Abstract

We present an analysis of the diffraction efficiency of diffractive lenses displayed on spatial light modulators that depends on the modulation response of the display. An ideal display would produce continuous phase-only modulation, reaching a maximum phase-modulation depth of 2π. We introduce the concept of modulation diffraction efficiency that accounts for the effect of nonlinearities only in the phase modulation of the display. We review a diffractive model with which to evaluate this modulation efficiency, including modulation defects such as nonlinear phase modulation, coupled amplitude modulation, phase quantization, and a limited modulation depth. We apply this diffractive model to Fresnel lenses and show that these modulation defects produce a lens multiplex effect. Finally we demonstrate that the application of a minimum Euclidean projection principle leads to high modulation diffraction efficiency even if the phase-modulation depth is much less than 2π. We demonstrate that the modulation efficiency can exceed 90% for a modulation depth of 1.4π and can exceed 40% (the equivalent for a binary phase element) for a modulation depth of only 0.7π. Experimental results from use of a twisted nematic liquid-crystal display are presented to confirm these conclusions.

© 2004 Optical Society of America

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References

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  1. K. Miyamoto, “The phase Fresnel lens,” J. Opt. Soc. Am. 51, 17–20 (1961).
    [CrossRef]
  2. J. A. Davis, D. M. Cottrell, J. E. Davis, R. A. Lilly, “Fresnel lens encoded binary phase-only filters for optical pattern recognition,” Opt. Lett. 14, 659–661 (1989).
    [CrossRef] [PubMed]
  3. Y. Takaki, H. Ohzu, “Liquid-crystal active lens: a reconfigurable lens employing a phase modulator,” Opt. Commun. 126, 123–134 (1996).
    [CrossRef]
  4. V. Laude, “Twisted-nematic liquid crystal pixelated active lens,” Opt. Commun. 153, 134–152 (1998).
    [CrossRef]
  5. A. Márquez, C. Iemmi, J. C. Escalera, J. Campos, S. Ledesma, J. A. Davis, M. J. Yzuel, “Amplitude apodizers encoded onto Fresnel lenses implemented on a phase-only spatial light modulator,” Appl. Opt. 40, 2316–2322 (2001).
    [CrossRef]
  6. D. A. Buralli, G. M. Morris, “Effects of diffraction efficiency on the modulation transfer function of diffractive lenses,” Appl. Opt. 31, 4389–4396 (1992).
    [CrossRef] [PubMed]
  7. W. Singer, H. Tiziani, “Born approximation for the nonparaxial scalar treatment of thick phase gratings,” Appl. Opt. 37, 1249–1555 (1998).
    [CrossRef]
  8. U. Levy, E. Marom, D. Mendlovic, “Thin element approximation for the analysis of blazed gratings: simplified model and validity limits,” Opt. Commun. 229, 11–21 (2004).
    [CrossRef]
  9. E. G. Loewen, M. Nevière, D. Maystre, “Grating efficiency theory as it applies to blazed and holographic gratings,” Appl. Opt. 16, 2711–2721 (1977).
    [CrossRef] [PubMed]
  10. D. A. Pommet, M. G. Moharam, E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A 11, 1827–1834 (1994).
    [CrossRef]
  11. P. Lalanne, S. Astilean, P. Chavel, E. Cambril, H. Launois, “Design and fabrication of blazed binary diffractive elements with sampling periods smaller than the structural cutoff,” J. Opt. Soc. Am. A 16, 1143–1156 (1999).
    [CrossRef]
  12. E. Noponen, J. Turunen, A. Vasara, “Electromagnetic theory and design of diffractive-lens arrays,” J. Opt. Soc. Am. A 10, 434–443 (1993).
    [CrossRef]
  13. M. Rossi, G. L. Bona, E. E. Kunz, “Arrays of anamorphic phase-matched Fresnel elements for diode-to-fiber coupling,” Appl. Opt. 34, 2483–2488 (1995).
    [CrossRef] [PubMed]
  14. M. T. Gale, M. Rossi, “Continuous-relief diffractive lenses and microlens arrays,” in Diffractive Optics for Industrial and Commercial Applications, J. Turunen, F. Wyroswki, eds. (Akademie-Verlag, Berlin, 1997), Chap. 4.
  15. H. Dammann, “Blazed synthetic phase-only holograms,” Optik 31, 95–104 (1970).
  16. T. Yatagai, M. Takeda, “Effect of phase nonlinearity in kinoform,” Optik 43, 337–352 (1975).
  17. J. A. Davis, D. M. Cottrell, R. A. Lilly, S. W. Connely, “Multiplexed phase-encoded lenses written on spatial light modulators,” Opt. Lett. 14, 420–422 (1989).
    [CrossRef] [PubMed]
  18. E. Carcolé, J. Campos, S. Bosch, “Diffraction theory of Fresnel lenses encoded in low-resolution devices,” Appl. Opt. 33, 162–174 (1994).
    [CrossRef] [PubMed]
  19. M. Kuittinen, H. P. Herzig, “Encoding of efficient diffractive microlenses,” Opt. Lett. 20, 2156–2158 (1995).
    [CrossRef] [PubMed]
  20. J. Fan, D. Zaleta, K. S. Urquhart, S. H. Lee, “Efficient encoding algorithms for computer-aided design of diffractive optical elements by the use of electron-beam fabrication,” Appl. Opt. 34, 2522–2533 (1995).
    [CrossRef] [PubMed]
  21. U. Levy, D. Mendlovic, E. Marom, “Efficiency analysis of diffractive lenses,” J. Opt. Soc. Am. A 18, 86–93 (2001).
    [CrossRef]
  22. C. Chen, A. A. Sawchuk, “Nonlinear least-squares and phase-shifting quantization methods for diffractive optical element design,” Appl. Opt. 36, 7297–7306 (1997).
    [CrossRef]
  23. V. Arrizón, S. Sinzinger, “Modified quantization schemes for Fourier-type array generator,” Opt. Commun. 140, 309–315 (1997).
    [CrossRef]
  24. U. Levy, N. Cohen, D. Mendlovic, “Analytic approach for optimal quantization of diffractive optical elements,” Appl. Opt. 38, 5527–5532 (1999).
    [CrossRef]
  25. J. L. 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]
  26. A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
    [CrossRef]
  27. A. Márquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays,” Opt. Eng. 40, 2558–2564 (2001).
    [CrossRef]
  28. R. D. Juday, “Optical realizable filters and the minimum Euclidean distance principle,” Appl. Opt. 32, 5100–5111 (1993).
    [CrossRef] [PubMed]
  29. V. Laude, P. Réfrégier, “Multicriteria characterization of coding domains with optimal Fourier SLM filters,” Appl. Opt. 33, 4465–4471 (1994).
    [CrossRef] [PubMed]
  30. R. D. Juday, “Generality of matched filtering and minimum Euclidean distance projection for optical pattern recognition,” J. Opt. Soc. Am. A 18, 1882–1896 (2001).
    [CrossRef]
  31. I. Moreno, J. Campos, C. Gorecki, M. J. Yzuel, “Effects of amplitude and phase mismatching errors in the generation of a kinoform for pattern recognition,” Jpn. J. Appl. Phys. 34, 6423–6432 (1995).
    [CrossRef]
  32. W. J. Dallas, “Computer generated holograms,” in The Computer in Optical Research, B. R. Frieden, ed., Vol. 41 of Topics in Applied Physics (Springer-Verlag, Berlin, 1980), Chap. 6.

2004 (1)

U. Levy, E. Marom, D. Mendlovic, “Thin element approximation for the analysis of blazed gratings: simplified model and validity limits,” Opt. Commun. 229, 11–21 (2004).
[CrossRef]

2001 (4)

2000 (1)

A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

1999 (2)

1998 (2)

1997 (3)

1996 (1)

Y. Takaki, H. Ohzu, “Liquid-crystal active lens: a reconfigurable lens employing a phase modulator,” Opt. Commun. 126, 123–134 (1996).
[CrossRef]

1995 (4)

1994 (3)

1993 (2)

1992 (1)

1989 (2)

1977 (1)

1975 (1)

T. Yatagai, M. Takeda, “Effect of phase nonlinearity in kinoform,” Optik 43, 337–352 (1975).

1970 (1)

H. Dammann, “Blazed synthetic phase-only holograms,” Optik 31, 95–104 (1970).

1961 (1)

Arrizón, V.

V. Arrizón, S. Sinzinger, “Modified quantization schemes for Fourier-type array generator,” Opt. Commun. 140, 309–315 (1997).
[CrossRef]

Astilean, S.

Bona, G. L.

Bosch, S.

Bougrenet de la Tocnaye, J. L.

Buralli, D. A.

Cambril, E.

Campos, J.

A. Márquez, C. Iemmi, J. C. Escalera, J. Campos, S. Ledesma, J. A. Davis, M. J. Yzuel, “Amplitude apodizers encoded onto Fresnel lenses implemented on a phase-only spatial light modulator,” Appl. Opt. 40, 2316–2322 (2001).
[CrossRef]

A. Márquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays,” Opt. Eng. 40, 2558–2564 (2001).
[CrossRef]

A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

I. Moreno, J. Campos, C. Gorecki, M. J. Yzuel, “Effects of amplitude and phase mismatching errors in the generation of a kinoform for pattern recognition,” Jpn. J. Appl. Phys. 34, 6423–6432 (1995).
[CrossRef]

E. Carcolé, J. Campos, S. Bosch, “Diffraction theory of Fresnel lenses encoded in low-resolution devices,” Appl. Opt. 33, 162–174 (1994).
[CrossRef] [PubMed]

Carcolé, E.

Chavel, P.

Chen, C.

Cohen, N.

Connely, S. W.

Cottrell, D. M.

Dallas, W. J.

W. J. Dallas, “Computer generated holograms,” in The Computer in Optical Research, B. R. Frieden, ed., Vol. 41 of Topics in Applied Physics (Springer-Verlag, Berlin, 1980), Chap. 6.

Dammann, H.

H. Dammann, “Blazed synthetic phase-only holograms,” Optik 31, 95–104 (1970).

Davis, J. A.

Davis, J. E.

Dupont, L.

Escalera, J. C.

Fan, J.

Gale, M. T.

M. T. Gale, M. Rossi, “Continuous-relief diffractive lenses and microlens arrays,” in Diffractive Optics for Industrial and Commercial Applications, J. Turunen, F. Wyroswki, eds. (Akademie-Verlag, Berlin, 1997), Chap. 4.

Gorecki, C.

I. Moreno, J. Campos, C. Gorecki, M. J. Yzuel, “Effects of amplitude and phase mismatching errors in the generation of a kinoform for pattern recognition,” Jpn. J. Appl. Phys. 34, 6423–6432 (1995).
[CrossRef]

Grann, E. B.

Herzig, H. P.

Iemmi, C.

A. Márquez, C. Iemmi, J. C. Escalera, J. Campos, S. Ledesma, J. A. Davis, M. J. Yzuel, “Amplitude apodizers encoded onto Fresnel lenses implemented on a phase-only spatial light modulator,” Appl. Opt. 40, 2316–2322 (2001).
[CrossRef]

A. Márquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays,” Opt. Eng. 40, 2558–2564 (2001).
[CrossRef]

A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

Juday, R. D.

Kuittinen, M.

Kunz, E. E.

Lalanne, P.

Laude, V.

Launois, H.

Ledesma, S.

Lee, S. H.

Levy, U.

Lilly, R. A.

Loewen, E. G.

Marom, E.

U. Levy, E. Marom, D. Mendlovic, “Thin element approximation for the analysis of blazed gratings: simplified model and validity limits,” Opt. Commun. 229, 11–21 (2004).
[CrossRef]

U. Levy, D. Mendlovic, E. Marom, “Efficiency analysis of diffractive lenses,” J. Opt. Soc. Am. A 18, 86–93 (2001).
[CrossRef]

Márquez, A.

A. Márquez, C. Iemmi, J. C. Escalera, J. Campos, S. Ledesma, J. A. Davis, M. J. Yzuel, “Amplitude apodizers encoded onto Fresnel lenses implemented on a phase-only spatial light modulator,” Appl. Opt. 40, 2316–2322 (2001).
[CrossRef]

A. Márquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays,” Opt. Eng. 40, 2558–2564 (2001).
[CrossRef]

A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

Maystre, D.

Mendlovic, D.

Miyamoto, K.

Moharam, M. G.

Moreno, A.

A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

Moreno, I.

A. Márquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays,” Opt. Eng. 40, 2558–2564 (2001).
[CrossRef]

A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

I. Moreno, J. Campos, C. Gorecki, M. J. Yzuel, “Effects of amplitude and phase mismatching errors in the generation of a kinoform for pattern recognition,” Jpn. J. Appl. Phys. 34, 6423–6432 (1995).
[CrossRef]

Morris, G. M.

Nevière, M.

Noponen, E.

Ohzu, H.

Y. Takaki, H. Ohzu, “Liquid-crystal active lens: a reconfigurable lens employing a phase modulator,” Opt. Commun. 126, 123–134 (1996).
[CrossRef]

Pommet, D. A.

Réfrégier, P.

Robert, A.

A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

Rossi, M.

M. Rossi, G. L. Bona, E. E. Kunz, “Arrays of anamorphic phase-matched Fresnel elements for diode-to-fiber coupling,” Appl. Opt. 34, 2483–2488 (1995).
[CrossRef] [PubMed]

M. T. Gale, M. Rossi, “Continuous-relief diffractive lenses and microlens arrays,” in Diffractive Optics for Industrial and Commercial Applications, J. Turunen, F. Wyroswki, eds. (Akademie-Verlag, Berlin, 1997), Chap. 4.

Sawchuk, A. A.

Singer, W.

Sinzinger, S.

V. Arrizón, S. Sinzinger, “Modified quantization schemes for Fourier-type array generator,” Opt. Commun. 140, 309–315 (1997).
[CrossRef]

Takaki, Y.

Y. Takaki, H. Ohzu, “Liquid-crystal active lens: a reconfigurable lens employing a phase modulator,” Opt. Commun. 126, 123–134 (1996).
[CrossRef]

Takeda, M.

T. Yatagai, M. Takeda, “Effect of phase nonlinearity in kinoform,” Optik 43, 337–352 (1975).

Tiziani, H.

Turunen, J.

Urquhart, K. S.

Vasara, A.

Yatagai, T.

T. Yatagai, M. Takeda, “Effect of phase nonlinearity in kinoform,” Optik 43, 337–352 (1975).

Yzuel, M. J.

A. Márquez, C. Iemmi, J. C. Escalera, J. Campos, S. Ledesma, J. A. Davis, M. J. Yzuel, “Amplitude apodizers encoded onto Fresnel lenses implemented on a phase-only spatial light modulator,” Appl. Opt. 40, 2316–2322 (2001).
[CrossRef]

A. Márquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays,” Opt. Eng. 40, 2558–2564 (2001).
[CrossRef]

A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

I. Moreno, J. Campos, C. Gorecki, M. J. Yzuel, “Effects of amplitude and phase mismatching errors in the generation of a kinoform for pattern recognition,” Jpn. J. Appl. Phys. 34, 6423–6432 (1995).
[CrossRef]

Zaleta, D.

Appl. Opt. (12)

A. Márquez, C. Iemmi, J. C. Escalera, J. Campos, S. Ledesma, J. A. Davis, M. J. Yzuel, “Amplitude apodizers encoded onto Fresnel lenses implemented on a phase-only spatial light modulator,” Appl. Opt. 40, 2316–2322 (2001).
[CrossRef]

D. A. Buralli, G. M. Morris, “Effects of diffraction efficiency on the modulation transfer function of diffractive lenses,” Appl. Opt. 31, 4389–4396 (1992).
[CrossRef] [PubMed]

W. Singer, H. Tiziani, “Born approximation for the nonparaxial scalar treatment of thick phase gratings,” Appl. Opt. 37, 1249–1555 (1998).
[CrossRef]

E. G. Loewen, M. Nevière, D. Maystre, “Grating efficiency theory as it applies to blazed and holographic gratings,” Appl. Opt. 16, 2711–2721 (1977).
[CrossRef] [PubMed]

M. Rossi, G. L. Bona, E. E. Kunz, “Arrays of anamorphic phase-matched Fresnel elements for diode-to-fiber coupling,” Appl. Opt. 34, 2483–2488 (1995).
[CrossRef] [PubMed]

E. Carcolé, J. Campos, S. Bosch, “Diffraction theory of Fresnel lenses encoded in low-resolution devices,” Appl. Opt. 33, 162–174 (1994).
[CrossRef] [PubMed]

J. Fan, D. Zaleta, K. S. Urquhart, S. H. Lee, “Efficient encoding algorithms for computer-aided design of diffractive optical elements by the use of electron-beam fabrication,” Appl. Opt. 34, 2522–2533 (1995).
[CrossRef] [PubMed]

U. Levy, N. Cohen, D. Mendlovic, “Analytic approach for optimal quantization of diffractive optical elements,” Appl. Opt. 38, 5527–5532 (1999).
[CrossRef]

J. L. 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]

R. D. Juday, “Optical realizable filters and the minimum Euclidean distance principle,” Appl. Opt. 32, 5100–5111 (1993).
[CrossRef] [PubMed]

V. Laude, P. Réfrégier, “Multicriteria characterization of coding domains with optimal Fourier SLM filters,” Appl. Opt. 33, 4465–4471 (1994).
[CrossRef] [PubMed]

C. Chen, A. A. Sawchuk, “Nonlinear least-squares and phase-shifting quantization methods for diffractive optical element design,” Appl. Opt. 36, 7297–7306 (1997).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Jpn. J. Appl. Phys. (1)

I. Moreno, J. Campos, C. Gorecki, M. J. Yzuel, “Effects of amplitude and phase mismatching errors in the generation of a kinoform for pattern recognition,” Jpn. J. Appl. Phys. 34, 6423–6432 (1995).
[CrossRef]

Opt. Commun. (4)

V. Arrizón, S. Sinzinger, “Modified quantization schemes for Fourier-type array generator,” Opt. Commun. 140, 309–315 (1997).
[CrossRef]

Y. Takaki, H. Ohzu, “Liquid-crystal active lens: a reconfigurable lens employing a phase modulator,” Opt. Commun. 126, 123–134 (1996).
[CrossRef]

V. Laude, “Twisted-nematic liquid crystal pixelated active lens,” Opt. Commun. 153, 134–152 (1998).
[CrossRef]

U. Levy, E. Marom, D. Mendlovic, “Thin element approximation for the analysis of blazed gratings: simplified model and validity limits,” Opt. Commun. 229, 11–21 (2004).
[CrossRef]

Opt. Eng. (2)

A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

A. Márquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays,” Opt. Eng. 40, 2558–2564 (2001).
[CrossRef]

Opt. Lett. (3)

Optik (2)

H. Dammann, “Blazed synthetic phase-only holograms,” Optik 31, 95–104 (1970).

T. Yatagai, M. Takeda, “Effect of phase nonlinearity in kinoform,” Optik 43, 337–352 (1975).

Other (2)

M. T. Gale, M. Rossi, “Continuous-relief diffractive lenses and microlens arrays,” in Diffractive Optics for Industrial and Commercial Applications, J. Turunen, F. Wyroswki, eds. (Akademie-Verlag, Berlin, 1997), Chap. 4.

W. J. Dallas, “Computer generated holograms,” in The Computer in Optical Research, B. R. Frieden, ed., Vol. 41 of Topics in Applied Physics (Springer-Verlag, Berlin, 1980), Chap. 6.

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Figures (5)

Fig. 1
Fig. 1

Generation of multiple harmonic lenses when a diffractive lens is displayed on a LCD, producing an imperfect phase-only modulation. F i , image focal points of the i-order harmonic lens.

Fig. 2
Fig. 2

Intensity of orders α = 0, ±1 versus mismatch parameter c. Dotted-dashed curve, orders α = ±1; continuous curve, order α = 0.

Fig. 3
Fig. 3

(a) Two examples of phase mismatching versus the addressed phase. Diagonal line, perfect matching. Model 1, linear mismatch; Model 2, saturated mismatch. (b) Intensity of the orders α = 0, ±1 versus mismatch parameter c. Dotted-dashed curves, Model 1; continuous curves, Model 2.

Fig. 4
Fig. 4

Complex modulation provided by the twisted nematic LCD for wavelength λ = 458 nm in the phase-mostly configuration. A theoretical prediction based on a Jones model and the experimental measured modulation obtained with uniformly distributed addressed gray levels are shown.

Fig. 5
Fig. 5

Experimentally measured value of the light efficiency of horizontal cylindrical lenses as a function of mismatch parameter c for linear, saturated, and binary mismatching encodings. Curves, theoretical diffraction efficiency; symbols, experimental values.

Equations (18)

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

mφx, y=aφx, yexpipφx, y,
mφ=aφexpipφ=α=- Gα expiαφ,
Gα=12π02π aφexpipφexp-iαφdφ,
φx, y=-πr2/λf,
mr=α=-+ Gα expjαφr=α=-+ Gα exp-j πr2λf/α,
mφ=t=0N-1 a2πt/Nexpip2πt/N×rectpφ2π/N-t,
rectpφ2π/N-t=α=-1Nexp-i2π αN t×sincαNexpiαφ,
Gα=1NsincαNt=0N-1 a2πt/Nexpip2πt/N×exp-i2παt/N.
pφ=0φπ2π1-cπ<φ2π,
|G0|2=cos2π1-c,
|G1|2=2π2 cos2π12-c,
|G-1|2=2π2 cos2π32-c.
mφ=expi1-cφ.
|Gα|2=sinc21-c-α.
mφ=expiφφ<εexpiεε<φ<ε/2+π1φ>ε/2+π.
|G0|2=1π2sinε2+π-ε2cosε22,
ηm=|G1|2=ε2π+1πsinε22,
|G-1|2=sinε2π-2 cosε/4sin3ε/4π2.

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