Abstract

We present a general formulation based on the Jones-matrix theory for reciprocal nonabsorbing polarization devices, including polarization interference filters and liquid-crystal displays. The development of this formulation is based on general symmetry conditions that relate the Jones matrix when the device is illuminated from the front side and from the back side. The application to liquid-crystal displays results in a constraint of the Jones-matrix elements, which represents a generalization of the existing models that explain their modulation properties.

© 2000 Optical Society of America

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References

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  1. R. C. Jones, “New calculus for the treatment of optical systems,” J. Opt. Soc. Am. 31, 488–493 (1941).
    [CrossRef]
  2. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
  3. B. E. A. Saleh, M. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
  4. C. Brosseau, Fundamentals of Polarized Light (Wiley, New York, 1998).
  5. U. Efron, ed., Spatial Light Modulator Technology (Marcel Dekker, New York, 1995).
  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. J. A. Coy, M. Zaldarriaga, D. F. Grosz, O. E. Martı́nez, “Characterization of a liquid crystal television as a programmable spatial light modulator,” Opt. Eng. 35, 15–19 (1996).
    [CrossRef]
  8. J. L. de Bougrenet de la Tocnaye, L. Dupont, “Complex amplitude modulation by use of liquid-crystal spatial light modulators,” Appl. Opt. 36, 1730–1741 (1997).
    [CrossRef] [PubMed]
  9. I. Moreno, N. Bennis, J. A. Davis, C. Ferreira, “Twist angle determination in liquid crystal displays by location of local adiabatic points,” Opt. Commun. 158, 231–238 (1998).
    [CrossRef]
  10. K. Ohkubo, J. Ohtsubo, “Evaluation of LCTV as a spatial light modulator,” Opt. Commun. 102, 116–124 (1993).
    [CrossRef]
  11. S. Huard, Polarisation de la lumière (Masson, Paris, 1994).
  12. J. A. Davis, I. Moreno, P. Tsai, “Polarization eigenstates for twisted-nematic liquid crystal displays,” Appl. Opt. 37, 937–945 (1998).
    [CrossRef]
  13. I. Moreno, J. A. Davis, K. G. D’Nelly, D. B. Allison, “Transmission and phase measurements for polarization eigenvectors in twisted-nematic liquid-crystal spatial light modulators,” Opt. Eng. 37, 3048–3052 (1998).
    [CrossRef]
  14. D. B. Taber, J. A. Davis, L. A. Holloway, O. Almagor, “Optically controlled Fabry–Perot interferometer using a liquid crystal light valve,” Appl. Opt. 29, 2624–2631 (1990).
    [CrossRef]

1998 (3)

I. Moreno, N. Bennis, J. A. Davis, C. Ferreira, “Twist angle determination in liquid crystal displays by location of local adiabatic points,” Opt. Commun. 158, 231–238 (1998).
[CrossRef]

J. A. Davis, I. Moreno, P. Tsai, “Polarization eigenstates for twisted-nematic liquid crystal displays,” Appl. Opt. 37, 937–945 (1998).
[CrossRef]

I. Moreno, J. A. Davis, K. G. D’Nelly, D. B. Allison, “Transmission and phase measurements for polarization eigenvectors in twisted-nematic liquid-crystal spatial light modulators,” Opt. Eng. 37, 3048–3052 (1998).
[CrossRef]

1997 (1)

1996 (1)

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

1993 (1)

K. Ohkubo, J. Ohtsubo, “Evaluation of LCTV as a spatial light modulator,” Opt. Commun. 102, 116–124 (1993).
[CrossRef]

1990 (2)

D. B. Taber, J. A. Davis, L. A. Holloway, O. Almagor, “Optically controlled Fabry–Perot interferometer using a liquid crystal light valve,” Appl. Opt. 29, 2624–2631 (1990).
[CrossRef]

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]

1941 (1)

Allison, D. B.

I. Moreno, J. A. Davis, K. G. D’Nelly, D. B. Allison, “Transmission and phase measurements for polarization eigenvectors in twisted-nematic liquid-crystal spatial light modulators,” Opt. Eng. 37, 3048–3052 (1998).
[CrossRef]

Almagor, O.

D. B. Taber, J. A. Davis, L. A. Holloway, O. Almagor, “Optically controlled Fabry–Perot interferometer using a liquid crystal light valve,” Appl. Opt. 29, 2624–2631 (1990).
[CrossRef]

Bennis, N.

I. Moreno, N. Bennis, J. A. Davis, C. Ferreira, “Twist angle determination in liquid crystal displays by location of local adiabatic points,” Opt. Commun. 158, 231–238 (1998).
[CrossRef]

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light (Wiley, New York, 1998).

Coy, J. A.

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

D’Nelly, K. G.

I. Moreno, J. A. Davis, K. G. D’Nelly, D. B. Allison, “Transmission and phase measurements for polarization eigenvectors in twisted-nematic liquid-crystal spatial light modulators,” Opt. Eng. 37, 3048–3052 (1998).
[CrossRef]

Davis, J. A.

I. Moreno, J. A. Davis, K. G. D’Nelly, D. B. Allison, “Transmission and phase measurements for polarization eigenvectors in twisted-nematic liquid-crystal spatial light modulators,” Opt. Eng. 37, 3048–3052 (1998).
[CrossRef]

J. A. Davis, I. Moreno, P. Tsai, “Polarization eigenstates for twisted-nematic liquid crystal displays,” Appl. Opt. 37, 937–945 (1998).
[CrossRef]

I. Moreno, N. Bennis, J. A. Davis, C. Ferreira, “Twist angle determination in liquid crystal displays by location of local adiabatic points,” Opt. Commun. 158, 231–238 (1998).
[CrossRef]

D. B. Taber, J. A. Davis, L. A. Holloway, O. Almagor, “Optically controlled Fabry–Perot interferometer using a liquid crystal light valve,” Appl. Opt. 29, 2624–2631 (1990).
[CrossRef]

de Bougrenet de la Tocnaye, J. L.

Dupont, L.

Ferreira, C.

I. Moreno, N. Bennis, J. A. Davis, C. Ferreira, “Twist angle determination in liquid crystal displays by location of local adiabatic points,” Opt. Commun. 158, 231–238 (1998).
[CrossRef]

Grosz, D. F.

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

Holloway, L. A.

D. B. Taber, J. A. Davis, L. A. Holloway, O. Almagor, “Optically controlled Fabry–Perot interferometer using a liquid crystal light valve,” Appl. Opt. 29, 2624–2631 (1990).
[CrossRef]

Huard, S.

S. Huard, Polarisation de la lumière (Masson, Paris, 1994).

Jones, R. C.

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]

Marti´nez, O. E.

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

Moreno, I.

I. Moreno, N. Bennis, J. A. Davis, C. Ferreira, “Twist angle determination in liquid crystal displays by location of local adiabatic points,” Opt. Commun. 158, 231–238 (1998).
[CrossRef]

I. Moreno, J. A. Davis, K. G. D’Nelly, D. B. Allison, “Transmission and phase measurements for polarization eigenvectors in twisted-nematic liquid-crystal spatial light modulators,” Opt. Eng. 37, 3048–3052 (1998).
[CrossRef]

J. A. Davis, I. Moreno, P. Tsai, “Polarization eigenstates for twisted-nematic liquid crystal displays,” Appl. Opt. 37, 937–945 (1998).
[CrossRef]

Ohkubo, K.

K. Ohkubo, J. Ohtsubo, “Evaluation of LCTV as a spatial light modulator,” Opt. Commun. 102, 116–124 (1993).
[CrossRef]

Ohtsubo, J.

K. Ohkubo, J. Ohtsubo, “Evaluation of LCTV as a spatial light modulator,” Opt. Commun. 102, 116–124 (1993).
[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]

B. E. A. Saleh, M. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

Taber, D. B.

D. B. Taber, J. A. Davis, L. A. Holloway, O. Almagor, “Optically controlled Fabry–Perot interferometer using a liquid crystal light valve,” Appl. Opt. 29, 2624–2631 (1990).
[CrossRef]

Teich, M.

B. E. A. Saleh, M. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

Tsai, P.

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Zaldarriaga, M.

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

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

Opt. Commun. (2)

I. Moreno, N. Bennis, J. A. Davis, C. Ferreira, “Twist angle determination in liquid crystal displays by location of local adiabatic points,” Opt. Commun. 158, 231–238 (1998).
[CrossRef]

K. Ohkubo, J. Ohtsubo, “Evaluation of LCTV as a spatial light modulator,” Opt. Commun. 102, 116–124 (1993).
[CrossRef]

Opt. Eng. (3)

I. Moreno, J. A. Davis, K. G. D’Nelly, D. B. Allison, “Transmission and phase measurements for polarization eigenvectors in twisted-nematic liquid-crystal spatial light modulators,” Opt. Eng. 37, 3048–3052 (1998).
[CrossRef]

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]

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

Other (5)

S. Huard, Polarisation de la lumière (Masson, Paris, 1994).

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

B. E. A. Saleh, M. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

C. Brosseau, Fundamentals of Polarized Light (Wiley, New York, 1998).

U. Efron, ed., Spatial Light Modulator Technology (Marcel Dekker, New York, 1995).

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Figures (4)

Fig. 1
Fig. 1

Polarization device inserted between a polarizer–analyzer pair.

Fig. 2
Fig. 2

(a) Propagation of an arbitrary elliptical state of polarization. Coordinate system S is used to describe forward propagation, and system S˜ is used to describe backward propagation. (b) Polarization ellipse for forward propagation. The dot indicates that propagation is in the sense of leaving the page. (c) Polarization ellipse for backward propagation. The cross indicates propagation in the sense of entering the page. (d) Same as (c) but considering propagation as leaving the page.

Fig. 3
Fig. 3

Arbitrary elliptical state of polarization passing through a polarization device. Coordinate system S is used to describe forward propagation, and system S˜ is used to describe backward propagation. If the device is reciprocal, V1 is transformed into V2 in the forward way, while V˜2 is transformed into V˜1 in the backward way.

Fig. 4
Fig. 4

(a) Twist angle and (b) effective birefringence for a LCD versus depth of the cell: assumptions of the Lu and Saleh model6 (solid lines), the Coy et al. model7 (dotted–dashed curves), and a realistic behavior (dashed curves).

Equations (53)

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Md=exp(-iβ)AB-B*A*,
|A|2+|B|2=1,
Md=exp(-iβ)X-iYZ-iW-Z-iWX+iY,
A=X-iY,
B=Z-iW.
X2+Y2+Z2+W2=1.
VOUT=R(-φ2)P(0)R(+φ2)MdR(-φ1)10,
P(0)=1000,
R(φ)=cos φsin φ-sin φcos φ,
VIN=R(-φ1)10=cos φ1sin φ1.
I=[X cos(φ1-φ2)+Z sin(φ1-φ2)]2+[Y cos(φ1+φ2)+W sin(φ1+φ2)]2.
ϕ=β+tan-1Y cos(φ1+φ2)+W sin(φ1+φ2)X cos(φ1-φ2)+Z sin(φ1-φ2).
EyEx=1-X2+YZ2+W2 i expi tan-1WZ,
EyEx=1-X2-YZ2+W2 (-i)expi tan-1WZ,
V=Exexp(iδ)Ey.
V˜=Exexp[i(π-δ)]Ey=-Exexp(-iδ)Ey.
V˜=JV*,
J=-100+1.
V2=MdV1.
V˜1=M˜dV˜2,
M˜d=JMdtJ.
M˜d=JMdJ.
MWP(δ, θ)
=cosδ2+i sinδ2cos(2θ)i sinδ2sin(2θ)i sinδ2sin 2(θ)cosδ2-i sinδ2cos(2θ),
 
Md=JMdtJ,
MSolc(Γ, ρ)=A sin(mKΛ)-sin[(m-1)KΛ]sin(KΛ)B sin(mKΛ)sin(KΛ)-B sin(mKΛ)sin(KΛ)A*sin(mKΛ)-sin[(m-1)KΛ]sin(KΛ),
A=[cos(12 Γ)-i cos(2ρ)sin(12Γ)]2+sin2(2ρ)sin2(12Γ),
B=sin(4ρ)sin2(12Γ),
cos(KΛ)=12(A+A*)=1-2 cos2(2ρ)sin2(12Γ).
X=cos212 Γ-sin212 Γcos(4ρ)sin(mKΛ)sin(KΛ)-sin[(m-1)KΛ]sin(KΛ),
Y=cos(2ρ)sin Γ sin(mKΛ)sin(KΛ),
Z=sin(4ρ)sin212 Γsin(mKΛ)sin(KΛ),
W=0.
M˜d=R(-π/2)MdR(+π/2).
M90 Twist(β)
=exp(-iβ)π2γsin γcos γ+i βγsin γ-cos γ+i βγsin γπ2γsin γ,
 
MLCD=exp(-iβ)R(-α)M,
I=[X cos(φ1-φ2+α)+Z sin(φ1-φ2+α)]2+[Y cos(φ1+φ2-α)+W sin(φ1+φ2-α)]2,
ϕ=β+tan-1×Y cos(φ1+φ2-α)+W sin(φ1+φ2-α)X cos(φ1-φ2+α)+Z sin(φ1-φ2+α),
M˜LCD=R(+α)MLCDR(-α).
M=JMtJ.
MLCDLu&Saleh(α, β)
=exp(-iβ)R(-α)×cos γ-i βγsin γαγsin γ-αγsin γcos γ+i βγsin γ,
MLCDCoy et al.(α, β, δ)=[R(-α)W0(δ)R(+α)]×MLCDLu&Saleh(α, β)W0(δ),
W0(δ)=100exp(iδ),
MLCDCoy et al.(α, β, δ)=exp(-iβ)R(-α)X-iYZ-ZX+iY,
 
X=cos γ cos δ-βγsin γ sin δ,
Y=cos γ sin δ+βγsin γ cos δ,
Z=αγsin γ.
MLCD(α, β, X, Y, Z)=exp(-iβ)R(-α)×X-iYZ-ZX+iY,

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