Abstract

An approximate analytical expression is calculated for the transmission of thin twisted nematic layers situated between a polarizer/analyzer pair. The approximation assumes that the twist angle of the nematic liquid crystal is smaller than the maximum retardation of the cell. The direction of the incident light is assumed to be parallel to the normal of the electrode. This configuration is analyzed for a general arrangement of polarizer and analyzer; the general result is evaluated for the case of the polarizer parallel and analyzer perpendicular to the liquid-crystal optical axis on the input and output electrodes, respectively. The results show that in the case of a thin twisted nematic layer the transmission depends on the thickness of the layer, on the birefringence of the liquid crystal, and on the wavelength of the light. This is a departure from the well-known independence of the transmission on these parameters for a thick twisted nematic layer. The analysis also shows that the transmission through thin layers can be minimized by tuning the layer thickness to particular values that depend on wavelength and on liquid-crystal birefringence. In this way, high image contrast can be obtained even for thin liquid-crystal layers.

© 1976 Optical Society of America

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References

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  1. M. Schadt and W. Helfrich, Appl. Phys. Lett. 18, 127–128 (1971).
    [Crossref]
  2. F. Pockels, Lehrbuch der Krystalloptic (B. G. Teubner, Berlin, 1906).
  3. S. Ramaseshan, Proc. Ind. Acad. Sci. 34A, 32 (1951).
  4. H. de Vries, Acta Crystallogr. 4, 219 (1951).
    [Crossref]
  5. H. Mada and S. Kobayaski, Rev. Phys. Appl. 10, 147 (1975).
    [Crossref]
  6. C. H. Gooch and H. A. Tarry, Electron. Lett. 10,(1973).
  7. R. Azzam and N. Bashara, J. Opt. Soc. Am. 62, 1252 (1972).
    [Crossref]
  8. R. Dreher and G. Meier, Phys. Rev. A 8, 1616 (1973).
    [Crossref]
  9. R. Dreher and G. Meier, Solid State Commun. 13, 607 (1973).
    [Crossref]
  10. G. Conners, J. Opt. Soc. Am. 58, 875 (1968).
    [Crossref]
  11. S. Chandrasekhar and K. Rao, Acta. Crystallogr. A 24, 445 (1968).
    [Crossref]
  12. M. Mauguin, Bull. Soc. Franc. Miner. Crisf. 34, 71 (1911).
  13. J. Billard, Mol. Cryst. 3, 227 (1967).
    [Crossref]
  14. T. J. Scheffer, J. Appl. Phys. 44, 4799–4803 (1973).
    [Crossref]
  15. I. A. Shanks, Electron. Lett. 10, 7 (1974).
    [Crossref]
  16. E. P. Raynes and I. A. Shanks, Electron. Lett. 10, 7 (1974).
    [Crossref]
  17. S. Sato and M. Wada, IEEE Trans. Electron. Devices ED-21, 312 (1974).
    [Crossref]
  18. J. Grinberg and et al., IEEE Trans. Electron. Devices ED-22, 775–783 (1975).
    [Crossref]
  19. J. Grinberg and et al., J. SPIE 14, 217–225 (1975).
  20. See, for example, A. L. Goodman, J. Vac. Sci. Technol. 10, 804–823 (1973).
    [Crossref]
  21. Dwight W. Berreman, J. Opt. Soc. Am. 43, 1374 (1973).
    [Crossref]
  22. J. E. Bigelow and R. A. Kashnow, G. E. Technical Information Series Report No. 74, CDR 090, June1974 (unpublished), and the references included therein.

1975 (3)

H. Mada and S. Kobayaski, Rev. Phys. Appl. 10, 147 (1975).
[Crossref]

J. Grinberg and et al., IEEE Trans. Electron. Devices ED-22, 775–783 (1975).
[Crossref]

J. Grinberg and et al., J. SPIE 14, 217–225 (1975).

1974 (3)

I. A. Shanks, Electron. Lett. 10, 7 (1974).
[Crossref]

E. P. Raynes and I. A. Shanks, Electron. Lett. 10, 7 (1974).
[Crossref]

S. Sato and M. Wada, IEEE Trans. Electron. Devices ED-21, 312 (1974).
[Crossref]

1973 (6)

T. J. Scheffer, J. Appl. Phys. 44, 4799–4803 (1973).
[Crossref]

See, for example, A. L. Goodman, J. Vac. Sci. Technol. 10, 804–823 (1973).
[Crossref]

Dwight W. Berreman, J. Opt. Soc. Am. 43, 1374 (1973).
[Crossref]

C. H. Gooch and H. A. Tarry, Electron. Lett. 10,(1973).

R. Dreher and G. Meier, Phys. Rev. A 8, 1616 (1973).
[Crossref]

R. Dreher and G. Meier, Solid State Commun. 13, 607 (1973).
[Crossref]

1972 (1)

1971 (1)

M. Schadt and W. Helfrich, Appl. Phys. Lett. 18, 127–128 (1971).
[Crossref]

1968 (2)

G. Conners, J. Opt. Soc. Am. 58, 875 (1968).
[Crossref]

S. Chandrasekhar and K. Rao, Acta. Crystallogr. A 24, 445 (1968).
[Crossref]

1967 (1)

J. Billard, Mol. Cryst. 3, 227 (1967).
[Crossref]

1951 (2)

S. Ramaseshan, Proc. Ind. Acad. Sci. 34A, 32 (1951).

H. de Vries, Acta Crystallogr. 4, 219 (1951).
[Crossref]

1911 (1)

M. Mauguin, Bull. Soc. Franc. Miner. Crisf. 34, 71 (1911).

Azzam, R.

Bashara, N.

Berreman, Dwight W.

Dwight W. Berreman, J. Opt. Soc. Am. 43, 1374 (1973).
[Crossref]

Bigelow, J. E.

J. E. Bigelow and R. A. Kashnow, G. E. Technical Information Series Report No. 74, CDR 090, June1974 (unpublished), and the references included therein.

Billard, J.

J. Billard, Mol. Cryst. 3, 227 (1967).
[Crossref]

Chandrasekhar, S.

S. Chandrasekhar and K. Rao, Acta. Crystallogr. A 24, 445 (1968).
[Crossref]

Conners, G.

de Vries, H.

H. de Vries, Acta Crystallogr. 4, 219 (1951).
[Crossref]

Dreher, R.

R. Dreher and G. Meier, Phys. Rev. A 8, 1616 (1973).
[Crossref]

R. Dreher and G. Meier, Solid State Commun. 13, 607 (1973).
[Crossref]

Gooch, C. H.

C. H. Gooch and H. A. Tarry, Electron. Lett. 10,(1973).

Goodman, A. L.

See, for example, A. L. Goodman, J. Vac. Sci. Technol. 10, 804–823 (1973).
[Crossref]

Grinberg, J.

J. Grinberg and et al., IEEE Trans. Electron. Devices ED-22, 775–783 (1975).
[Crossref]

J. Grinberg and et al., J. SPIE 14, 217–225 (1975).

Helfrich, W.

M. Schadt and W. Helfrich, Appl. Phys. Lett. 18, 127–128 (1971).
[Crossref]

Kashnow, R. A.

J. E. Bigelow and R. A. Kashnow, G. E. Technical Information Series Report No. 74, CDR 090, June1974 (unpublished), and the references included therein.

Kobayaski, S.

H. Mada and S. Kobayaski, Rev. Phys. Appl. 10, 147 (1975).
[Crossref]

Mada, H.

H. Mada and S. Kobayaski, Rev. Phys. Appl. 10, 147 (1975).
[Crossref]

Mauguin, M.

M. Mauguin, Bull. Soc. Franc. Miner. Crisf. 34, 71 (1911).

Meier, G.

R. Dreher and G. Meier, Phys. Rev. A 8, 1616 (1973).
[Crossref]

R. Dreher and G. Meier, Solid State Commun. 13, 607 (1973).
[Crossref]

Pockels, F.

F. Pockels, Lehrbuch der Krystalloptic (B. G. Teubner, Berlin, 1906).

Ramaseshan, S.

S. Ramaseshan, Proc. Ind. Acad. Sci. 34A, 32 (1951).

Rao, K.

S. Chandrasekhar and K. Rao, Acta. Crystallogr. A 24, 445 (1968).
[Crossref]

Raynes, E. P.

E. P. Raynes and I. A. Shanks, Electron. Lett. 10, 7 (1974).
[Crossref]

Sato, S.

S. Sato and M. Wada, IEEE Trans. Electron. Devices ED-21, 312 (1974).
[Crossref]

Schadt, M.

M. Schadt and W. Helfrich, Appl. Phys. Lett. 18, 127–128 (1971).
[Crossref]

Scheffer, T. J.

T. J. Scheffer, J. Appl. Phys. 44, 4799–4803 (1973).
[Crossref]

Shanks, I. A.

I. A. Shanks, Electron. Lett. 10, 7 (1974).
[Crossref]

E. P. Raynes and I. A. Shanks, Electron. Lett. 10, 7 (1974).
[Crossref]

Tarry, H. A.

C. H. Gooch and H. A. Tarry, Electron. Lett. 10,(1973).

Wada, M.

S. Sato and M. Wada, IEEE Trans. Electron. Devices ED-21, 312 (1974).
[Crossref]

Acta Crystallogr. (1)

H. de Vries, Acta Crystallogr. 4, 219 (1951).
[Crossref]

Acta. Crystallogr. A (1)

S. Chandrasekhar and K. Rao, Acta. Crystallogr. A 24, 445 (1968).
[Crossref]

Appl. Phys. Lett. (1)

M. Schadt and W. Helfrich, Appl. Phys. Lett. 18, 127–128 (1971).
[Crossref]

Bull. Soc. Franc. Miner. Crisf. (1)

M. Mauguin, Bull. Soc. Franc. Miner. Crisf. 34, 71 (1911).

Electron. Lett. (3)

I. A. Shanks, Electron. Lett. 10, 7 (1974).
[Crossref]

E. P. Raynes and I. A. Shanks, Electron. Lett. 10, 7 (1974).
[Crossref]

C. H. Gooch and H. A. Tarry, Electron. Lett. 10,(1973).

IEEE Trans. Electron. Devices (2)

S. Sato and M. Wada, IEEE Trans. Electron. Devices ED-21, 312 (1974).
[Crossref]

J. Grinberg and et al., IEEE Trans. Electron. Devices ED-22, 775–783 (1975).
[Crossref]

J. Appl. Phys. (1)

T. J. Scheffer, J. Appl. Phys. 44, 4799–4803 (1973).
[Crossref]

J. Opt. Soc. Am. (3)

J. SPIE (1)

J. Grinberg and et al., J. SPIE 14, 217–225 (1975).

J. Vac. Sci. Technol. (1)

See, for example, A. L. Goodman, J. Vac. Sci. Technol. 10, 804–823 (1973).
[Crossref]

Mol. Cryst. (1)

J. Billard, Mol. Cryst. 3, 227 (1967).
[Crossref]

Phys. Rev. A (1)

R. Dreher and G. Meier, Phys. Rev. A 8, 1616 (1973).
[Crossref]

Proc. Ind. Acad. Sci. (1)

S. Ramaseshan, Proc. Ind. Acad. Sci. 34A, 32 (1951).

Rev. Phys. Appl. (1)

H. Mada and S. Kobayaski, Rev. Phys. Appl. 10, 147 (1975).
[Crossref]

Solid State Commun. (1)

R. Dreher and G. Meier, Solid State Commun. 13, 607 (1973).
[Crossref]

Other (2)

F. Pockels, Lehrbuch der Krystalloptic (B. G. Teubner, Berlin, 1906).

J. E. Bigelow and R. A. Kashnow, G. E. Technical Information Series Report No. 74, CDR 090, June1974 (unpublished), and the references included therein.

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Figures (7)

FIG. 1
FIG. 1

Schematic of liquid-crystal molecule orientation in twisted nematic device.

FIG. 2
FIG. 2

Twisted nematic layer model (α = 90°).

FIG. 3
FIG. 3

Change in the polarization splitting between the (n − 1) and n axes.

FIG. 4
FIG. 4

Buildup of the perpendicular polarization by the rotation of liquid-crystal optical axis.

FIG. 5
FIG. 5

Locus of the bk vectors. (a) General case of the overall retardation. (b) When the retardation is π[or (2K − 1)π], the PyN is maximum. (c) When the retardation is 2π (or 2), the PyN = 0.

FIG. 6
FIG. 6

Transmission of thin twisted nematic cells.

FIG. 7
FIG. 7

Directions of the input light polarizer and the analyzer (output light polarizer) with respect to the liquid-crystal optical axes: (a) at the input electrode; (b) at the output electrode.

Equations (38)

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P x n = P x ( n - 1 ) cos - P y ( n - 1 ) sin , P y n = P y ( n - 1 ) cos + P x ( n - 1 ) sin .
P x n = P x ( n - 1 ) cos - P y ( n - 1 ) sin , P y n = ( P y ( n - 1 ) cos + P x ( n - 1 ) sin ) × exp ( i 2 π Δ n 0 Δ z / λ ) .
R n = P y n / P x n ,
T = R N R N * .
tan a ,
exp ( i 2 π Δ n 0 Δ z / λ ) = e i ϕ b ,
R n = ( R n - 1 + a ) b / ( 1 - a R n - 1 ) .
P y 0 = 0 ,             P x 0 = 1 ,
R 0 = 0 ,
R 1 = a b ,
R 2 = ( a b 2 + a b ) / ( 1 - a 2 b ) ,
R 3 = ( a b 3 + a b 2 + a b - a 3 b 2 ) / ( 1 - 2 a 2 b - a 2 b 2 ) ,
R 4 = [ a ( k = 1 4 b k ) - 2 a 3 b 2 - 2 a 3 b 3 ] × ( 1 - 3 a 2 b - 2 a 2 b 2 - a 2 b 3 + a 4 b 2 ) - 1 ,
R 5 = [ a ( k = 1 5 b k ) - 3 a 3 a 4 - 4 a 3 b 3 - 3 a 3 b 2 + a 5 b 3 ] × [ 1 - a 2 ( k = 1 4 ( 5 - k ) b k ) + 3 a 4 b 2 + 2 a 4 b 3 ] - 1 .
R n a ( k = 1 n b k ) / [ 1 - a 2 ( k = 1 n - 1 ( n - k ) b k ) ] .
R n ( 0 ) = 0
R n ( 1 ) = a k = 1 n b k .
- a 3 l = 1 n - 2 b n - l - 1 k = 1 l ( l - k + 1 ) b k .
A = lim η cos n ,
ln A = lim n ln ( 1 - α 2 / 2 n 2 ) 1 / n ,
k = 1 n b k = b b n - 1 b - 1
k = 1 n - 1 ( n - k ) b k = m = 1 n - 1 k = 1 m b k = b ( b - 1 ) 2 [ b n - 1 - n ( b - 1 ) ] ,
R n = [ a b / ( b - 1 ) ] ( b n - 1 ) 1 - [ a 2 b / ( b - 1 ) 2 ] [ b 2 - 1 - n ( b - 1 ) ] .
a α N ,             N ϕ = 2 π Δ n 0 d / λ β
R N ( α / i β ) ( cos β - 1 + i sin β ) 1 + ( α / β ) 2 [ cos ( β - 1 ) + i ( sin β - β ) ] ,
T 2 ( α / β ) 2 ( 1 - cos β ) 1 - 2 ( α / β ) 2 ( 1 - cos β ) + ( α / β ) 4 ( 2 - 2 cos β - 2 β sin β + β 2 ) .
β 0 = 2 π K Δ n 0 d 0 / λ ,
P x N = cos δ C N ( 1 - a 2 k = 1 N - 1 ( N - k ) b k - a D k = 1 N b k - 1 ) , P y N = cos δ C N [ a k = 1 N b k + D ( b N - a 2 k = 1 N - 1 k b k ) ] .
P x N = C - N cos δ { 1 + X 2 ( cos β - 1 ) + i X ( X sin β - α ) + D ( cos β + i sin β ) [ - X sin β + i X ( 1 - cos β ) ] } , P y N = C - N cos δ { X sin β + i X ( 1 - cos β ) + D ( cos β + i sin β ) [ 1 + X 2 ( cos β - 1 ) + i X ( α - X sin β ) ] } .
P γ = P x N cos γ + P y N sin γ ,
P ( γ + 90 ° ) = P y N cos γ + P x N sin γ .
P γ = cos γ ( P x N + e P y N ) , P ( γ + 90 ° ) = cos γ ( P y N + e P x N ) .
P γ = C - N cos δ cos γ { 1 + X 2 ( cos β - 1 ) - D X sin β + e X sin β + e D [ cos β - α X sin β - X 2 ( cos β - 1 ) ] + i [ X 2 ( sin β - β ) + D X ( cos β - 1 ) + e X ( 1 - cos β ) + e D ( sin β + α X cos β - X 2 sin β ) ] } P ( γ + 90 ° ) = C - N cos δ cos γ { X sin β + D [ cos β - α X sin β - X 2 ( cos β - 1 ) ] + e + e X 2 ( cos β - 1 ) - e D X sin β + i [ X ( 1 - cos β ) + D ( sin β + α X cos β - X 2 sin β ) + e X 2 ( sin β - β ) + e D X ( cos β - 1 ) ] } .
I γ = P γ P γ * , I ( γ + 90 ° ) = P ( γ + 90 ° ) P ( γ + 90 ° ) * .
I γ = E { ( 1 + e 2 D 2 ) ( 1 + α 2 X 2 ) + 2 ( D - e ) 2 X 2 ( 1 - cos β ) + 2 e D [ cos β ( 1 - α 2 X 2 ) - 2 α X sin β ] + 2 ( e D + 1 ) ( D - e ) X [ α X ( 1 - cos β ) - sin β ] } , I ( γ + 90 ° ) = E { ( e 2 + D 2 ) ( 1 + α 2 X 2 ) + 2 ( 1 - e D ) 2 X 2 ( 1 - cos β ) + 2 e D [ cos β ( 1 - α 2 X 2 ) - 2 α X sin β ] + 2 ( e D - 1 ) ( e + D ) [ α X ( 1 - cos β ) - sin β ] } ,
I ( γ + 90 ° ) = 2 E { D 4 X 2 ( 1 - cos β ) + 2 D 3 X [ α X ( 1 - cos β ) - sin β ] + D 2 [ 1 + α 2 X 2 - 2 X 2 ( 1 - cos β ) + cos β ( 1 - α 2 X 2 ) - 2 α X sin β ] - 2 D X [ α X ( 1 - cos β ) - sin β ] + X 2 ( 1 - cos β ) } .
D = X ( 1 - cos β ) 2 [ X ( 1 - cos β ) - sin β ] .
D - X ( 1 - cos β ) 2 sin β - X Δ β 4