. 10
Fig. 10

Distribution of the deformation angles γ, Φ(see Fig. 7) for a 10-μm twisted nematic layer. The subscripts on the family of curves indicate the ratios of applied voltage to critical voltage, as follows: (1) V/Vc ≤ 1; (2) V/Vc = 1.7; and (3) V/Vc = 4.1 (From Berreman11).

Fig. 11
Fig. 11

Construction, on an equatorial projection, for transmission through a twisted nematic layer in the field-activated case.

Fig. 12
Fig. 12

Reflection mode projector using thin 45° twisted liquid crystal layer.14

Fig. 13
Fig. 13

Poincaré sphere construction for thin, reflective, 45° twisted liquid crystal layer.

Fig. 14
Fig. 14

Multicolor liquid crystal device.

Fig. 15
Fig. 15

Construction on Poincaré sphere of operation of multicolor liquid crystal device.

Fig. 16
Fig. 16

Construction for measurement of cone half-angle ω near the boundaries of a 90° twisted nematic layer.

Fig. 17
Fig. 17

Construction for measurement of ellipticity in a twisted nematic layer when the azimuthal rotation (total twist angle) is unknown.

Tables (1)

Tables Icon

Table I Color Dispersion on Poincaré Sphere

Equations (8)

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

θ = 2 π Δ n d / λ ,
Δ 0 2 = ( 2 π Δ n λ ) 2 + ( 2 ρ 0 ) 2 .
tan 2 γ = 2 ρ 0 / 2 π Δ n / λ = ρ 0 λ / π Δ n .
n ^ ( z = 0 ) = y ^ ,             n ^ ( z = d ) = x ^ ,
ω = arctan 2 λ / Δ n p ,
V c 2 = [ k 11 ( π / 2 ) 2 + ( k 33 - 2 k 22 ) ( π / 4 ) 2 ] Δ ,
K π sin ω = π / 2 ,
arcsin [ 1 / ( 2 K ) ] = arctan [ λ / ( 4 Δ n d ) ] .

Metrics