Abstract

We examine the diffraction properties of one- and two-dimensional binary-phase gratings encoded onto pixelated liquid crystal displays (LCDs). We find that the first-order diffracted intensity from these binary-phase patterns can reach 100% of the zero-order intensity when the period of the grating approaches the Nyquist limit of the LCD. Experimental results show excellent agreement with theoretical predictions. This is a surprising result that has a number of implications for the encoding of diffractive optical elements.

© 2008 Optical Society of America

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  1. J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. 38, 5004-5013 (1999).
    [Crossref]
  2. J. A. Davis, D. A. Smith, D. E. McNamara, D. M. Cottrell, and J. Campos, “Fractional derivatives--Analysis and experimental implementation,” Appl. Opt. 40, 5943-5948 (2001).
    [Crossref]
  3. J. B. Bentley, J. A. Davis, M. A. Bandres, and J. C. Gutiérrez-Vega, “Generation of helical Ince-Gaussian beams with a liquid crystal display,” Opt. Lett. 31, 649-651 (2006).
    [Crossref] [PubMed]
  4. J. A. Davis, C. S. Tuvey, O. López-Coronado, J. Campos, M. J. Yzuel, and C. Iemmi, “Tailoring the depth of focus for optical imaging systems using a Fourier transform approach,” Opt. Lett. 32, 844-846 (2007).
    [Crossref] [PubMed]
  5. H. Dammann, “Blazed synthetic phase-only holograms,” Optik (Jena) 31, 95-104 (1970).
  6. J. A. Davis, E. A. Merrill, D. M. Cottrell, and R. M. Bunch, “Effects of sampling, nonlinear recording, and binarization on the output of the joint Fourier transform correlator,” Opt. Eng. 29, 1094-1100 (1990).
    [Crossref]
  7. S. Bhattacharya and R. S. Sirohi, “Amplitude checker grating from one-dimensional Ronchi grating and its application to array generation,” Appl. Opt. 36, 3745-3752 (1997).
    [Crossref] [PubMed]
  8. N. McArdle and M. R. Taghizadeh, “Real-time reconfigurable interconnections for parallel optical processing,” Opt. Rev. 2, 189-193 (1995).
    [Crossref]
  9. V. Arrizon and E. Lopez-Olazagasti, “Binary phase grating for array generation at 1/16 of the Talbot length,” J. Opt. Soc. Am. A 12, 801-804 (1995).
    [Crossref]
  10. M. Zhu, G. Carbone, and C. Rosenblatt, “Electrically switchable, polarization-independent diffraction grating based on negative dielectric anisotropy liquid crystal,” Appl. Phys. Lett. 88, 253502 (2006).
    [Crossref]
  11. C. Soutar and K. Lu, “Determination of the physical properties of an arbitrary twisted-nematic liquid crystal cell,” Opt. Eng. 33, 2704-2712 (1994).
    [Crossref]
  12. J. A. Davis, D. B. Allison, K. G. D'Nelly, M. L. Wilson, and I. Moreno, “Ambiguities in measuring the physical parameters for twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 38, 705-709 (1999).
    [Crossref]
  13. J. A. Davis, I. Moreno, and P. Tsai, “Polarization eigenstates for twisted-nematic liquid crystal displays,” Appl. Opt. . 37, 937-945 (1998).
    [Crossref]
  14. I. Moreno, J. A. Davis, K. G. D'Nelly, and D. B. Allison, “Transmission and phase measurement for polarization eigenvectors in twisted nematic liquid crystal displays,” Opt. Eng. 37, 3048-3052 (1998).
    [Crossref]
  15. J. A. Davis, J. Nicolas, and A. Marquez, “Phasor analysis of eigenvectors generated in liquid crystal displays,” Appl. Opt. 41, 4579-4584 (2002).
    [Crossref] [PubMed]
  16. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978), Chap. 7.4.

2007 (1)

2006 (2)

J. B. Bentley, J. A. Davis, M. A. Bandres, and J. C. Gutiérrez-Vega, “Generation of helical Ince-Gaussian beams with a liquid crystal display,” Opt. Lett. 31, 649-651 (2006).
[Crossref] [PubMed]

M. Zhu, G. Carbone, and C. Rosenblatt, “Electrically switchable, polarization-independent diffraction grating based on negative dielectric anisotropy liquid crystal,” Appl. Phys. Lett. 88, 253502 (2006).
[Crossref]

2002 (1)

2001 (1)

1999 (2)

J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. 38, 5004-5013 (1999).
[Crossref]

J. A. Davis, D. B. Allison, K. G. D'Nelly, M. L. Wilson, and I. Moreno, “Ambiguities in measuring the physical parameters for twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 38, 705-709 (1999).
[Crossref]

1998 (2)

I. Moreno, J. A. Davis, K. G. D'Nelly, and D. B. Allison, “Transmission and phase measurement for polarization eigenvectors in twisted nematic liquid crystal displays,” Opt. Eng. 37, 3048-3052 (1998).
[Crossref]

J. A. Davis, I. Moreno, and P. Tsai, “Polarization eigenstates for twisted-nematic liquid crystal displays,” Appl. Opt. . 37, 937-945 (1998).
[Crossref]

1997 (1)

1995 (2)

V. Arrizon and E. Lopez-Olazagasti, “Binary phase grating for array generation at 1/16 of the Talbot length,” J. Opt. Soc. Am. A 12, 801-804 (1995).
[Crossref]

N. McArdle and M. R. Taghizadeh, “Real-time reconfigurable interconnections for parallel optical processing,” Opt. Rev. 2, 189-193 (1995).
[Crossref]

1994 (1)

C. Soutar and K. Lu, “Determination of the physical properties of an arbitrary twisted-nematic liquid crystal cell,” Opt. Eng. 33, 2704-2712 (1994).
[Crossref]

1990 (1)

J. A. Davis, E. A. Merrill, D. M. Cottrell, and R. M. Bunch, “Effects of sampling, nonlinear recording, and binarization on the output of the joint Fourier transform correlator,” Opt. Eng. 29, 1094-1100 (1990).
[Crossref]

1970 (1)

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

Allison, D. B.

J. A. Davis, D. B. Allison, K. G. D'Nelly, M. L. Wilson, and I. Moreno, “Ambiguities in measuring the physical parameters for twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 38, 705-709 (1999).
[Crossref]

I. Moreno, J. A. Davis, K. G. D'Nelly, and D. B. Allison, “Transmission and phase measurement for polarization eigenvectors in twisted nematic liquid crystal displays,” Opt. Eng. 37, 3048-3052 (1998).
[Crossref]

Arrizon, V.

Bandres, M. A.

Bentley, J. B.

Bhattacharya, S.

Bunch, R. M.

J. A. Davis, E. A. Merrill, D. M. Cottrell, and R. M. Bunch, “Effects of sampling, nonlinear recording, and binarization on the output of the joint Fourier transform correlator,” Opt. Eng. 29, 1094-1100 (1990).
[Crossref]

Campos, J.

Carbone, G.

M. Zhu, G. Carbone, and C. Rosenblatt, “Electrically switchable, polarization-independent diffraction grating based on negative dielectric anisotropy liquid crystal,” Appl. Phys. Lett. 88, 253502 (2006).
[Crossref]

Cottrell, D. M.

Dammann, H.

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

Davis, J. A.

J. A. Davis, C. S. Tuvey, O. López-Coronado, J. Campos, M. J. Yzuel, and C. Iemmi, “Tailoring the depth of focus for optical imaging systems using a Fourier transform approach,” Opt. Lett. 32, 844-846 (2007).
[Crossref] [PubMed]

J. B. Bentley, J. A. Davis, M. A. Bandres, and J. C. Gutiérrez-Vega, “Generation of helical Ince-Gaussian beams with a liquid crystal display,” Opt. Lett. 31, 649-651 (2006).
[Crossref] [PubMed]

J. A. Davis, J. Nicolas, and A. Marquez, “Phasor analysis of eigenvectors generated in liquid crystal displays,” Appl. Opt. 41, 4579-4584 (2002).
[Crossref] [PubMed]

J. A. Davis, D. A. Smith, D. E. McNamara, D. M. Cottrell, and J. Campos, “Fractional derivatives--Analysis and experimental implementation,” Appl. Opt. 40, 5943-5948 (2001).
[Crossref]

J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. 38, 5004-5013 (1999).
[Crossref]

J. A. Davis, D. B. Allison, K. G. D'Nelly, M. L. Wilson, and I. Moreno, “Ambiguities in measuring the physical parameters for twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 38, 705-709 (1999).
[Crossref]

I. Moreno, J. A. Davis, K. G. D'Nelly, and D. B. Allison, “Transmission and phase measurement for polarization eigenvectors in twisted nematic liquid crystal displays,” Opt. Eng. 37, 3048-3052 (1998).
[Crossref]

J. A. Davis, I. Moreno, and P. Tsai, “Polarization eigenstates for twisted-nematic liquid crystal displays,” Appl. Opt. . 37, 937-945 (1998).
[Crossref]

J. A. Davis, E. A. Merrill, D. M. Cottrell, and R. M. Bunch, “Effects of sampling, nonlinear recording, and binarization on the output of the joint Fourier transform correlator,” Opt. Eng. 29, 1094-1100 (1990).
[Crossref]

D'Nelly, K. G.

J. A. Davis, D. B. Allison, K. G. D'Nelly, M. L. Wilson, and I. Moreno, “Ambiguities in measuring the physical parameters for twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 38, 705-709 (1999).
[Crossref]

I. Moreno, J. A. Davis, K. G. D'Nelly, and D. B. Allison, “Transmission and phase measurement for polarization eigenvectors in twisted nematic liquid crystal displays,” Opt. Eng. 37, 3048-3052 (1998).
[Crossref]

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978), Chap. 7.4.

Gutiérrez-Vega, J. C.

Iemmi, C.

López-Coronado, O.

Lopez-Olazagasti, E.

Lu, K.

C. Soutar and K. Lu, “Determination of the physical properties of an arbitrary twisted-nematic liquid crystal cell,” Opt. Eng. 33, 2704-2712 (1994).
[Crossref]

Marquez, A.

McArdle, N.

N. McArdle and M. R. Taghizadeh, “Real-time reconfigurable interconnections for parallel optical processing,” Opt. Rev. 2, 189-193 (1995).
[Crossref]

McNamara, D. E.

Merrill, E. A.

J. A. Davis, E. A. Merrill, D. M. Cottrell, and R. M. Bunch, “Effects of sampling, nonlinear recording, and binarization on the output of the joint Fourier transform correlator,” Opt. Eng. 29, 1094-1100 (1990).
[Crossref]

Moreno, I.

J. A. Davis, D. B. Allison, K. G. D'Nelly, M. L. Wilson, and I. Moreno, “Ambiguities in measuring the physical parameters for twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 38, 705-709 (1999).
[Crossref]

J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. 38, 5004-5013 (1999).
[Crossref]

I. Moreno, J. A. Davis, K. G. D'Nelly, and D. B. Allison, “Transmission and phase measurement for polarization eigenvectors in twisted nematic liquid crystal displays,” Opt. Eng. 37, 3048-3052 (1998).
[Crossref]

J. A. Davis, I. Moreno, and P. Tsai, “Polarization eigenstates for twisted-nematic liquid crystal displays,” Appl. Opt. . 37, 937-945 (1998).
[Crossref]

Nicolas, J.

Rosenblatt, C.

M. Zhu, G. Carbone, and C. Rosenblatt, “Electrically switchable, polarization-independent diffraction grating based on negative dielectric anisotropy liquid crystal,” Appl. Phys. Lett. 88, 253502 (2006).
[Crossref]

Sirohi, R. S.

Smith, D. A.

Soutar, C.

C. Soutar and K. Lu, “Determination of the physical properties of an arbitrary twisted-nematic liquid crystal cell,” Opt. Eng. 33, 2704-2712 (1994).
[Crossref]

Taghizadeh, M. R.

N. McArdle and M. R. Taghizadeh, “Real-time reconfigurable interconnections for parallel optical processing,” Opt. Rev. 2, 189-193 (1995).
[Crossref]

Tsai, P.

Tuvey, C. S.

Wilson, M. L.

J. A. Davis, D. B. Allison, K. G. D'Nelly, M. L. Wilson, and I. Moreno, “Ambiguities in measuring the physical parameters for twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 38, 705-709 (1999).
[Crossref]

Yzuel, M. J.

Zhu, M.

M. Zhu, G. Carbone, and C. Rosenblatt, “Electrically switchable, polarization-independent diffraction grating based on negative dielectric anisotropy liquid crystal,” Appl. Phys. Lett. 88, 253502 (2006).
[Crossref]

Appl. Opt. (5)

Appl. Phys. Lett. (1)

M. Zhu, G. Carbone, and C. Rosenblatt, “Electrically switchable, polarization-independent diffraction grating based on negative dielectric anisotropy liquid crystal,” Appl. Phys. Lett. 88, 253502 (2006).
[Crossref]

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

Opt. Eng. (4)

J. A. Davis, E. A. Merrill, D. M. Cottrell, and R. M. Bunch, “Effects of sampling, nonlinear recording, and binarization on the output of the joint Fourier transform correlator,” Opt. Eng. 29, 1094-1100 (1990).
[Crossref]

C. Soutar and K. Lu, “Determination of the physical properties of an arbitrary twisted-nematic liquid crystal cell,” Opt. Eng. 33, 2704-2712 (1994).
[Crossref]

J. A. Davis, D. B. Allison, K. G. D'Nelly, M. L. Wilson, and I. Moreno, “Ambiguities in measuring the physical parameters for twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 38, 705-709 (1999).
[Crossref]

I. Moreno, J. A. Davis, K. G. D'Nelly, and D. B. Allison, “Transmission and phase measurement for polarization eigenvectors in twisted nematic liquid crystal displays,” Opt. Eng. 37, 3048-3052 (1998).
[Crossref]

Opt. Lett. (2)

Opt. Rev. (1)

N. McArdle and M. R. Taghizadeh, “Real-time reconfigurable interconnections for parallel optical processing,” Opt. Rev. 2, 189-193 (1995).
[Crossref]

Optik (Jena) (1)

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

Other (1)

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978), Chap. 7.4.

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Figures (4)

Fig. 1
Fig. 1

Patterns corresponding to (a) one-dimensional grating (b) two-dimensional grating, and (c) the unit cell used for the analysis of the two-dimensional grating. The dark areas are encoded with one phase value, while the white areas are encoded with a different phase value.

Fig. 2
Fig. 2

Diffraction patterns for one-dimensional gratings with (a)–(c)  N = 4 pixels (d)–(f)  N = 2 pixels, and (g)–(i)  N = 1 pixel. Phase values are (a), (d), (g)  ϕ = 0 rad; (b), (e), (h)  ϕ = π / 2 rad; and (c), (f), (i)  ϕ = π rad.

Fig. 3
Fig. 3

Diffraction patterns for two-dimensional gratings with (a)–(c)  N = 4 pixels, (d)–(f)  N = 2 pixels, and (g)–(i)  N = 1 pixel. Phase values are (a), (d), (g)  ϕ = 0 rad; (b), (e), (h)  ϕ = π / 2 rad; and (c), (f), (i)  ϕ = π rad.

Fig. 4
Fig. 4

Experimental intensities and theory for the principal and first-order diffracted beams as a function of phase for two- dimensional binary-phase gratings with (a)  N = 4 pixels/square and (b) Nyquist limit with N = 1 pixel/square.

Equations (9)

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t ( x , y ) = 1 4 N 2 [ m = n = k = 0 N 1 = 0 N 1 δ ( x 2 m N Δ k Δ ) δ ( y 2 n N Δ Δ ) e i ϕ / 2 + m = n = k = 0 N 1 = 0 N 1 δ ( x 2 m N Δ k Δ N Δ ) δ ( y 2 n N Δ Δ N Δ ) e i ϕ / 2 + m = n = k = 0 N 1 = 0 N 1 δ ( x 2 m N Δ k Δ N Δ ) δ ( y 2 n N Δ Δ ) e i ϕ / 2 + m = n = k = 0 N 1 = 0 N 1 δ ( x 2 m N Δ k Δ ) δ ( y 2 n N Δ Δ N Δ ) e i ϕ / 2 ] .
T ( p , q ) = 1 4 N 2 [ m = n = k = 0 N 1 = 0 N 1 δ ( p 2 π m 2 N Δ ) δ ( q 2 π n 2 N Δ ) e i p k Δ e i q Δ e i ϕ / 2 + m = n = k = 0 N 1 = 0 N 1 δ ( p 2 π m 2 N Δ ) δ ( q 2 π n 2 N Δ ) e i p k Δ e i p N Δ e i q Δ e i q N Δ e i ϕ / 2 + m = n = k = 0 N 1 = 0 N 1 δ ( p 2 π m 2 N Δ ) δ ( q 2 π n 2 N Δ ) e i p k Δ e i p N Δ e i q Δ e i ϕ / 2 + m = n = k = 0 N 1 = 0 N 1 δ ( p 2 π m 2 N Δ ) δ ( q 2 π n 2 N Δ ) e i p k Δ e i q Δ e i q N Δ e i ϕ / 2 ] .
T ( p , q ) = 1 4 N 2 m = n = k = 0 N 1 = 0 N 1 δ ( p 2 π m 2 N Δ ) δ ( q 2 π n 2 N Δ ) e i p k Δ e i q Δ × ( e i ϕ / 2 + e i p N Δ e i q N Δ e i ϕ / 2 + e i p N Δ e i ϕ / 2 + e i q N Δ e i ϕ / 2 ) .
T ( p , q ) = 1 4 N 2 m = n = δ ( p 2 π m 2 N Δ ) δ ( q 2 π n 2 N Δ ) ( 1 e i p N Δ 1 e i p Δ ) ( 1 e i q N Δ 1 e i q Δ ) × ( e i ϕ / 2 + e i p N Δ e i q N Δ e i ϕ / 2 + e i p N Δ e i ϕ / 2 + e i q N Δ e i ϕ / 2 ) .
T ( p , q ) = 1 4 N 2 m = n = δ ( p 2 π m 2 N Δ ) δ ( q 2 π n 2 N Δ ) ( 1 e i m π e i n π + e i ( m + n ) π 1 e i m π / N e i n π / N + e i ( m + n ) π / N ) × ( e i ϕ / 2 + e i m π e i n π e i ϕ / 2 + e i m π e i ϕ / 2 + e i n π e i ϕ / 2 ) .
T ( p , q ) m / N , n / N even = m = n = δ ( p 2 π m 2 N Δ ) δ ( q - 2 π n 2 N Δ ) cos ( ϕ / 2 ) .
I m / N , n / N even = cos 2 ( ϕ / 2 )
T ( p , q ) m , n odd = i N 2 m = n = δ ( p 2 π m 2 N Δ ) δ ( q 2 π n 2 N Δ ) e i ( m + n ) π 2 N sin ( ϕ / 2 ) sin ( n π 2 N ) sin ( m π 2 N ) .
I m , n odd = sin 2 ( ϕ / 2 ) N 4 sin 2 ( n π 2 N ) sin 2 ( m π 2 N ) .

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