Abstract

This work presents a detailed analysis of a liquid crystal (LC) phase diffraction grating based on a approach combining vector theory of scattering and coupled mode analysis. In general, the coupled mode analysis gives a solution for the diffracted field regardless of aperture and the polarization state of the incident light. However, the aperture of the incident light defines the angular selectivity of the diffraction grating as well as the distribution of the intensity of the diffractive maximums. The solution of the vector theory of scattering in combination with the coupled mode analysis for diffraction of the light beam with finite aperture has allowed one to optimize the parameters of the high efficiency diffractive LC grating. The analytic solutions here were verified with experimental results for a reverse-twisted LC grating and a comparison with the standard Gooch-Tarry’s method, which typically applied for a twisted nematic LC display.

© 2010 Optical Society of America

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References

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  1. Y. Hori, K. Asai, and M. Fukai, “Field Controllable Liquid-Crystal Light Valves,” IEEE Trans. Electron. Dev. 26(11), 1734–1737 (1979).
    [CrossRef]
  2. M. Fritsch, H. Wohler, G. Haas, and D. Mlynski, “Liquid-Crystal Phase Modulator for Large Screen Projection,” IEEE Trans. Electron. Dev. 36(9), 1882–1887 (1989).
    [CrossRef]
  3. P. Shannon, W. Gibbons, S. Sun, and B. Swetlin, “Surface-Mediated Alignment of Nematic Liquid Crystals with Polarized Laser Light,” Nature 351, 351–352 (1991).
  4. P. J. Bos, J. Chen, J. W. Doane, B. Smith, C. Holton, and W. Glenn, “An Optically Active Diffractive Device for a High Efficiency Light Valve”, Digest of Technical Papers, Society for Information Display International Symposium, pp. 601–604 (1995).
  5. W. E. Glenn, “The Use of Optical Diffraction to Produce Images,” J. SID 5(3), (1997).
  6. C. E. Holton, P. Bos, M. Miller, and W. Glenn, “Patterned Alignment Liquid Crystal Diffractive Spatial Light Modulators and Devices,” SPIE Proc. 3292–04, Photonics West, San Jose, CA, Jan., (1998).
    [CrossRef]
  7. G. Kreymerman, “Liquid crystal diffractive phase grating as light modulator for projection display,” Opt. Eng. 45(11), 116202 (2006).
    [CrossRef]
  8. P. Yeh, “Extended Jones matrix method,” J. Opt. Soc. Am. 72(4), 507–513 (1982).
    [CrossRef]
  9. K. H. Yang, and M. Lu, “Nematic LC modes and LC phase grating for reflective spatial light modulators,” IBM J. Res. Develop. High-resolution displays, 42, 401 (1998).
    [CrossRef]
  10. L. D. Landau, and E. M. Lifschitz, Electrodynamics of Continuous Media (Oxford, England: Pergamon Press, 1984).
  11. A. Sommerfeld, Partial Differential Equations in Physics, (Academic Press, New York, New York, 1949).
  12. C. H. Gooch, and H. A. Tarry, “The optical properties of twisted nematic liquid crystal structures with twist angles,” J. Phys. D Appl. Phys. 8(13), 1575–1584 (1975).
    [CrossRef]

2006

G. Kreymerman, “Liquid crystal diffractive phase grating as light modulator for projection display,” Opt. Eng. 45(11), 116202 (2006).
[CrossRef]

1997

W. E. Glenn, “The Use of Optical Diffraction to Produce Images,” J. SID 5(3), (1997).

1991

P. Shannon, W. Gibbons, S. Sun, and B. Swetlin, “Surface-Mediated Alignment of Nematic Liquid Crystals with Polarized Laser Light,” Nature 351, 351–352 (1991).

1989

M. Fritsch, H. Wohler, G. Haas, and D. Mlynski, “Liquid-Crystal Phase Modulator for Large Screen Projection,” IEEE Trans. Electron. Dev. 36(9), 1882–1887 (1989).
[CrossRef]

1982

1979

Y. Hori, K. Asai, and M. Fukai, “Field Controllable Liquid-Crystal Light Valves,” IEEE Trans. Electron. Dev. 26(11), 1734–1737 (1979).
[CrossRef]

1975

C. H. Gooch, and H. A. Tarry, “The optical properties of twisted nematic liquid crystal structures with twist angles,” J. Phys. D Appl. Phys. 8(13), 1575–1584 (1975).
[CrossRef]

Asai, K.

Y. Hori, K. Asai, and M. Fukai, “Field Controllable Liquid-Crystal Light Valves,” IEEE Trans. Electron. Dev. 26(11), 1734–1737 (1979).
[CrossRef]

Fritsch, M.

M. Fritsch, H. Wohler, G. Haas, and D. Mlynski, “Liquid-Crystal Phase Modulator for Large Screen Projection,” IEEE Trans. Electron. Dev. 36(9), 1882–1887 (1989).
[CrossRef]

Fukai, M.

Y. Hori, K. Asai, and M. Fukai, “Field Controllable Liquid-Crystal Light Valves,” IEEE Trans. Electron. Dev. 26(11), 1734–1737 (1979).
[CrossRef]

Gibbons, W.

P. Shannon, W. Gibbons, S. Sun, and B. Swetlin, “Surface-Mediated Alignment of Nematic Liquid Crystals with Polarized Laser Light,” Nature 351, 351–352 (1991).

Glenn, W. E.

W. E. Glenn, “The Use of Optical Diffraction to Produce Images,” J. SID 5(3), (1997).

Gooch, C. H.

C. H. Gooch, and H. A. Tarry, “The optical properties of twisted nematic liquid crystal structures with twist angles,” J. Phys. D Appl. Phys. 8(13), 1575–1584 (1975).
[CrossRef]

Haas, G.

M. Fritsch, H. Wohler, G. Haas, and D. Mlynski, “Liquid-Crystal Phase Modulator for Large Screen Projection,” IEEE Trans. Electron. Dev. 36(9), 1882–1887 (1989).
[CrossRef]

Hori, Y.

Y. Hori, K. Asai, and M. Fukai, “Field Controllable Liquid-Crystal Light Valves,” IEEE Trans. Electron. Dev. 26(11), 1734–1737 (1979).
[CrossRef]

Kreymerman, G.

G. Kreymerman, “Liquid crystal diffractive phase grating as light modulator for projection display,” Opt. Eng. 45(11), 116202 (2006).
[CrossRef]

Mlynski, D.

M. Fritsch, H. Wohler, G. Haas, and D. Mlynski, “Liquid-Crystal Phase Modulator for Large Screen Projection,” IEEE Trans. Electron. Dev. 36(9), 1882–1887 (1989).
[CrossRef]

Shannon, P.

P. Shannon, W. Gibbons, S. Sun, and B. Swetlin, “Surface-Mediated Alignment of Nematic Liquid Crystals with Polarized Laser Light,” Nature 351, 351–352 (1991).

Sun, S.

P. Shannon, W. Gibbons, S. Sun, and B. Swetlin, “Surface-Mediated Alignment of Nematic Liquid Crystals with Polarized Laser Light,” Nature 351, 351–352 (1991).

Swetlin, B.

P. Shannon, W. Gibbons, S. Sun, and B. Swetlin, “Surface-Mediated Alignment of Nematic Liquid Crystals with Polarized Laser Light,” Nature 351, 351–352 (1991).

Tarry, H. A.

C. H. Gooch, and H. A. Tarry, “The optical properties of twisted nematic liquid crystal structures with twist angles,” J. Phys. D Appl. Phys. 8(13), 1575–1584 (1975).
[CrossRef]

Wohler, H.

M. Fritsch, H. Wohler, G. Haas, and D. Mlynski, “Liquid-Crystal Phase Modulator for Large Screen Projection,” IEEE Trans. Electron. Dev. 36(9), 1882–1887 (1989).
[CrossRef]

Yeh, P.

IEEE Trans. Electron. Dev.

Y. Hori, K. Asai, and M. Fukai, “Field Controllable Liquid-Crystal Light Valves,” IEEE Trans. Electron. Dev. 26(11), 1734–1737 (1979).
[CrossRef]

M. Fritsch, H. Wohler, G. Haas, and D. Mlynski, “Liquid-Crystal Phase Modulator for Large Screen Projection,” IEEE Trans. Electron. Dev. 36(9), 1882–1887 (1989).
[CrossRef]

J. Opt. Soc. Am.

J. Phys. D Appl. Phys.

C. H. Gooch, and H. A. Tarry, “The optical properties of twisted nematic liquid crystal structures with twist angles,” J. Phys. D Appl. Phys. 8(13), 1575–1584 (1975).
[CrossRef]

J. SID

W. E. Glenn, “The Use of Optical Diffraction to Produce Images,” J. SID 5(3), (1997).

Nature

P. Shannon, W. Gibbons, S. Sun, and B. Swetlin, “Surface-Mediated Alignment of Nematic Liquid Crystals with Polarized Laser Light,” Nature 351, 351–352 (1991).

Opt. Eng.

G. Kreymerman, “Liquid crystal diffractive phase grating as light modulator for projection display,” Opt. Eng. 45(11), 116202 (2006).
[CrossRef]

Other

K. H. Yang, and M. Lu, “Nematic LC modes and LC phase grating for reflective spatial light modulators,” IBM J. Res. Develop. High-resolution displays, 42, 401 (1998).
[CrossRef]

L. D. Landau, and E. M. Lifschitz, Electrodynamics of Continuous Media (Oxford, England: Pergamon Press, 1984).

A. Sommerfeld, Partial Differential Equations in Physics, (Academic Press, New York, New York, 1949).

P. J. Bos, J. Chen, J. W. Doane, B. Smith, C. Holton, and W. Glenn, “An Optically Active Diffractive Device for a High Efficiency Light Valve”, Digest of Technical Papers, Society for Information Display International Symposium, pp. 601–604 (1995).

C. E. Holton, P. Bos, M. Miller, and W. Glenn, “Patterned Alignment Liquid Crystal Diffractive Spatial Light Modulators and Devices,” SPIE Proc. 3292–04, Photonics West, San Jose, CA, Jan., (1998).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

The element (or pixel) of a reverse twisted LC phase diffractive grating with pre-tilt angle ϕ and zero voltage applied to the cell.

Fig. 2.
Fig. 2.

The angular distribution of the diffraction maximums for a reverse TN-LC (E-7) diffractive grating; solid line (aperture of beam 0.5mm) and doted line (aperture of beam 0.2mm). The stems with boxes on top are experimental result. The grating parameters are; ne = 1.7765, no=1.5225, b=2µm, L=75µm, pre-tilt angle 5°, wavelength of light 543nm.

Fig. 3.
Fig. 3.

The diffraction efficiency as function of alternative voltage (1 kHz) applied to electrodes of reverse TN-LC diffraction grating for two orthogonal polarization of incident light. The circles are for polarization parallel to vector director of LC molecules, while the diamonds are for polarization orthogonal to vector director. L=75µm, b=2µm, pre-tilt angle 5°.

Fig. 4.
Fig. 4.

The diffraction transmission N∣B(γ)∣2 of a reverse TN LC (E-7) grating (solid line) and the transmission T(γ), of a 90°TN cell (dashed line) placed between parallel polarizers at zero voltage between electrodes as functions of optical retardation γ.

Fig. 5.
Fig. 5.

The diffraction efficiency for the cells of a reverse TN-LC (E-7) grating for different pre-tilt angles 5° (L=75µm), 10° (L=50µm), 25° (L=24µm), 39° (L=50µm). The circles are for incident light with polarization parallel to vector director, the diamonds are for polarization orthogonal to vector director of LC molecules on entry to cells (b=2µm for all cells).

Fig. 6.
Fig. 6.

The diffraction efficiency of reverse TN-LC (E-7) grating as function of incident angle for two cells with pre-tilt angles 5° - solid lines (L=75µm, b=2µm) and 10° - doted lines (L=50µm, b=2µm). The circles are for incident light with polarization parallel to vector director; the diamonds are for polarization orthogonal to vector director of LC molecules on entry to cell.

Fig. 7.
Fig. 7.

The diffraction efficiency of a reverse TN-LC (E-7) grating as function of incident angle for the cell with a pre-tilt angle 5 degree at 2 volts AC (1 kHz) applied to electrodes of cell. The circles are for incident light with polarization parallel to vector director; the diamonds are for polarization orthogonal to vector director of LC molecules on entry to LC grating (L=75µm, b=2µm).

Equations (26)

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ε ik = ε 0 δ ik + ε a d i d k
D i + D i = ε ik ( E k + E k )
D i = ε 0 E i + δ ε ik E k
2 D + k 2 D = × × ( δ ε E )
D ( R ) = V G ( R r ) F ( r ) dr
D ( R ) = 1 4 π × × ( exp ( jk R R ) ( δ ε E ) exp ( j k · r ) dV
E = k × ( k × Φ ) exp ( jk R ) 4 π R ε
Φ i = ε a d i d k e k E 0 ( r ) exp ( j q · r ) dV
h = E 2 R 2 d Ω V E ( 0 ) 2
h = k 4 sin 2 ( ψ ) 16 π 2 ε 2 V E ( 0 ) 2 Φ 2 d Ω
E y = ( k ) 2 exp ( jk R ) 4 π ε R ε a d y d x E 0 x ( r ) exp ( j q · r ) dV
E x = ( k ) 2 exp ( jk R ) 4 π ε R ε a d x d y E 0 y ( r ) exp ( j q · r ) dV
( x ) = 4 π [ n = 1 ( 1 ) n + 1 2 n 1 cos [ ( 2 n 1 ) Kx ] ]
E ( r ) = 1 2 m E m ( r ) exp ( j k m · r ) + C . C .
Δ E + ( ε ω 2 c 2 ) E = 0
m [ 1 2 ( 2 j k m · E m k m 2 E m ) exp ( j k m · r ) + C . C . ]
+ ω 2 c 2 [ ε 0 + ε a sin ( π z b ) π n = 1 ( 1 ) n + 1 2 n 1 exp [ j ( 2 n 1 ) Kx ] + C . C . ]
× m [ 1 2 E m exp ( j k m · r ) + C . C . ] = 0
cos θ m E m z + sin θ m E m x = j ε a sin ( π z b ) λ ε 0 1 2 [ E m + 1 exp ( j Kx ) exp [ j ( k m k m + 1 ) · r ] + E m 1 exp ( j Kx ) exp [ j ( k m k m 1 ) · r ] ]
E m z = j ε a sin ( π z b ) λ ε o 1 2 [ E m + 1 exp ( jk 0 ( cos θ m cos θ m + 1 ) z ) + E m 1 exp ( jk 0 ( cos θ m cos θ m 1 ) z ) ]
E m z = j α sin ( dz ) ( E m + 1 + E m 1 )
E 0 = E ( 0 ) J 0 ( u )
E 0 ( x , y , z ) d y d x exp ( j q · r ) dV
= E 0 ( x , y , 0 ) J 0 [ u ( z ) ] [ sin ( π z b ) 2 ] ( x ) exp [ j ( q x x + q z z ) ] dx dy dz
a a dy a a ( x ) exp ( j q x x ) dx A 0 b J 0 ( u ) sin ( π z b ) exp ( j q z z ) dz B = 2 aAB
A = 8 L m = 1 ( 1 ) m + 1 P 2 ( m ) q x 2 [ 2 cos ( q x a ) sin [ P ( m ) a ] q x P ( m ) 2 sin ( q x a ) cos [ P ( m ) a ] ]

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