Abstract

We derive an extended Jones matrix method to treat the transmission of light through birefringent networks, where the incident angle of light and the optical axis of the birefringent media are arbitrary. As an example, we employ the method to analyze the leakage problem of a twisted nematic liquid-crystal display and to suggest its possible solutions. A generalization of the method covers all dielectric media, including uniaxial and biaxial crystals and also gyrotropic materials that exhibit optical rotation and Faraday rotation.

© 1993 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. See, for example, P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69, 742–756 (1979).
    [CrossRef]
  2. D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4-matrix formulation,”J. Opt. Soc. Am. 62, 502–510 (1972).
    [CrossRef]
  3. P. Yeh, “Extended Jones matrix method,”J. Opt. Soc. Am. 72, 507–513 (1982).
    [CrossRef]
  4. See, for example, P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
  5. A. R. MacGregor, “Method for computing homogeneous liquid-crystal conoscopic figures,” J. Opt. Soc. Am. A 7, 337–347 (1990).
    [CrossRef]
  6. A. Lien, “The general and simplified Jones matrix representations for the high pretilt twisted nematic cell,” J. Appl. Phys. 67, 2853–2856 (1990).
    [CrossRef]
  7. A. Lien, “Extended Jones matrix representation for the twisted nematic liquid-crystal display at oblique incidence,” Appl. Phys. Lett. 57, 2767–2769 (1990).
    [CrossRef]
  8. H. L. Ong, “Electro-optics of electrically controlled birefringence liquid-crystal displays by 2 × 2 propagation matrix and analytic expression at oblique angle,” Appl. Phys. Lett. 59, 155–157 (1991).
    [CrossRef]
  9. H. L. Ong, “Electro-optics of a twisted nematic liquid-crystal display by 2 × 2 propagation matrix at oblique angle,” Jpn. J. Appl. Phys. Lett. 30, L1028–L1031 (1991).
    [CrossRef]
  10. D. Taber, Department of Optical Devices, Rockwell International Science Center, Thousand Oaks, California 91360 (personal communication, 1992).

1991 (2)

H. L. Ong, “Electro-optics of electrically controlled birefringence liquid-crystal displays by 2 × 2 propagation matrix and analytic expression at oblique angle,” Appl. Phys. Lett. 59, 155–157 (1991).
[CrossRef]

H. L. Ong, “Electro-optics of a twisted nematic liquid-crystal display by 2 × 2 propagation matrix at oblique angle,” Jpn. J. Appl. Phys. Lett. 30, L1028–L1031 (1991).
[CrossRef]

1990 (3)

A. R. MacGregor, “Method for computing homogeneous liquid-crystal conoscopic figures,” J. Opt. Soc. Am. A 7, 337–347 (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]

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)

1972 (1)

Berreman, D. W.

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]

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

MacGregor, A. R.

Ong, H. L.

H. L. Ong, “Electro-optics of electrically controlled birefringence liquid-crystal displays by 2 × 2 propagation matrix and analytic expression at oblique angle,” Appl. Phys. Lett. 59, 155–157 (1991).
[CrossRef]

H. L. Ong, “Electro-optics of a twisted nematic liquid-crystal display by 2 × 2 propagation matrix at oblique angle,” Jpn. J. Appl. Phys. Lett. 30, L1028–L1031 (1991).
[CrossRef]

Taber, D.

D. Taber, Department of Optical Devices, Rockwell International Science Center, Thousand Oaks, California 91360 (personal communication, 1992).

Yeh, P.

Appl. Phys. Lett. (2)

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

H. L. Ong, “Electro-optics of electrically controlled birefringence liquid-crystal displays by 2 × 2 propagation matrix and analytic expression at oblique angle,” Appl. Phys. Lett. 59, 155–157 (1991).
[CrossRef]

J. Appl. Phys. (1)

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. Opt. Soc. Am. (3)

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

Jpn. J. Appl. Phys. Lett. (1)

H. L. Ong, “Electro-optics of a twisted nematic liquid-crystal display by 2 × 2 propagation matrix at oblique angle,” Jpn. J. Appl. Phys. Lett. 30, L1028–L1031 (1991).
[CrossRef]

Other (2)

D. Taber, Department of Optical Devices, Rockwell International Science Center, Thousand Oaks, California 91360 (personal communication, 1992).

See, for example, P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

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 (8)

Fig. 1
Fig. 1

Reflection and refraction of light at an interface between two media. The coordinates are chosen such that the (x, y) plane contains the interface and the z direction is perpendicular to the interface.

Fig. 2
Fig. 2

Orientation of the c axis. θc and ϕc are the angle between the c axis and the z direction and the angle between the projection of the c axis on the (x, y) plane and the x direction, respectively.

Fig. 3
Fig. 3

LCD medium divided into N layers. Each layer can be considered as a uniaxial medium. The orientation of the c axis may change from layer to layer (e.g., a TNLCD). The ordinary and extraordinary indices of refraction, no and ne, respectively, are constants for all layers.

Fig. 4
Fig. 4

TNLCD with a TNLC sandwiched between two crossed polarizers.

Fig. 5
Fig. 5

Angles θc and ϕc of the c axis of each plate when the applied voltage is 8 V (OFF state).

Fig. 6
Fig. 6

Transmission contour plot for the TNLCD in the OFF state. The incident angle varies from θ = 0° to 90° and from ϕ = 0° to 360°. The transmittances for the contour lines are 0.025, 0.05, 0.075, 0.1, 0.125, 0.15, 0.175, 0.2, 0.225, 0.25, 0.275, and 0.3. The following parameters are chosen: no = 1.52 for the two ideal O-type polarizers; thickness d = 5.9 μm, no = 1.487 and ne = 1.568 for the TNLC; and λ = 0.55 μm for the incident light.

Fig. 7
Fig. 7

TNLCD with a second, negatively birefringent TNLC that acts as a compensator. The compensator TNLC twists in the opposite direction to that of the original TNLC.

Fig. 8
Fig. 8

(a) Contour plot for the transmittance with arbitrary incident angles (with the compensator TNLC). The incident angle varies from ϕ = 0° to 90° and from ϕ = 0° to 360°. The transmittances for the contour lines are 0.0025, 0.005, 0.0075, 0.01, 0.0125, 0.015, 0.0175, 0.02, 0.0225, and 0.025. (b) Contour plot for the crossed polarizers. The transmittances for the contour lines are 0.0025, 0.005, 0.0075, 0.01, 0.0125, 0.015, and 0.0175.

Equations (42)

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

Incident : E = [ A 1 p ^ i 1 exp ( - i k i 1 · r ) + A 2 p ^ i 2 exp ( - i k i 2 · r ) ] exp ( i ω t ) , Reflected : E = [ B 1 p ^ r 1 exp ( - i k r 1 · r ) + B 2 p ^ r 2 exp ( - i k r 2 · r ) ] exp ( i ω t ) , Refracted : E = [ C 1 p ^ t 1 exp ( - i k t 1 · r ) + C 2 p ^ t 2 exp ( - i k t 2 · r ) ] exp ( i ω t ) ,
H = ( i / ω μ ) × E
Incident : H = ( 1 / ω μ ) [ A 1 k i 1 × p ^ i 1 exp ( - i k i 1 · r ) + A 2 k i 2 × p ^ i 2 exp ( - i k i 2 · r ) ] × exp ( i ω t ) , Reflected : H = ( 1 / ω μ ) [ B 1 k r 1 × p ^ r 1 exp ( - i k r 1 · r ) + B 2 k r 2 × p ^ r 2 exp ( - i k r 2 · r ) ] × exp ( i ω t ) , Refracted : H = ( 1 / ω μ ) [ C 1 k t 1 × p ^ t 1 exp ( - i k t 1 · r ) + C 2 k t 2 × p ^ t 2 exp ( - i k t 2 · r ) ] × exp ( i ω t ) ,
( k i 1 · x ^ ) = ( k i 2 · x ^ ) = ( k r 1 · x ^ ) = ( k r 2 · x ^ ) = ( k t 1 · x ^ ) = ( k t 2 · x ^ ) = α , ( k i 1 · y ^ ) = ( k i 2 · y ^ ) = ( k r 1 · y ^ ) = ( k r 2 · y ^ ) = ( k t 1 · y ^ ) = ( k t 2 · y ^ ) = β ,
C o = A s t s o + A p t p o ,             C e = A s t s e + A p t p e ,
t s o = 2 k z D / ( A D - B C ) t p o = - 2 k z B / ( A D - B C ) , t s e = - 2 k z C / ( A D - B C ) , t p e = 2 k z A / ( A D - B C ) ,
A = o ^ · ( g ^ × k ) + o ^ · ( g ^ × k o ) , B = e ^ · ( g ^ × k ) + e ^ · ( g ^ × k e ) , C = k o ^ · g ^ - [ ( g ^ × k ) · ( k o × o ^ ) / k ] , D = k e ^ · g ^ - [ ( g ^ × k ) · ( k e × e ^ ) / k ] ,
k z = [ ( n ω / c ) 2 - α 2 - β 2 ] 1 / 2 ,
c ^ = ( x ^ cos ϕ c + y ^ sin ϕ c ) sin θ c + z ^ cos θ c .
k e a = ( α cos ϕ c + β sin ϕ c ) cos θ c - k e z sin θ c , k e b = - α sin ϕ c + β cos ϕ c , k e c = ( α cos ϕ c + β sin ϕ c ) sin θ c + k e z cos θ c ,
k e a 2 + k e b 2 n e 2 + k e c 2 n o 2 = ( ω c ) 2 ,
u k e z 2 - v k e z + w = 0 ,
u = sin 2 θ c n e 2 + cos 2 θ c n o 2 , v = k d sin ( 2 θ c ) ( 1 n e 2 - 1 n o 2 ) , w = k d 2 cos 2 θ c + k e b 2 n e 2 + k d 2 sin 2 θ c n o 2 - ( ω c ) 2 ,
k d = α cos ϕ c + β sin ϕ c .
k e z = [ v + ( v 2 - 4 u w ) 1 / 2 ] / 2 u ,
k o z = [ ( n o ω / c ) 2 - α 2 - β 2 ] 1 / 2 .
[ C o C e ] = [ t s o t p o t s e t p e ] [ A s A p ] ,
[ A s A p ] = [ t o s t e s t o p t e p ] [ exp ( - i k o z d ) 0 0 exp ( - i k e z d ) ] × [ t s o t p o t s e t p e ] [ A s A p ] ,
[ A s A p ] = D o P D i [ A s A p ] ,
P = [ exp ( - i k o z d ) 0 0 exp ( - i k e z d ) ] , D o = [ t o s t e s t o p t e p ] , D i = [ t s o t p o t s e t p e ] .
[ A s A p ] = [ M 11 M 12 M 21 M 22 ] [ A s A p ] .
T = ( A s 2 + A p 2 ) / ( A s 2 + A p 2 ) .
T = 1 / 2 ( M 11 2 + M 12 2 + M 21 2 + M 22 2 ) .
o ^ = ( c ^ × k o ) / c ^ × k o ,
e ^ ( k o × o ^ ) / k o × o ^ .
t s o s ^ · o ^ t s , t p o p ^ o · o ^ t p , t s e s ^ · e ^ t s , t p e p ^ o · e ^ t p t o s o ^ · s ^ t s , t e s e ^ · s ^ t s , t o p o ^ · p ^ o t p t e p e ^ · p ^ o t p ,
p ^ o = ( k o × s ^ ) / k o × s ^ ,
t s = 2 n cos θ n cos θ + n o cos θ o , t p = 2 n cos θ n cos θ o + n o cos θ , t s = 2 n o cos θ o n o cos θ o + n cos θ , t p = 2 n o cos θ o n o cos θ + n cos θ o ,
( A o A e ) n + 1 = [ t s o t p o t s e t p e ] n + 1 [ t o s t e s t o p t e p ] n [ A o A e ] n ,
[ A o A e ] n + 1 = [ s ^ · o ^ p ^ o · o s ^ · e ^ p ^ o · e ^ ] n + 1 [ o ^ · s ^ e ^ · s ^ o ^ · p ^ o e ^ · p ^ o ] n [ A o A e ] n = [ o ^ n · o ^ n + 1 e ^ n · o ^ n + 1 o ^ n · e ^ n + 1 e ^ n · e ^ n + 1 ] [ A o A e ] n ,
D n , n + 1 = [ o ^ n · o ^ n + 1 e ^ n · o ^ n + 1 o ^ n · e ^ n + 1 e ^ n · e ^ n + 1 ] .
[ A s A p ] = M [ A s A p ] ,
M = D o P N D N - 1 , N P N - 1 D N - 2 , N - 1 D 2 , 3 P 2 D 1 , 2 P 1 D i ,
M = M ¯ N M ¯ N - 1 M ¯ 2 M ¯ 1 M N M N - 1 M 2 M 1 ,
k × ( k × E ) + ω 2 μ E = 0 ,
[ ω 2 μ x x - k y 2 - k z 2 ω 2 μ x y + k x k y ω 2 μ x z + k x k z ω 2 μ y x + k y k x ω 2 μ y y - k x 2 - k z 2 ω 2 μ y z + k y k z ω 2 μ z x + k z k x ω 2 μ z y + k z k y ω 2 μ z z - k x 2 - k y 2 ] [ E x E y E z ] = 0 ,
det [ ω 2 μ x x - k y 2 - k z 2 ω 2 μ x y + k x k y ω 2 μ x z + k x k z ω 2 μ y x + k y k x ω 2 μ y y - k x 2 - k z 2 ω 2 μ y z + k y k z ω 2 μ z x + k z k x ω 2 μ z y + k z k y ω 2 μ z z - k x 2 - k y 2 ] = 0.
P = [ exp ( - i k 1 z d ) 0 0 exp ( - i k 2 z d ) ] .
[ C 1 C 2 ] = D 12 [ A 1 A 2 ] = [ t 11 t 21 t 12 t 22 ] [ A 1 A 2 ] .
t s = 2 n 1 cos θ 1 n 1 cos θ 1 + n 2 cos θ 2 ,             t p = 2 n 1 cos θ 1 n 1 cos θ 2 + n 2 cos θ 1 ,
D 12 = [ s ^ · p ^ t 1 p ^ 2 · p ^ t 1 s ^ · p ^ t 2 p ^ 2 · p ^ t 2 ] [ t s 0 0 t p ] [ p ^ i 1 · s ^ p ^ i 2 · s ^ p ^ i 1 · p ^ 1 p ^ i 2 · p ^ 1 ] = [ t s ( p ^ i 1 · s ^ ) ( s ^ · p ^ t 1 ) + t p ( p ^ i 1 · p ^ 1 ) ( p ^ 2 · p ^ t 1 ) t s ( p ^ i 2 · s ^ ) ( s ^ · p ^ t 1 ) + t p ( p ^ i 2 · p ^ 1 ) ( p ^ 2 · p ^ t 1 ) t s ( p ^ i 1 · s ^ ) ( s ^ · p ^ t 2 ) + t p ( p ^ i 1 · p ^ 1 ) ( p ^ 2 · p ^ 2 ) t s ( p ^ i 2 · s ^ ) ( s ^ · p ^ t 2 ) + t p ( p ^ i 2 · p ^ 1 ) ( p ^ 2 · p ^ t 2 ) ] ,
D 12 = [ p ^ i 1 · p ^ t 1 p ^ i 2 · p ^ t 1 p ^ i 1 · p ^ t 2 p ^ i 2 · p ^ t 2 ] .

Metrics