Abstract

A detailed simulation of the fringing-field effect in liquid-crystal (LC)-based blazed-grating structures has been carried out. These studies are aimed at clarifying the relationship between the width of the fringing-field-broadened phase profile of the blazed grating and the LC cell thickness. This fringing-field broadening of the blazed grating’s phase profile is shown to affect mostly the 2π phase-step zone (fly-back zone) of the blazed grating. The results of the simulations carried out on the blazed-grating structure indicate two main effects of the fringing field: (1) reduction in the attainable diffraction efficiency and (2) limitation of the maximum deflection angle of the device. Both effects are shown to be directly linked to the broadening of the fly-back zone, which is shown to be proportional to the LC cell thickness.

© 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. W. Klaus, M. Ide, S. Morokawa, M. Tsuchiya, T. Kamiya, “Angle-independent beam steering using a liquid crystal grating with multi-resistive electrodes,” Opt. Commun. 138, 151–157 (1997).
    [CrossRef]
  5. X. Wang, D. Wilson, R. Muller, P. Maker, D. Psaltis, “Liquid-crystal blazed-grating beam deflector,” Appl. Opt. 39, 6545–6555 (2000).
    [CrossRef]
  6. M. L. Jepsen, H. J. Gerritsen, “Liquid-crystal-filled grating with high diffraction efficiency,” Opt. Lett. 21, 1081–1083 (1996).
    [CrossRef] [PubMed]
  7. M. Bouvier, T. Scharf, “Analysis of nematic-liquid-crystal binary gratings with high spatial frequency,” Opt. Eng. 39, 2129–2137 (2000).
    [CrossRef]
  8. 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]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  12. 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 programmable liquid-crystal grating,” Appl. Opt. 35, 6236–6240 (1996).
    [CrossRef] [PubMed]
  13. 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. 35, 3371–3379 (1996).
    [CrossRef]
  14. T. Scharf, M. Bouvier, R. Dändliker, “Multilevel nematic liquid crystal phase grating,” 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]
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    [CrossRef] [PubMed]
  16. B. Veweire, R. Defever, “Limitation of resolution of LCOS-based projection displays by diffraction effects,” in Proceedings of the 19th International Display Research Conference (EuroDisplay ’99, Berlin, Germany) (Society of Information Displays, San Jose, Calif., 1999), pp. 489–492.
  17. H. De Smet, J. Van Der Steen, A. Van Calster, “Microdisplays with high pixel count,” in Proceedings of the Society for Information Displays (SID) Digest (Society of Information Displays, San Jose, Calif., 2001), pp. 968–971.
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    [CrossRef]
  19. 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 (2001).
    [CrossRef]
  20. Autronics-Melcher’s 2dimMOS software, http://www.autronic-melchers.com/index.htm .
  21. F. C. Frank, “On the theory of liquid crystals,” Discuss. Faraday Soc. 25, 19–28 (1958).
    [CrossRef]
  22. C. W. Oseen, “The theory of liquid crystals,” Trans. Faraday Soc. 29, 883–889 (1933).
    [CrossRef]
  23. R. Magnussen, T. K. Gaylord, “Diffraction efficiencies of thin phase gratings with arbitrary grating shape,” J. Opt. Soc. Am. 68, 806–809 (1978).
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    [CrossRef] [PubMed]

2002

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

M. Oh-e, K. Kondo, “Electro-optical characteristics and switching behavior of the in-plane switching mode,” Appl. Phys. Lett. 67, 3895–3897 (2001).
[CrossRef]

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 (2001).
[CrossRef]

C. M. Titus, J. R. Kelly, E. C. Garland, S. V. Shiyanovskii, J. A. Anderson, P. J. Bos, “Asymmetric transmissive behavior of liquid-crystal diffraction gratings,” Opt. Lett. 26, 1188–1190 (2001).
[CrossRef]

2000

X. Wang, D. Wilson, R. Muller, P. Maker, D. Psaltis, “Liquid-crystal blazed-grating beam deflector,” Appl. Opt. 39, 6545–6555 (2000).
[CrossRef]

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

1998

1997

W. Klaus, M. Ide, S. Morokawa, M. Tsuchiya, T. Kamiya, “Angle-independent beam steering using a liquid crystal grating with multi-resistive electrodes,” Opt. Commun. 138, 151–157 (1997).
[CrossRef]

1996

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]

M. L. Jepsen, H. J. Gerritsen, “Liquid-crystal-filled grating with high diffraction efficiency,” Opt. Lett. 21, 1081–1083 (1996).
[CrossRef] [PubMed]

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. 35, 3371–3379 (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 programmable liquid-crystal grating,” Appl. Opt. 35, 6236–6240 (1996).
[CrossRef] [PubMed]

1995

1982

1979

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

1958

F. C. Frank, “On the theory of liquid crystals,” Discuss. Faraday Soc. 25, 19–28 (1958).
[CrossRef]

1933

C. W. Oseen, “The theory of liquid crystals,” Trans. Faraday Soc. 29, 883–889 (1933).
[CrossRef]

Anderson, J. A.

Bos, P. J.

Bouvier, M.

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

T. Scharf, M. Bouvier, R. Dändliker, “Multilevel nematic liquid crystal phase grating,” 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. 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]

Corkum, D. L.

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 programmable liquid-crystal grating,” Appl. Opt. 35, 6236–6240 (1996).
[CrossRef] [PubMed]

Dändliker, R.

T. Scharf, M. Bouvier, R. Dändliker, “Multilevel nematic liquid crystal phase grating,” 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 Displays (SID) Digest (Society of Information Displays, 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 (EuroDisplay ’99, Berlin, Germany) (Society of Information Displays, 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. 35, 3371–3379 (1996).
[CrossRef]

Dorschner, T. A.

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]

Frank, F. C.

F. C. Frank, “On the theory of liquid crystals,” Discuss. Faraday Soc. 25, 19–28 (1958).
[CrossRef]

Friedman, L. J.

Friends, M. W.

Fujita, T.

Garland, E. C.

Gaylord, T. K.

Gerritsen, H. J.

Hard, S.

Hobbs, D. S.

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]

Ide, M.

W. Klaus, M. Ide, S. Morokawa, M. Tsuchiya, T. Kamiya, “Angle-independent beam steering using a liquid crystal grating with multi-resistive electrodes,” Opt. Commun. 138, 151–157 (1997).
[CrossRef]

Jarem, J. M.

Jepsen, M. L.

Kamiya, T.

W. Klaus, M. Ide, S. Morokawa, M. Tsuchiya, T. Kamiya, “Angle-independent beam steering using a liquid crystal grating with multi-resistive electrodes,” Opt. Commun. 138, 151–157 (1997).
[CrossRef]

Kelly, J. R.

Klaus, W.

W. Klaus, M. Ide, S. Morokawa, M. Tsuchiya, T. Kamiya, “Angle-independent beam steering using a liquid crystal grating with multi-resistive electrodes,” Opt. Commun. 138, 151–157 (1997).
[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 (2001).
[CrossRef]

M. Oh-e, K. Kondo, “Electro-optical characteristics and switching behavior of the in-plane switching mode,” Appl. Phys. Lett. 67, 3895–3897 (2001).
[CrossRef]

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.

Lindquist, R. G.

Löfving, B.

Magnussen, R.

Maker, P.

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]

Morokawa, S.

W. Klaus, M. Ide, S. Morokawa, M. Tsuchiya, T. Kamiya, “Angle-independent beam steering using a liquid crystal grating with multi-resistive electrodes,” Opt. Commun. 138, 151–157 (1997).
[CrossRef]

Muller, R.

Nguyen, H. Q.

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 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]

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 (2001).
[CrossRef]

M. Oh-e, K. Kondo, “Electro-optical characteristics and switching behavior of the in-plane switching mode,” Appl. Phys. Lett. 67, 3895–3897 (2001).
[CrossRef]

Oseen, C. W.

C. W. Oseen, “The theory of liquid crystals,” Trans. Faraday Soc. 29, 883–889 (1933).
[CrossRef]

Psaltis, D.

Resler, D. P.

Scharf, T.

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

T. Scharf, M. Bouvier, R. Dändliker, “Multilevel nematic liquid crystal phase grating,” 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.

Shiyanovskii, S. V.

Titus, C. M.

Tsuchiya, M.

W. Klaus, M. Ide, S. Morokawa, M. Tsuchiya, T. Kamiya, “Angle-independent beam steering using a liquid crystal grating with multi-resistive electrodes,” Opt. Commun. 138, 151–157 (1997).
[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 Displays (SID) Digest (Society of Information Displays, 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 Displays (SID) Digest (Society of Information Displays, 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 (EuroDisplay ’99, Berlin, Germany) (Society of Information Displays, San Jose, Calif., 1999), pp. 489–492.

Wang, X.

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. 35, 3371–3379 (1996).
[CrossRef]

Wilson, D.

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 (2001).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

M. Oh-e, K. Kondo, “Electro-optical characteristics and switching behavior of the in-plane switching mode,” Appl. Phys. Lett. 67, 3895–3897 (2001).
[CrossRef]

Discuss. Faraday Soc.

F. C. Frank, “On the theory of liquid crystals,” Discuss. Faraday Soc. 25, 19–28 (1958).
[CrossRef]

J. Appl. Phys.

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 (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.

Mol. Cryst. Liq. Cryst.

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. Commun.

W. Klaus, M. Ide, S. Morokawa, M. Tsuchiya, T. Kamiya, “Angle-independent beam steering using a liquid crystal grating with multi-resistive electrodes,” Opt. Commun. 138, 151–157 (1997).
[CrossRef]

Opt. Eng.

M. Bouvier, T. Scharf, “Analysis of nematic-liquid-crystal binary gratings with high spatial frequency,” Opt. Eng. 39, 2129–2137 (2000).
[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. 35, 3371–3379 (1996).
[CrossRef]

Opt. Lett.

Proc. IEEE

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]

Trans. Faraday Soc.

C. W. Oseen, “The theory of liquid crystals,” Trans. Faraday Soc. 29, 883–889 (1933).
[CrossRef]

Other

T. Scharf, M. Bouvier, R. Dändliker, “Multilevel nematic liquid crystal phase grating,” 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]

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]

B. Veweire, R. Defever, “Limitation of resolution of LCOS-based projection displays by diffraction effects,” in Proceedings of the 19th International Display Research Conference (EuroDisplay ’99, Berlin, Germany) (Society of Information Displays, 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 Displays (SID) Digest (Society of Information Displays, San Jose, Calif., 2001), pp. 968–971.

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

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Figures (10)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

Computed phase retardation versus voltage for a 4.8-μm-thick LC cell; the cell parameters are given in Table 1, and λ = 633 nm.

Fig. 3
Fig. 3

Phase profile (left) and resulting diffraction efficiency (right) of a 48-μm-period blazed grating for LC cell thicknesses of (from bottom to top) 4.8, 6.3, 7.9, 9.5, and 11.1 μm; λ = 633 nm.

Fig. 4
Fig. 4

Broadening kernels for LC thicknesses of 4.8, 7.9, and 11.1 μm (from left to right): results of numerical deconvolution (crosses) and corresponding biexponential functions (solid curves).

Fig. 5
Fig. 5

Dependence of the biexponential kernel width σ on the LC thickness for a 48-μm blazed-grating period.

Fig. 6
Fig. 6

Diffraction efficiency of the 48-μm-period blazed grating versus LC thickness. The direct computation is based on simulation results (squares), and the approximate thickness dependence is given by expression (11) and Eq. (12), with β = 2 and γ = 1.5 (dashed curve).

Fig. 7
Fig. 7

Blazed-grating phase profiles for 4.8-μm-thick LC cells (left) and the diffraction efficiency (right) for grating periods of 48, 30, 20, 16, 12, 10, and 8 μm (from top to bottom); λ = 633 nm. The LC cell parameters are given in Table 1.

Fig. 8
Fig. 8

Electrode voltage distribution, equipotential lines. The LC director distribution (top) and the corresponding retardation phase profile (bottom) for a 4.8-μm-thick, 12-μm-period, six-electrode blazed-grating structure. The design phase profile is illustrated by the dashed curve. The LC cell parameters are given in Table 1.

Fig. 9
Fig. 9

Diffraction efficiency of the 4.8-μm-thick blazed grating versus grating period. Squares, direct computation based on the simulation results; dashed curve, approximate dependence on the fly-back-zone width as given by expression (11) and Eq. (12), with β = 2 and γ = 1.5.

Fig. 10
Fig. 10

Diffraction efficiency versus grating period for various LC thicknesses.

Tables (1)

Tables Icon

Table 1 Parameters of the LC Layer, Used for Simulation

Equations (17)

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

nx, z=noneno2Nx2x, z+ne2Nz2x, z1/2,
Δnx=1d0dne-nx, zdz,
Δϕx=2πλ dΔnx.
η-1=1/Λ20ΛexpiΔϕxexp2πix/Λdx2,
B0x=2π/λne-nod-2π/Λx, 0xΛ,
Bx= B0xkx-xdx,
kx=1/σexpx/ασ,x<01/σexp-x/1-ασ,x0,
σdd-0.5dmin,
Δϕmax=2πλ dΔnmax=2π.
dmin=λ/Δnmax,
η1-ΔXFB/Λ2,
ΔXFB=βd-γdmin.
ΔXFB=2d-1.5dmin.
θmax=λ/Λmax.
λ/Λmax=λ/Λmin,
θmax=λ/Λminλ/2ΔXFB=λ/4d-3dmin,
θmax=λ4d-3λ/Δnmax,

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