Abstract

The polarization properties of a nematic zero-twist liquid-crystal (NLC) spatial light modulator (SLM) were studied. A large ratio between the liquid-crystal (LC) layer thickness and the pixel pitch combined with spatial variations in the applied electric field causes fringing fields between pixels. Depending on the LC alignment, the electric field components within the LC layer can result in a twist deformation. The produced inhomogeneous optical anisotropy affects the polarization of light propagating through the device. We experimentally examined polarization effects in different diffraction orders for both binary and blazed phase gratings. Simulations of the LC deformation together with finite-difference time-domain simulations for the optical propagation were used to calculate the corresponding far-field intensities. It was demonstrated how rigorous simulations of the NLC SLM properties can be used to understand the polarization features of different diffraction orders.

© 2005 Optical Society of America

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References

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  1. U. Efron, Spatial Light Modulator Technology (Marcel Dekker, New York, 1995).
  2. P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nquyen, D. P. Resler, R. C. Sharp, E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268–298 (1996).
    [CrossRef]
  3. S. Serati, J. Stockley, “Advanced liquid crystal on silicon optical phased arrays,” in Proceedings of the 2002 IEEE Aerospace Conference (IEEE Press, Piscataway, N.J., 2002), Vol. 3, pp. 1395–1402 (2002).
  4. J. Stockley, D. Subacius, S. Serati, “Influence of the interpixel region in liquid crystal diffraction gratings,” in Liquid Crystal Materials, Devices, and Applications VII, R. Shashidhar ed., Proc. SPIE3635, 127–136 (1999).
    [CrossRef]
  5. M. Bouvier, T. Scharf, “Analysis of nematic-liquid-crystal binary gratings with high spatial frequency,” Opt. Eng. 39, 2129–2137 (2000).
    [CrossRef]
  6. G. Haas, H. Wöhler, M. Fritch, D. Mlynski, “Simulation of two-dimensional nematic director structures in inhomogeneous electric fields,” Mol. Cryst. Liq. Cryst. 198, 15–28 (1991).
    [CrossRef]
  7. Z. He, T. Nose, S. Sato, “Polarization properties of an amplitude nematic liquid crystal grating,” Opt. Eng. 37, 2885–2898 (1998).
    [CrossRef]
  8. Z. He, T. Nose, S. Sato, “Diffraction and polarization properties of a liquid crystal grating,” Jpn. J. Appl. Phys., Part 1 35, 3529–3530 (1996).
    [CrossRef]
  9. S. Harris, “Polarization effects in nematic liquid crystal optical phased arrays,” in Liquid Crystals VII, I.-C. Khoo, ed., Proc. SPIE5213, 26–39 (2003).
    [CrossRef]
  10. E. Hällstig, J. Stigwall, T. Martin, L. Sjöqvist, M. Lindgren, “Fringing fields in a liquid-crystal spatial light modulator for beam steering,” J. Mod. Opt. 51, 1233–1247 (2004).
    [CrossRef]
  11. P. G. de Gennes, J. Prost, The Physics of Liquid Crystals, 2nd ed. (Oxford U. Press, Oxford, UK, 1993).
  12. H. Mori, E. C. Gartland, J. R. Reilly, P. J. Bois, “Multidimensional director modeling using the Q tensor representation in a liquid crystal cell and its application to the π cell with patterned electrodes,” Jpn. J. Appl. Phys., Part 1 38, 135–146 (1999).
    [CrossRef]
  13. B. Witzigmann, P. Regli, W. Fichtner, “Rigorous electromagnetic simulation of liquid-crystal displays,” J. Opt. Soc. Am. A 15, 753–757 (1998).
    [CrossRef]
  14. E. E. Kriezis, S. J. Elston, “Finite-difference time domain method for light wave propagation within liquid crystal devices,” Opt. Commun. 165, 99–105 (1999).
    [CrossRef]
  15. E. E. Kriezis, S. J. Elston, “Light wave propagation in liquid crystal displays by the 2-D finite-difference time-domain method,” Opt. Commun. 177, 69–77 (2000).
    [CrossRef]
  16. E. E. Kriezis, “Numerical modelling of light wave propagation in reflective liquid crystal microdisplay devices,” J. Mod. Opt. 49, 2065–2081 (2002).
    [CrossRef]
  17. C. Titus, J. Kelly, E. Gartland, S. Shiyanovskii, J. Anderson, P. Bos, “Asymmetric transmissive behavior of liquid-crystal diffraction gratings,” Opt. Lett. 26, 1188–1190 (2001).
    [CrossRef]
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    [CrossRef]
  20. A. Taflove, Computational Electrodynamics (Artech House, Norwood, Mass., 1995).
  21. T. Martin, “An improved near- to far-zone transformation for the finite-difference time-domain method,” IEEE Trans. Antennas Propag. 46, 1263–1271 (1998).
    [CrossRef]
  22. E. Collett, Polarized Light Fundamentals and Applications (Marcel Dekker, New York, 1993).

2004 (1)

E. Hällstig, J. Stigwall, T. Martin, L. Sjöqvist, M. Lindgren, “Fringing fields in a liquid-crystal spatial light modulator for beam steering,” J. Mod. Opt. 51, 1233–1247 (2004).
[CrossRef]

2002 (1)

E. E. Kriezis, “Numerical modelling of light wave propagation in reflective liquid crystal microdisplay devices,” J. Mod. Opt. 49, 2065–2081 (2002).
[CrossRef]

2001 (1)

2000 (2)

E. E. Kriezis, S. J. Elston, “Light wave propagation in liquid crystal displays by the 2-D finite-difference time-domain method,” Opt. Commun. 177, 69–77 (2000).
[CrossRef]

M. Bouvier, T. Scharf, “Analysis of nematic-liquid-crystal binary gratings with high spatial frequency,” Opt. Eng. 39, 2129–2137 (2000).
[CrossRef]

1999 (2)

H. Mori, E. C. Gartland, J. R. Reilly, P. J. Bois, “Multidimensional director modeling using the Q tensor representation in a liquid crystal cell and its application to the π cell with patterned electrodes,” Jpn. J. Appl. Phys., Part 1 38, 135–146 (1999).
[CrossRef]

E. E. Kriezis, S. J. Elston, “Finite-difference time domain method for light wave propagation within liquid crystal devices,” Opt. Commun. 165, 99–105 (1999).
[CrossRef]

1998 (3)

B. Witzigmann, P. Regli, W. Fichtner, “Rigorous electromagnetic simulation of liquid-crystal displays,” J. Opt. Soc. Am. A 15, 753–757 (1998).
[CrossRef]

Z. He, T. Nose, S. Sato, “Polarization properties of an amplitude nematic liquid crystal grating,” Opt. Eng. 37, 2885–2898 (1998).
[CrossRef]

T. Martin, “An improved near- to far-zone transformation for the finite-difference time-domain method,” IEEE Trans. Antennas Propag. 46, 1263–1271 (1998).
[CrossRef]

1996 (2)

Z. He, T. Nose, S. Sato, “Diffraction and polarization properties of a liquid crystal grating,” Jpn. J. Appl. Phys., Part 1 35, 3529–3530 (1996).
[CrossRef]

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nquyen, D. P. Resler, R. C. Sharp, E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268–298 (1996).
[CrossRef]

1991 (1)

G. Haas, H. Wöhler, M. Fritch, D. Mlynski, “Simulation of two-dimensional nematic director structures in inhomogeneous electric fields,” Mol. Cryst. Liq. Cryst. 198, 15–28 (1991).
[CrossRef]

1972 (1)

1941 (1)

Anderson, J.

Berreman, D.

Bois, P. J.

H. Mori, E. C. Gartland, J. R. Reilly, P. J. Bois, “Multidimensional director modeling using the Q tensor representation in a liquid crystal cell and its application to the π cell with patterned electrodes,” Jpn. J. Appl. Phys., Part 1 38, 135–146 (1999).
[CrossRef]

Bos, P.

Bouvier, M.

M. Bouvier, T. Scharf, “Analysis of nematic-liquid-crystal binary gratings with high spatial frequency,” Opt. Eng. 39, 2129–2137 (2000).
[CrossRef]

Collett, E.

E. Collett, Polarized Light Fundamentals and Applications (Marcel Dekker, New York, 1993).

Corkum, D. L.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nquyen, D. P. Resler, R. C. Sharp, E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268–298 (1996).
[CrossRef]

de Gennes, P. G.

P. G. de Gennes, J. Prost, The Physics of Liquid Crystals, 2nd ed. (Oxford U. Press, Oxford, UK, 1993).

Dorschner, T. A.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nquyen, D. P. Resler, R. C. Sharp, E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268–298 (1996).
[CrossRef]

Efron, U.

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

Elston, S. J.

E. E. Kriezis, S. J. Elston, “Light wave propagation in liquid crystal displays by the 2-D finite-difference time-domain method,” Opt. Commun. 177, 69–77 (2000).
[CrossRef]

E. E. Kriezis, S. J. Elston, “Finite-difference time domain method for light wave propagation within liquid crystal devices,” Opt. Commun. 165, 99–105 (1999).
[CrossRef]

Fichtner, W.

Friedman, L. J.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nquyen, D. P. Resler, R. C. Sharp, E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268–298 (1996).
[CrossRef]

Fritch, M.

G. Haas, H. Wöhler, M. Fritch, D. Mlynski, “Simulation of two-dimensional nematic director structures in inhomogeneous electric fields,” Mol. Cryst. Liq. Cryst. 198, 15–28 (1991).
[CrossRef]

Gartland, E.

Gartland, E. C.

H. Mori, E. C. Gartland, J. R. Reilly, P. J. Bois, “Multidimensional director modeling using the Q tensor representation in a liquid crystal cell and its application to the π cell with patterned electrodes,” Jpn. J. Appl. Phys., Part 1 38, 135–146 (1999).
[CrossRef]

Haas, G.

G. Haas, H. Wöhler, M. Fritch, D. Mlynski, “Simulation of two-dimensional nematic director structures in inhomogeneous electric fields,” Mol. Cryst. Liq. Cryst. 198, 15–28 (1991).
[CrossRef]

Hällstig, E.

E. Hällstig, J. Stigwall, T. Martin, L. Sjöqvist, M. Lindgren, “Fringing fields in a liquid-crystal spatial light modulator for beam steering,” J. Mod. Opt. 51, 1233–1247 (2004).
[CrossRef]

Harris, S.

S. Harris, “Polarization effects in nematic liquid crystal optical phased arrays,” in Liquid Crystals VII, I.-C. Khoo, ed., Proc. SPIE5213, 26–39 (2003).
[CrossRef]

He, Z.

Z. He, T. Nose, S. Sato, “Polarization properties of an amplitude nematic liquid crystal grating,” Opt. Eng. 37, 2885–2898 (1998).
[CrossRef]

Z. He, T. Nose, S. Sato, “Diffraction and polarization properties of a liquid crystal grating,” Jpn. J. Appl. Phys., Part 1 35, 3529–3530 (1996).
[CrossRef]

Hobbs, D. S.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nquyen, D. P. Resler, R. C. Sharp, E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268–298 (1996).
[CrossRef]

Holz, M.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nquyen, D. P. Resler, R. C. Sharp, E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268–298 (1996).
[CrossRef]

Jones, R. C.

Kelly, J.

Kriezis, E. E.

E. E. Kriezis, “Numerical modelling of light wave propagation in reflective liquid crystal microdisplay devices,” J. Mod. Opt. 49, 2065–2081 (2002).
[CrossRef]

E. E. Kriezis, S. J. Elston, “Light wave propagation in liquid crystal displays by the 2-D finite-difference time-domain method,” Opt. Commun. 177, 69–77 (2000).
[CrossRef]

E. E. Kriezis, S. J. Elston, “Finite-difference time domain method for light wave propagation within liquid crystal devices,” Opt. Commun. 165, 99–105 (1999).
[CrossRef]

Liberman, S.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nquyen, D. P. Resler, R. C. Sharp, E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268–298 (1996).
[CrossRef]

Lindgren, M.

E. Hällstig, J. Stigwall, T. Martin, L. Sjöqvist, M. Lindgren, “Fringing fields in a liquid-crystal spatial light modulator for beam steering,” J. Mod. Opt. 51, 1233–1247 (2004).
[CrossRef]

Martin, T.

E. Hällstig, J. Stigwall, T. Martin, L. Sjöqvist, M. Lindgren, “Fringing fields in a liquid-crystal spatial light modulator for beam steering,” J. Mod. Opt. 51, 1233–1247 (2004).
[CrossRef]

T. Martin, “An improved near- to far-zone transformation for the finite-difference time-domain method,” IEEE Trans. Antennas Propag. 46, 1263–1271 (1998).
[CrossRef]

McManamon, P. F.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nquyen, D. P. Resler, R. C. Sharp, E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268–298 (1996).
[CrossRef]

Mlynski, D.

G. Haas, H. Wöhler, M. Fritch, D. Mlynski, “Simulation of two-dimensional nematic director structures in inhomogeneous electric fields,” Mol. Cryst. Liq. Cryst. 198, 15–28 (1991).
[CrossRef]

Mori, H.

H. Mori, E. C. Gartland, J. R. Reilly, P. J. Bois, “Multidimensional director modeling using the Q tensor representation in a liquid crystal cell and its application to the π cell with patterned electrodes,” Jpn. J. Appl. Phys., Part 1 38, 135–146 (1999).
[CrossRef]

Nose, T.

Z. He, T. Nose, S. Sato, “Polarization properties of an amplitude nematic liquid crystal grating,” Opt. Eng. 37, 2885–2898 (1998).
[CrossRef]

Z. He, T. Nose, S. Sato, “Diffraction and polarization properties of a liquid crystal grating,” Jpn. J. Appl. Phys., Part 1 35, 3529–3530 (1996).
[CrossRef]

Nquyen, H. Q.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nquyen, D. P. Resler, R. C. Sharp, E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268–298 (1996).
[CrossRef]

Prost, J.

P. G. de Gennes, J. Prost, The Physics of Liquid Crystals, 2nd ed. (Oxford U. Press, Oxford, UK, 1993).

Regli, P.

Reilly, J. R.

H. Mori, E. C. Gartland, J. R. Reilly, P. J. Bois, “Multidimensional director modeling using the Q tensor representation in a liquid crystal cell and its application to the π cell with patterned electrodes,” Jpn. J. Appl. Phys., Part 1 38, 135–146 (1999).
[CrossRef]

Resler, D. P.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nquyen, D. P. Resler, R. C. Sharp, E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268–298 (1996).
[CrossRef]

Sato, S.

Z. He, T. Nose, S. Sato, “Polarization properties of an amplitude nematic liquid crystal grating,” Opt. Eng. 37, 2885–2898 (1998).
[CrossRef]

Z. He, T. Nose, S. Sato, “Diffraction and polarization properties of a liquid crystal grating,” Jpn. J. Appl. Phys., Part 1 35, 3529–3530 (1996).
[CrossRef]

Scharf, T.

M. Bouvier, T. Scharf, “Analysis of nematic-liquid-crystal binary gratings with high spatial frequency,” Opt. Eng. 39, 2129–2137 (2000).
[CrossRef]

Serati, S.

J. Stockley, D. Subacius, S. Serati, “Influence of the interpixel region in liquid crystal diffraction gratings,” in Liquid Crystal Materials, Devices, and Applications VII, R. Shashidhar ed., Proc. SPIE3635, 127–136 (1999).
[CrossRef]

S. Serati, J. Stockley, “Advanced liquid crystal on silicon optical phased arrays,” in Proceedings of the 2002 IEEE Aerospace Conference (IEEE Press, Piscataway, N.J., 2002), Vol. 3, pp. 1395–1402 (2002).

Sharp, R. C.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nquyen, D. P. Resler, R. C. Sharp, E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268–298 (1996).
[CrossRef]

Shiyanovskii, S.

Sjöqvist, L.

E. Hällstig, J. Stigwall, T. Martin, L. Sjöqvist, M. Lindgren, “Fringing fields in a liquid-crystal spatial light modulator for beam steering,” J. Mod. Opt. 51, 1233–1247 (2004).
[CrossRef]

Stigwall, J.

E. Hällstig, J. Stigwall, T. Martin, L. Sjöqvist, M. Lindgren, “Fringing fields in a liquid-crystal spatial light modulator for beam steering,” J. Mod. Opt. 51, 1233–1247 (2004).
[CrossRef]

Stockley, J.

S. Serati, J. Stockley, “Advanced liquid crystal on silicon optical phased arrays,” in Proceedings of the 2002 IEEE Aerospace Conference (IEEE Press, Piscataway, N.J., 2002), Vol. 3, pp. 1395–1402 (2002).

J. Stockley, D. Subacius, S. Serati, “Influence of the interpixel region in liquid crystal diffraction gratings,” in Liquid Crystal Materials, Devices, and Applications VII, R. Shashidhar ed., Proc. SPIE3635, 127–136 (1999).
[CrossRef]

Subacius, D.

J. Stockley, D. Subacius, S. Serati, “Influence of the interpixel region in liquid crystal diffraction gratings,” in Liquid Crystal Materials, Devices, and Applications VII, R. Shashidhar ed., Proc. SPIE3635, 127–136 (1999).
[CrossRef]

Taflove, A.

A. Taflove, Computational Electrodynamics (Artech House, Norwood, Mass., 1995).

Titus, C.

Watson, E. A.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nquyen, D. P. Resler, R. C. Sharp, E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268–298 (1996).
[CrossRef]

Witzigmann, B.

Wöhler, H.

G. Haas, H. Wöhler, M. Fritch, D. Mlynski, “Simulation of two-dimensional nematic director structures in inhomogeneous electric fields,” Mol. Cryst. Liq. Cryst. 198, 15–28 (1991).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

T. Martin, “An improved near- to far-zone transformation for the finite-difference time-domain method,” IEEE Trans. Antennas Propag. 46, 1263–1271 (1998).
[CrossRef]

J. Mod. Opt. (2)

E. Hällstig, J. Stigwall, T. Martin, L. Sjöqvist, M. Lindgren, “Fringing fields in a liquid-crystal spatial light modulator for beam steering,” J. Mod. Opt. 51, 1233–1247 (2004).
[CrossRef]

E. E. Kriezis, “Numerical modelling of light wave propagation in reflective liquid crystal microdisplay devices,” J. Mod. Opt. 49, 2065–2081 (2002).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Jpn. J. Appl. Phys., Part 1 (2)

Z. He, T. Nose, S. Sato, “Diffraction and polarization properties of a liquid crystal grating,” Jpn. J. Appl. Phys., Part 1 35, 3529–3530 (1996).
[CrossRef]

H. Mori, E. C. Gartland, J. R. Reilly, P. J. Bois, “Multidimensional director modeling using the Q tensor representation in a liquid crystal cell and its application to the π cell with patterned electrodes,” Jpn. J. Appl. Phys., Part 1 38, 135–146 (1999).
[CrossRef]

Mol. Cryst. Liq. Cryst. (1)

G. Haas, H. Wöhler, M. Fritch, D. Mlynski, “Simulation of two-dimensional nematic director structures in inhomogeneous electric fields,” Mol. Cryst. Liq. Cryst. 198, 15–28 (1991).
[CrossRef]

Opt. Commun. (2)

E. E. Kriezis, S. J. Elston, “Finite-difference time domain method for light wave propagation within liquid crystal devices,” Opt. Commun. 165, 99–105 (1999).
[CrossRef]

E. E. Kriezis, S. J. Elston, “Light wave propagation in liquid crystal displays by the 2-D finite-difference time-domain method,” Opt. Commun. 177, 69–77 (2000).
[CrossRef]

Opt. Eng. (2)

Z. He, T. Nose, S. Sato, “Polarization properties of an amplitude nematic liquid crystal grating,” Opt. Eng. 37, 2885–2898 (1998).
[CrossRef]

M. Bouvier, T. Scharf, “Analysis of nematic-liquid-crystal binary gratings with high spatial frequency,” Opt. Eng. 39, 2129–2137 (2000).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE (1)

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nquyen, D. P. Resler, R. C. Sharp, E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268–298 (1996).
[CrossRef]

Other (7)

S. Serati, J. Stockley, “Advanced liquid crystal on silicon optical phased arrays,” in Proceedings of the 2002 IEEE Aerospace Conference (IEEE Press, Piscataway, N.J., 2002), Vol. 3, pp. 1395–1402 (2002).

J. Stockley, D. Subacius, S. Serati, “Influence of the interpixel region in liquid crystal diffraction gratings,” in Liquid Crystal Materials, Devices, and Applications VII, R. Shashidhar ed., Proc. SPIE3635, 127–136 (1999).
[CrossRef]

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

S. Harris, “Polarization effects in nematic liquid crystal optical phased arrays,” in Liquid Crystals VII, I.-C. Khoo, ed., Proc. SPIE5213, 26–39 (2003).
[CrossRef]

A. Taflove, Computational Electrodynamics (Artech House, Norwood, Mass., 1995).

P. G. de Gennes, J. Prost, The Physics of Liquid Crystals, 2nd ed. (Oxford U. Press, Oxford, UK, 1993).

E. Collett, Polarized Light Fundamentals and Applications (Marcel Dekker, New York, 1993).

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Figures (14)

Fig. 1
Fig. 1

NLC in a zero-twist cell (a) without and (b) with applied voltage to three electrodes. In these schematics the rubbing direction is parallel with the stripe-shaped electrodes, and an applied voltage on a pixel is denoted by a darker gray color.

Fig. 2
Fig. 2

Schematic of the computational grid used for the FDTD simulation.

Fig. 3
Fig. 3

Schematic of the experimental setup. The QW plate was inserted only during one part of the measurement of the Stokes parameters.

Fig. 4
Fig. 4

Measured (circles) energies in the first three negative and positive orders for (a) z polarization and (b) x polarization by use of the SLM with alignment parallel to z. Corresponding simulated intensities for the first four orders are shown as crosses. Note the different scales on the vertical axes of the two figures.

Fig. 5
Fig. 5

Measured energies in the first four negative and positive orders for (a) x polarization and (b) z polarization by use of the SLM with perpendicular alignment. Note the different scales on the vertical axes of the two figures.

Fig. 6
Fig. 6

Measured Stokes parameters S1 (solid curve with dots), S2 (dashed curve with crosses), and S3 (dashed curve with circles) for the first three negative and positive orders.

Fig. 7
Fig. 7

Measured polarization ellipses for the first three negative and positive orders. Note that these are the same results as in Fig. 6 but presented as ellipses instead of Stokes parameters.

Fig. 8
Fig. 8

Measured (circles) and simulated (crosses) intensities in the (a) z and (b) x polarizations for a blazed grating with a period of 33 pixels.

Fig. 9
Fig. 9

Equipotential lines of the electrostatic potential and arrows indicating the director distribution after the simulation. The gray region corresponds to the LC, and the white region corresponds to the mirror. The 16 pixels on the left side and at the bottom of the mirror were set to 3 V, and the 17 pixels on the right side were held at 2 V.

Fig. 10
Fig. 10

Simulated amplitudes of the outgoing optical wave at the top of the computational volume for the two polarization directions z (dashed curve) and x (solid curve). All values were normalized to the maximum amplitude.

Fig. 11
Fig. 11

Simulated phases of the outgoing optical wave at the top of the computational volume for the two polarization directions z (dashed curve) and x (solid curve). The ideal phase shift for the binary grating has also been indicated in the plot (dotted curve).

Fig. 12
Fig. 12

Simulated polarization ellipses for the outgoing optical wave as it leaves the LC. The ellipses at the right and the left borders was extracted from x=0 µm and x=60 µm. The rest of the ellipses correspond to evenly spaced positions in the computational volume. The arrows indicate the direction of diffraction due to the phase gradient.

Fig. 13
Fig. 13

Stokes parameters S1 (solid curve with dots), S2 (dashed curve with crosses), and S3 (dashed curve with circles) versus diffraction orders calculated from the simulated far field.

Fig. 14
Fig. 14

Simulated polarization ellipses of the first three negative and positive orders.

Equations (10)

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

=+Δnx2ΔnxnyΔnxnzΔnxny+Δny2ΔnynzΔnxnzΔnynz+Δnz2,
(+Δnx2) 2Φx2++Δ nx2x 2Φy2+2Δnxny 2Φxy+Δ2nx nxx+nx nyy+ny nxy Φx+Δ2ny nyy+ny nxx+nx nyx Φy=0.
fg=12K1(·n)2+12K2(n·×n)2+12K3(n××n)2-12D·E,
ξ t ni=-K2n-ni12 D·Eni+λni,
Js=nˆ×H, Ms=-nˆ×E,
A(r)=μ0 exp(-jkr)4πr SJs exp(jkrˆ·r)dS, F(r)=0 exp(-jkr)4πr SMs exp(jkrˆ·r)dS,
θˆ=xˆ cos θ cos ϕ-yˆ sin θ-zˆ cos θ sin ϕ, ϕˆ=-xˆ sin ϕ-zˆ cos ϕ.
ExEθ=-jω(Aθ+γFϕ), Ez-Eϕ=jω(Aϕ-γFθ),
A=2N n=1NI(nΔθ), B=4N n=1NI(nΔθ)sin(2nΔθ), C=4N n=1NI(nΔθ)cos(4nΔθ), D=4N n=1NI(nΔθ)sin(4nΔθ),
S0=A-C, S1=2C, S2=2D,S3=B.

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