Abstract

An approximate analytical model was developed that links the fringing-field broadening of the phase profile of a liquid-crystal (LC) beam-steering device, and the resulting diffraction efficiency, to the physical parameters of the device including the cell thickness as well as the dielectric, optical, and geometrical constants of the device. The analysis includes a full solution of the Laplace equation for the LC device in which the broadening of the initial voltage profile into an effective voltage-drop profile, due to the fringing-field effect, is derived. It is shown that within the linear approximation used, the broadening of the phase profile is identical to the broadening of the effective voltage profile in the presence of the fringing field. On the basis of this model, the resulting broadening kernel of the phase profile is found to be proportional to the LC cell thickness. These results are found to be in an excellent agreement with high-precision computer simulations performed on the LC beam-steering structure, thereby validating this approximate linear model.

© 2004 Optical Society of America

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References

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  1. R. M. Matic, “Blazed phase liquid crystal beam steering,” in Laser Beam Propagation and Control, H. Weichel, L. F. DeSandre, eds., Proc. SPIE2120, 194–205 (1994).
    [CrossRef]
  2. P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268–298 (1996).
    [CrossRef]
  3. D. P. Resler, D. S. Hobbs, R. C. Sharp, L. J. Friedman, T. A. Dorschner, “High-efficiency liquid-crystal optical phased-array beam steering,” Opt. Lett. 21, 689–691 (1996).
    [CrossRef] [PubMed]
  4. V. G. Chigrinov, I. N. Kompanets, A. A. Vasiliev , “Behaviour of nematic liquid crystals in inhomogeneous electric fields,” Mol. Cryst. Liq. Cryst. 55, 193–207 (1979).
    [CrossRef]
  5. L. J. Friedman, D. S. Hobbs, S. Lieberman, D. L. Corkum, H. Q. Nguyen, D. P. Resler, R. C. Sharp, T. A. Dorschner, “Spatially resolved phase imaging of a programmable liquid-crystal grating,” Appl. Opt. 35, 6236–6240 (1996).
    [CrossRef] [PubMed]
  6. V. G. Dominique, A. J. Carney, E. A. Watson, “Measurement and modeling of the angular dispersion in liquid crystal broadband beam steering devices,” Opt. Eng. (Bellingham) 35, 3371–3379 (1996).
    [CrossRef]
  7. T. Scharf, M. Bouvier, R. Dändliker, “Multilevel nematic liquid crystal phase gratings,” in Eighth International Conference on Nonlinear Optics of Liquid and Photorefractive Crystals, G. V. Klimusheva, A. G. Iljin, eds., Proc. SPIE4418, 31–37 (2001).
    [CrossRef]
  8. M. Bouvier, T. Scharf, “Analysis of nematic-liquid-crystal binary gratings with high spatial frequency,” Opt. Eng. (Bellingham) 39, 2129–2137 (2000).
    [CrossRef]
  9. J. H. Kulick, J. M. Jarem, R. G. Lindquist, S. T. Kowel, M. W. Friends, T. M. Leslie, “Electrostatic and diffraction analysis of a liquid-crystal device utilizing fringing fields: applications to three-dimensional displays,” Appl. Opt. 34, 1901–1922 (1995).
    [CrossRef] [PubMed]
  10. G. Panasyuk, D. W. Allender, J. Kelly, “Approximate description of the two-dimensional director field in a liquid crystal display,” J. Appl. Phys. 89, 4777–4786 (2001).
    [CrossRef]
  11. E. Hallstig, J. Stigwell, M. Lindgren, L. Sjokvist, “Laser beam steering and tracking using a liquid crystal spatial light modulator,” in Laser Systems Technology, W. E. Thompson, P. H. Merritt, eds., Proc. SPIE5087, 13–23 (2003).
    [CrossRef]
  12. M. Oh-e, K. Kondo, “Electro-optical characteristics and switching behavior of the in-plane switching mode,” Appl. Phys. Lett. 67, 3895–3897 (1995).
    [CrossRef]
  13. M. Oh-e, M. Yonea, K. Kondo, “Switching of negative and positive dielectro-anisotropic liquid crystals by in-plane electric fields,” J. Appl. Phys. 82, 528–535 (1997).
    [CrossRef]
  14. B. Veweire, R. Defever, “Limitation of resolution of LCOS-based projection displays by diffraction effects,” in Proceedings of the 19th International Display Research Conference (Society for Information Display, San Jose, Calif., 1999), pp. 489–492.
  15. H. De Smet, J. Van Der Steen, A. Van Calster, “Microdisplays with high pixel count,” in Proceedings of the Society for Information Display Digest (Society for Information Display, San Jose, Calif., 2001), pp. 968–971.
  16. B. Apter, U. Efron, E. Bahat-Treidel, “On the fringing-field effect in liquid-crystal beam-steering devices,” Appl. Opt. 43, 11–19 (2004).
    [CrossRef] [PubMed]
  17. See, e.g., V. G. Chigrinov, Liquid Crystal Devices: Physics and Applications (Artech House, Boston, Mass., 1999), Chap. 2.
  18. A. Yariv, P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, New York, 2003).
  19. See, e.g., A. V. Oppenheim, A. S. Willsky, Signals & Systems (Prentice Hall, Upper Saddle River, N.J., 1997), Chap. 6.
  20. Autronics-Melcher’s 2dimMOS software, http://www.autronic-melchers.com/index.htm .
  21. R. Magnussen, T. K. Gaylord, “Diffraction efficiency of thin phase gratings with arbitrary grating shape,” J. Opt. Soc. Am. 68, 806–809 (1978).
    [CrossRef]
  22. T. Fujita, H. Nishihara, J. Koyama, “Blazed gratings and Fresnel lenses fabricated by electron-beam lithography,” Opt. Lett. 7, 578–580 (1982).
    [CrossRef] [PubMed]
  23. G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), p. 819.
  24. C. V. Brown, Em. E. Kriezis, S. J. Elston, “Optical diffraction from a liquid crystal phase grating,” J. Appl. Phys. 91, 3495–3500 (2002).
    [CrossRef]
  25. B. Wang, P. J. Bos, C. D. Hoke, “Light propagation in variable-refractive-index materials with liquid-crystal-infiltrated microcavities,” J. Opt. Soc. Am. A 20, 2123–2130 (2003).
    [CrossRef]

2004 (1)

2003 (1)

2002 (1)

C. V. Brown, Em. E. Kriezis, S. J. Elston, “Optical diffraction from a liquid crystal phase grating,” J. Appl. Phys. 91, 3495–3500 (2002).
[CrossRef]

2001 (1)

G. Panasyuk, D. W. Allender, J. Kelly, “Approximate description of the two-dimensional director field in a liquid crystal display,” J. Appl. Phys. 89, 4777–4786 (2001).
[CrossRef]

2000 (1)

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

1997 (1)

M. Oh-e, M. Yonea, K. Kondo, “Switching of negative and positive dielectro-anisotropic liquid crystals by in-plane electric fields,” J. Appl. Phys. 82, 528–535 (1997).
[CrossRef]

1996 (4)

V. G. Dominique, A. J. Carney, E. A. Watson, “Measurement and modeling of the angular dispersion in liquid crystal broadband beam steering devices,” Opt. Eng. (Bellingham) 35, 3371–3379 (1996).
[CrossRef]

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

D. P. Resler, D. S. Hobbs, R. C. Sharp, L. J. Friedman, T. A. Dorschner, “High-efficiency liquid-crystal optical phased-array beam steering,” Opt. Lett. 21, 689–691 (1996).
[CrossRef] [PubMed]

L. J. Friedman, D. S. Hobbs, S. Lieberman, D. L. Corkum, H. Q. Nguyen, D. P. Resler, R. C. Sharp, T. A. Dorschner, “Spatially resolved phase imaging of a programmable liquid-crystal grating,” Appl. Opt. 35, 6236–6240 (1996).
[CrossRef] [PubMed]

1995 (2)

1982 (1)

1979 (1)

V. G. Chigrinov, I. N. Kompanets, A. A. Vasiliev , “Behaviour of nematic liquid crystals in inhomogeneous electric fields,” Mol. Cryst. Liq. Cryst. 55, 193–207 (1979).
[CrossRef]

1978 (1)

Allender, D. W.

G. Panasyuk, D. W. Allender, J. Kelly, “Approximate description of the two-dimensional director field in a liquid crystal display,” J. Appl. Phys. 89, 4777–4786 (2001).
[CrossRef]

Apter, B.

Bahat-Treidel, E.

Bos, P. J.

Bouvier, M.

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

T. Scharf, M. Bouvier, R. Dändliker, “Multilevel nematic liquid crystal phase gratings,” in Eighth International Conference on Nonlinear Optics of Liquid and Photorefractive Crystals, G. V. Klimusheva, A. G. Iljin, eds., Proc. SPIE4418, 31–37 (2001).
[CrossRef]

Brown, C. V.

C. V. Brown, Em. E. Kriezis, S. J. Elston, “Optical diffraction from a liquid crystal phase grating,” J. Appl. Phys. 91, 3495–3500 (2002).
[CrossRef]

Carney, A. J.

V. G. Dominique, A. J. Carney, E. A. Watson, “Measurement and modeling of the angular dispersion in liquid crystal broadband beam steering devices,” Opt. Eng. (Bellingham) 35, 3371–3379 (1996).
[CrossRef]

Chigrinov, V. G.

V. G. Chigrinov, I. N. Kompanets, A. A. Vasiliev , “Behaviour of nematic liquid crystals in inhomogeneous electric fields,” Mol. Cryst. Liq. Cryst. 55, 193–207 (1979).
[CrossRef]

See, e.g., V. G. Chigrinov, Liquid Crystal Devices: Physics and Applications (Artech House, Boston, Mass., 1999), Chap. 2.

Corkum, D. L.

L. J. Friedman, D. S. Hobbs, S. Lieberman, D. L. Corkum, H. Q. Nguyen, D. P. Resler, R. C. Sharp, T. A. Dorschner, “Spatially resolved phase imaging of a programmable liquid-crystal grating,” Appl. Opt. 35, 6236–6240 (1996).
[CrossRef] [PubMed]

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

Dändliker, R.

T. Scharf, M. Bouvier, R. Dändliker, “Multilevel nematic liquid crystal phase gratings,” in Eighth International Conference on Nonlinear Optics of Liquid and Photorefractive Crystals, G. V. Klimusheva, A. G. Iljin, eds., Proc. SPIE4418, 31–37 (2001).
[CrossRef]

De Smet, H.

H. De Smet, J. Van Der Steen, A. Van Calster, “Microdisplays with high pixel count,” in Proceedings of the Society for Information Display Digest (Society for Information Display, San Jose, Calif., 2001), pp. 968–971.

Defever, R.

B. Veweire, R. Defever, “Limitation of resolution of LCOS-based projection displays by diffraction effects,” in Proceedings of the 19th International Display Research Conference (Society for Information Display, San Jose, Calif., 1999), pp. 489–492.

Dominique, V. G.

V. G. Dominique, A. J. Carney, E. A. Watson, “Measurement and modeling of the angular dispersion in liquid crystal broadband beam steering devices,” Opt. Eng. (Bellingham) 35, 3371–3379 (1996).
[CrossRef]

Dorschner, T. A.

Efron, U.

Elston, S. J.

C. V. Brown, Em. E. Kriezis, S. J. Elston, “Optical diffraction from a liquid crystal phase grating,” J. Appl. Phys. 91, 3495–3500 (2002).
[CrossRef]

Friedman, L. J.

Friends, M. W.

Fujita, T.

Gaylord, T. K.

Hallstig, E.

E. Hallstig, J. Stigwell, M. Lindgren, L. Sjokvist, “Laser beam steering and tracking using a liquid crystal spatial light modulator,” in Laser Systems Technology, W. E. Thompson, P. H. Merritt, eds., Proc. SPIE5087, 13–23 (2003).
[CrossRef]

Hobbs, D. S.

Hoke, C. D.

Holz, M.

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

Jarem, J. M.

Kelly, J.

G. Panasyuk, D. W. Allender, J. Kelly, “Approximate description of the two-dimensional director field in a liquid crystal display,” J. Appl. Phys. 89, 4777–4786 (2001).
[CrossRef]

Kompanets, I. N.

V. G. Chigrinov, I. N. Kompanets, A. A. Vasiliev , “Behaviour of nematic liquid crystals in inhomogeneous electric fields,” Mol. Cryst. Liq. Cryst. 55, 193–207 (1979).
[CrossRef]

Kondo, K.

M. Oh-e, M. Yonea, K. Kondo, “Switching of negative and positive dielectro-anisotropic liquid crystals by in-plane electric fields,” J. Appl. Phys. 82, 528–535 (1997).
[CrossRef]

M. Oh-e, K. Kondo, “Electro-optical characteristics and switching behavior of the in-plane switching mode,” Appl. Phys. Lett. 67, 3895–3897 (1995).
[CrossRef]

Korn, G. A.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), p. 819.

Korn, T. M.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), p. 819.

Kowel, S. T.

Koyama, J.

Kriezis, Em. E.

C. V. Brown, Em. E. Kriezis, S. J. Elston, “Optical diffraction from a liquid crystal phase grating,” J. Appl. Phys. 91, 3495–3500 (2002).
[CrossRef]

Kulick, J. H.

Leslie, T. M.

Liberman, S.

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

Lieberman, S.

Lindgren, M.

E. Hallstig, J. Stigwell, M. Lindgren, L. Sjokvist, “Laser beam steering and tracking using a liquid crystal spatial light modulator,” in Laser Systems Technology, W. E. Thompson, P. H. Merritt, eds., Proc. SPIE5087, 13–23 (2003).
[CrossRef]

Lindquist, R. G.

Magnussen, R.

Matic, R. M.

R. M. Matic, “Blazed phase liquid crystal beam steering,” in Laser Beam Propagation and Control, H. Weichel, L. F. DeSandre, eds., Proc. SPIE2120, 194–205 (1994).
[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. Nguyen, D. P. Resler, R. C. Sharp, E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268–298 (1996).
[CrossRef]

Nguyen, H. Q.

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

L. J. Friedman, D. S. Hobbs, S. Lieberman, D. L. Corkum, H. Q. Nguyen, D. P. Resler, R. C. Sharp, T. A. Dorschner, “Spatially resolved phase imaging of a programmable liquid-crystal grating,” Appl. Opt. 35, 6236–6240 (1996).
[CrossRef] [PubMed]

Nishihara, H.

Oh-e, M.

M. Oh-e, M. Yonea, K. Kondo, “Switching of negative and positive dielectro-anisotropic liquid crystals by in-plane electric fields,” J. Appl. Phys. 82, 528–535 (1997).
[CrossRef]

M. Oh-e, K. Kondo, “Electro-optical characteristics and switching behavior of the in-plane switching mode,” Appl. Phys. Lett. 67, 3895–3897 (1995).
[CrossRef]

Oppenheim, A. V.

See, e.g., A. V. Oppenheim, A. S. Willsky, Signals & Systems (Prentice Hall, Upper Saddle River, N.J., 1997), Chap. 6.

Panasyuk, G.

G. Panasyuk, D. W. Allender, J. Kelly, “Approximate description of the two-dimensional director field in a liquid crystal display,” J. Appl. Phys. 89, 4777–4786 (2001).
[CrossRef]

Resler, D. P.

Scharf, T.

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

T. Scharf, M. Bouvier, R. Dändliker, “Multilevel nematic liquid crystal phase gratings,” in Eighth International Conference on Nonlinear Optics of Liquid and Photorefractive Crystals, G. V. Klimusheva, A. G. Iljin, eds., Proc. SPIE4418, 31–37 (2001).
[CrossRef]

Sharp, R. C.

Sjokvist, L.

E. Hallstig, J. Stigwell, M. Lindgren, L. Sjokvist, “Laser beam steering and tracking using a liquid crystal spatial light modulator,” in Laser Systems Technology, W. E. Thompson, P. H. Merritt, eds., Proc. SPIE5087, 13–23 (2003).
[CrossRef]

Stigwell, J.

E. Hallstig, J. Stigwell, M. Lindgren, L. Sjokvist, “Laser beam steering and tracking using a liquid crystal spatial light modulator,” in Laser Systems Technology, W. E. Thompson, P. H. Merritt, eds., Proc. SPIE5087, 13–23 (2003).
[CrossRef]

Van Calster, A.

H. De Smet, J. Van Der Steen, A. Van Calster, “Microdisplays with high pixel count,” in Proceedings of the Society for Information Display Digest (Society for Information Display, San Jose, Calif., 2001), pp. 968–971.

Van Der Steen, J.

H. De Smet, J. Van Der Steen, A. Van Calster, “Microdisplays with high pixel count,” in Proceedings of the Society for Information Display Digest (Society for Information Display, San Jose, Calif., 2001), pp. 968–971.

Vasiliev, A. A.

V. G. Chigrinov, I. N. Kompanets, A. A. Vasiliev , “Behaviour of nematic liquid crystals in inhomogeneous electric fields,” Mol. Cryst. Liq. Cryst. 55, 193–207 (1979).
[CrossRef]

Veweire, B.

B. Veweire, R. Defever, “Limitation of resolution of LCOS-based projection displays by diffraction effects,” in Proceedings of the 19th International Display Research Conference (Society for Information Display, San Jose, Calif., 1999), pp. 489–492.

Wang, B.

Watson, E. A.

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

V. G. Dominique, A. J. Carney, E. A. Watson, “Measurement and modeling of the angular dispersion in liquid crystal broadband beam steering devices,” Opt. Eng. (Bellingham) 35, 3371–3379 (1996).
[CrossRef]

Willsky, A. S.

See, e.g., A. V. Oppenheim, A. S. Willsky, Signals & Systems (Prentice Hall, Upper Saddle River, N.J., 1997), Chap. 6.

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, New York, 2003).

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, New York, 2003).

Yonea, M.

M. Oh-e, M. Yonea, K. Kondo, “Switching of negative and positive dielectro-anisotropic liquid crystals by in-plane electric fields,” J. Appl. Phys. 82, 528–535 (1997).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

M. Oh-e, K. Kondo, “Electro-optical characteristics and switching behavior of the in-plane switching mode,” Appl. Phys. Lett. 67, 3895–3897 (1995).
[CrossRef]

J. Appl. Phys. (3)

M. Oh-e, M. Yonea, K. Kondo, “Switching of negative and positive dielectro-anisotropic liquid crystals by in-plane electric fields,” J. Appl. Phys. 82, 528–535 (1997).
[CrossRef]

G. Panasyuk, D. W. Allender, J. Kelly, “Approximate description of the two-dimensional director field in a liquid crystal display,” J. Appl. Phys. 89, 4777–4786 (2001).
[CrossRef]

C. V. Brown, Em. E. Kriezis, S. J. Elston, “Optical diffraction from a liquid crystal phase grating,” J. Appl. Phys. 91, 3495–3500 (2002).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Mol. Cryst. Liq. Cryst. (1)

V. G. Chigrinov, I. N. Kompanets, A. A. Vasiliev , “Behaviour of nematic liquid crystals in inhomogeneous electric fields,” Mol. Cryst. Liq. Cryst. 55, 193–207 (1979).
[CrossRef]

Opt. Eng. (Bellingham) (2)

V. G. Dominique, A. J. Carney, E. A. Watson, “Measurement and modeling of the angular dispersion in liquid crystal broadband beam steering devices,” Opt. Eng. (Bellingham) 35, 3371–3379 (1996).
[CrossRef]

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

Opt. Lett. (2)

Proc. IEEE (1)

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

Other (10)

R. M. Matic, “Blazed phase liquid crystal beam steering,” in Laser Beam Propagation and Control, H. Weichel, L. F. DeSandre, eds., Proc. SPIE2120, 194–205 (1994).
[CrossRef]

T. Scharf, M. Bouvier, R. Dändliker, “Multilevel nematic liquid crystal phase gratings,” in Eighth International Conference on Nonlinear Optics of Liquid and Photorefractive Crystals, G. V. Klimusheva, A. G. Iljin, eds., Proc. SPIE4418, 31–37 (2001).
[CrossRef]

E. Hallstig, J. Stigwell, M. Lindgren, L. Sjokvist, “Laser beam steering and tracking using a liquid crystal spatial light modulator,” in Laser Systems Technology, W. E. Thompson, P. H. Merritt, eds., Proc. SPIE5087, 13–23 (2003).
[CrossRef]

B. Veweire, R. Defever, “Limitation of resolution of LCOS-based projection displays by diffraction effects,” in Proceedings of the 19th International Display Research Conference (Society for Information Display, San Jose, Calif., 1999), pp. 489–492.

H. De Smet, J. Van Der Steen, A. Van Calster, “Microdisplays with high pixel count,” in Proceedings of the Society for Information Display Digest (Society for Information Display, San Jose, Calif., 2001), pp. 968–971.

See, e.g., V. G. Chigrinov, Liquid Crystal Devices: Physics and Applications (Artech House, Boston, Mass., 1999), Chap. 2.

A. Yariv, P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, New York, 2003).

See, e.g., A. V. Oppenheim, A. S. Willsky, Signals & Systems (Prentice Hall, Upper Saddle River, N.J., 1997), Chap. 6.

Autronics-Melcher’s 2dimMOS software, http://www.autronic-melchers.com/index.htm .

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), p. 819.

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Figures (10)

Fig. 1
Fig. 1

Electrode voltage distribution, equal potential lines, and LC director distribution for a 48-μm-period blazed grating formed in a 4.8-μm LC cell (top). Resulting phase profile for λ=633 nm (bottom).

Fig. 2
Fig. 2

LC cell configuration.

Fig. 3
Fig. 3

LC voltage-dependent phase curve.

Fig. 4
Fig. 4

Dependence of the potential modulation factor at various values of ωmμ(d)d (right to left): ωmμ(d)d=0, 2, 4, 6, 8, and 10.

Fig. 5
Fig. 5

Behavior of the phase transfer function at various thicknesses: d=4 μm (crosses), d=8 μm (squares), and d=12 μm (circles).

Fig. 6
Fig. 6

Broadening kernel at various LC thicknesses: 4.8, 7.9, and 11.1 μm (from top to bottom). Charts on the left-hand side correspond to the 48-μm grating period, and charts on the right-hand side correspond to the 20-μm grating period. Solid curves, the analytically derived kernel; crosses, the numerically deconvolved kernel.

Fig. 7
Fig. 7

Broadening of the blazed phase profile. Top, results of convolution (solid curves) of the ideal blazed profile (dotted lines) with the analytical broadening kernel. Bottom, the numerically calculated phase profile (solid curves) in comparison with the ideal blazed profile (dotted lines). LC thicknesses: d=4.8, 7.9, and 11.1 μm (from bottom to top in both panels); grating period: 48 μm.

Fig. 8
Fig. 8

Broadening of the blazed phase profile. Top, results of convolution (solid curves) of the ideal blazed profile (dotted lines) with the analytical broadening kernel. Bottom, the numerically calculated phase profile (solid curves) in comparison with the ideal blazed profile (dotted lines). LC thicknesses: d=4.8, 7.9, and 11.1 μm (from bottom to top in both charts); grating period: 20 μm.

Fig. 9
Fig. 9

Diffraction efficiency of the 48-μm-period blazed grating versus the LC thickness. (a) Direct computation based on simulation results (squares); (b) thickness dependence given by expressions (7.4), (7.11), and (6.10) (solid curve).

Fig. 10
Fig. 10

Diffraction efficiency of the 4.8-μm-thickness blazed grating versus the grating period. (a) Direct computation based on simulation results (squares); (b) grating period and thickness dependence given by expressions (7.4), (7.11), and (6.10) (solid curve).

Equations (74)

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Δϕmax(d)=2πdΔnmax/λ,
Δϕdes(x)=m=-amexp(iωmx),
Vdrv(x)=m=-bmexp(iωmx),
(ˆE)=0,
{xˆ[Ex+Δ(Excos2 θ+Ezsin θ cos θ)]+zˆ[Ez+Δ(Ezsin2 θ+Exsin θ cos θ)]}=0,
(+Δ cos2 θeff) 2V(x, z)x2+Δ sin 2θeff2V(x, z)xz+(+Δ sin2 θeff) 2V(x, z)z2=0,
sin2 θeffθeff2,cos2 θeff1-θeff2,Δ sin 2θeff,
(-Δθeff2) 2V(x, z)x2+(+Δθeff2) 2V(x, z)z2=0.
Δn(x, z)=none[no2cos2 θ(x, z)+ne2sin2 θ(x, z)]1/2-no
Δn(x, z)Δnmax[1-νθ2(x, z)],
ν=ne(ne+no)/(2no2).
Δϕdes(x)=Δϕmax-(2π/Λ)x,0xΛ.
Δϕdes(x)=(2π/λ)0dΔn(x, z)dz=(2πΔnmax/λ)×0d[1-νθ2(x, z)]dz.
0dθ2(x, z)dz=[dmin/(νΛ)]x,
dmin=λ/Δnmax
θ2=dmin/(2νd),
θ2=(dΛ)-10Λ0dθ2(x, z)dxdz
-Δdmin2νd2V(x, z)x2++Δdmin2νd2V(x, z)z2
=0.
V(x, d)=0,V(x, 0)=Vdrv(x).
V(x, z)=m=-cm(z)exp(iωmx).
d2cm(z)dz2-ωm2μ2(d)cm(z)=0,
μ(d)=2νd-Δdmin2νd+Δdmin1/2.
cm(z)=Amexp[ωmμ(d)z]+Bmexp[-ωmμ(d)z],
V(x, z)=m=-sinh[ωmμ(d)(d-z)]sinh[ωmμ(d)d] bmexp(iωmx).
Vm(z)=sinh[ωmμ(d)(d-z)]sinh[ωmμ(d)d],
Vdrv(x)=VinBL(x).
Vlin(x, z)=m=-V˜(ωm)(1-z/d)bmexp(iωmx),
(1/d)0dV(x, z)dz=(1/d)0dVlin(x, z)dz.
V˜(ωm)=tanh[ωmμ(d)d/2]ωmμ(d)d/2.
Veff(x)=Vlin(x, 0)=m=-tanh[ωmμ(d)d/2]ωmμ(d)d/2 bmexp(iωmx).
V^eff(ωm)=V˜(ωm)V^drv(ωm),
Δϕ(V)=Δϕmax-ξ(V-Vth),
VthVVsat,
Vdrv=Vth+(Δϕmax-Δϕ)/ξ.
Δϕdes(x)=(Δϕmax+ξVth)-ξVdrv(x)
Φdes(ωm)=(Δϕmax+ξVth)δ(ωm)-ξV^drv(ωm),
Δϕout(x)=(Δϕmax+ξVth)-ξVeff(x),
Φout(ωm)=(Δϕmax+ξVth)δ(ωm)-ξV^eff(ωm),
V^eff(ωm)=HVV(ωm)V^drv(ωm),
Φout(ωm)=(Δϕmax+ξVth)δ(ωm)-ξHVV(ωm)V^drv(ωm)
Δϕout(x)=-hPP(x)Δϕdes(x-x)dx,
Φout(ωm)=HPP(ωm)Φdes(ωm),
Φout(ωm)=HPP(ωm)[(Δϕmax+ξVth)δ(ωm)-ξV^drv(ωm)].
HPP(ωm=0)=1,
HVV(ωm)=HPP(ωm).
V˜(ωm)=tanh[ωmμ(d)d/2]ωmμ(d)d/2.
HPP(ωm)=K(ωm)=tanh[ωmμ(d)d/2]ωmμ(d)d/2.
Λ=48μm,λ=0.63μm,ne=1.7,no=1.5,=14.1,=4.4.
H˜PP(ωm)=(1+ωm2/ωa2)-1,
HPP(ωm)1-112 μ2(d)d2ωm2,
H˜PP(ωm)1-ωm2/ωa2.
ωa=12/[μ(d)d].
k(x)σ-1exp(-2|x|/σ),
σ=2/ωa=μ(d)d/3=2νd-Δdmin6νd+3Δdmin1/2d
σ(d)pd+q,
p=limdσ(d)d,q=limd[σ(d)-pd].
σ(d)31/2d-2-24ν dmin.
σ(d)31/2d-(2-2)no22(ne2-no2)ne λ.
d>1.85dmin/ν.
Δϕdes(x)=B0(x)=(2π/λ)(ne-no)d-(2π/Λ)x,0xΛ,
B(x)=-B0(x)k(x-x)dx,
k(x)=(1/σ)exp[(x/ασ)],x<0(1/σ)exp{-x/[(1-α)σ]},x0,
Λ=48,20μm;λ=0.63μm;ne=1.7;no=1.5;=14.1;=4.4.
η(1-ΔXFB/Λ)2,
Δϕout(x)=(Δϕmax-2π)-2πΛ x+π exp(2x/σ),-Λ/2x0Δϕmax-2πΛ x-π exp(-2x/σ),0<xΛ/2.
η=(1/Λ2)-Λ/2Λ/2exp[iΔϕout(x)]exp(2πix/Λ)dx2,
η=σΛ2{Ci[π exp(-Λ/σ)]-Ci(π)}2,
Ci(x)=C+ln(x)-x222!+x444!-x666!+ ,
η=1-σΛm=1(-1)m+1π2m2m(2m)! [1-exp(-2mΛ/σ)]2.
η=(1-1.648σ/Λ)2.
ΔXFB=1.648σ.
Δϕ(V)=(Δϕmax+γVth)-γV,
Δϕ(x)=(Δϕmax+γVth)-γV(x).

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