Abstract

We report a new self-interferometric technique for visualizing phase patterns that are encoded onto a phase-only liquid-crystal display (LCD). In our approach, the LCD generates both the desired object beam as well as the reference beam. Normally the phase patterns are encoded with a phase depth of 2π radians, and all of the incident energy is diffracted into the first-order beam. However, by reducing this phase depth, we can generate an additional zero-order diffracted beam, which acts as the reference beam. We work at distances such that these two patterns spatially interfere, producing an interference pattern that displays the encoded phase pattern. This approach was used recently to display the phase vortices of helical Ince–Gaussian beams. Here we show additional experimental results and analyze the process.

© 2006 Optical Society of America

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References

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  1. D. Malacara, Optical Shop Testing (Wiley, 1992).
  2. A. Bergeron, J. Gauvin, F. Gagnon, D. Gingras, H. H. Arsenault, and M. Douchet, "Phase calibration and applications of a liquid-crystal spatial light modulator," Appl. Opt. 34, 5133-5139 (1995).
    [CrossRef] [PubMed]
  3. V. Laude, S. Maze, P. Chavel, and Ph. Réfrégier, "Amplitude and phase coding measurements of a liquid crystal television," Opt. Commun. 103, 28-33 (1993).
    [CrossRef]
  4. 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]
  5. J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, "Bessel function output from an optical correlator with a phase-only encoded inverse filter," Appl. Opt. 38, 6709-6713 (1999).
    [CrossRef]
  6. J. A. Davis, D. E. McNamara, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, "Encoding complex diffractive optical elements onto a phase-only liquid-crystal spatial light modulator," Opt. Eng. 40, 327-329 (2001).
    [CrossRef]
  7. A. Marquez, C. Iemmi, J. C. Escalera, J. Campos, S. Ledesma, J. A. Davis, and M. J. Yzuel, "Amplitude apodizers encoded onto Fresnel lenses implemented on a phase-only spatial light modulator," Appl. Opt. 40, 2316-2322 (2001).
    [CrossRef]
  8. 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]
  9. 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]
  10. J. A. Davis, I. Moreno, and P. Tsai, "Polarization eigenstates for twisted-nematic liquid crystal displays," Appl. Opt. 37, 937-945 (1998).
    [CrossRef]
  11. I. Moreno, J. L. Martínez, and J. A. Davis, "Two-dimensional polarization rotator using a twisted-nematic liquid crystal display," Appl. Opt. (to be published).
    [PubMed]
  12. K. Crabtree, J. A. Davis, and I. Moreno, "Optical processing with vortex producing lenses," Appl. Opt. 43, 1360-1367 (2004).
    [CrossRef] [PubMed]

2006 (1)

2004 (1)

2001 (3)

1999 (2)

1998 (1)

1995 (1)

1993 (1)

V. Laude, S. Maze, P. Chavel, and Ph. Réfrégier, "Amplitude and phase coding measurements of a liquid crystal television," Opt. Commun. 103, 28-33 (1993).
[CrossRef]

1992 (1)

D. Malacara, Optical Shop Testing (Wiley, 1992).

Arsenault, H. H.

Bandres, M. A.

Bentley, J. B.

Bergeron, A.

Campos, J.

Chavel, P.

V. Laude, S. Maze, P. Chavel, and Ph. Réfrégier, "Amplitude and phase coding measurements of a liquid crystal television," Opt. Commun. 103, 28-33 (1993).
[CrossRef]

Cottrell, D. M.

Crabtree, K.

Davis, J. A.

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]

K. Crabtree, J. A. Davis, and I. Moreno, "Optical processing with vortex producing lenses," Appl. Opt. 43, 1360-1367 (2004).
[CrossRef] [PubMed]

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

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. E. McNamara, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, "Encoding complex diffractive optical elements onto a phase-only liquid-crystal spatial light modulator," Opt. Eng. 40, 327-329 (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. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, "Bessel function output from an optical correlator with a phase-only encoded inverse filter," Appl. Opt. 38, 6709-6713 (1999).
[CrossRef]

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

I. Moreno, J. L. Martínez, and J. A. Davis, "Two-dimensional polarization rotator using a twisted-nematic liquid crystal display," Appl. Opt. (to be published).
[PubMed]

Douchet, M.

Escalera, J. C.

Gagnon, F.

Gauvin, J.

Gingras, D.

Gutiérrez-Vega, J. C.

Iemmi, C.

Laude, V.

V. Laude, S. Maze, P. Chavel, and Ph. Réfrégier, "Amplitude and phase coding measurements of a liquid crystal television," Opt. Commun. 103, 28-33 (1993).
[CrossRef]

Ledesma, S.

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley, 1992).

Marquez, A.

Martínez, J. L.

I. Moreno, J. L. Martínez, and J. A. Davis, "Two-dimensional polarization rotator using a twisted-nematic liquid crystal display," Appl. Opt. (to be published).
[PubMed]

Maze, S.

V. Laude, S. Maze, P. Chavel, and Ph. Réfrégier, "Amplitude and phase coding measurements of a liquid crystal television," Opt. Commun. 103, 28-33 (1993).
[CrossRef]

McNamara, D. E.

J. A. Davis, D. E. McNamara, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, "Encoding complex diffractive optical elements onto a phase-only liquid-crystal spatial light modulator," Opt. Eng. 40, 327-329 (2001).
[CrossRef]

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]

Moreno, I.

Réfrégier, Ph.

V. Laude, S. Maze, P. Chavel, and Ph. Réfrégier, "Amplitude and phase coding measurements of a liquid crystal television," Opt. Commun. 103, 28-33 (1993).
[CrossRef]

Smith, D. A.

Tsai, P.

Yzuel, M. J.

Appl. Opt. (8)

Opt. Commun. (1)

V. Laude, S. Maze, P. Chavel, and Ph. Réfrégier, "Amplitude and phase coding measurements of a liquid crystal television," Opt. Commun. 103, 28-33 (1993).
[CrossRef]

Opt. Eng. (1)

J. A. Davis, D. E. McNamara, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, "Encoding complex diffractive optical elements onto a phase-only liquid-crystal spatial light modulator," Opt. Eng. 40, 327-329 (2001).
[CrossRef]

Opt. Lett. (1)

Other (1)

D. Malacara, Optical Shop Testing (Wiley, 1992).

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Figures (5)

Fig. 1
Fig. 1

Intensity of the main diffracted orders generated by a blazed diffraction grating as a function of the modulation depth parameter M.

Fig. 2
Fig. 2

Interference pattern generated for different values of the modulation depth parameter M:(a) M = 0.5 , (b) M = 0.25 , and (c) M = 0 .

Fig. 3
Fig. 3

Interference pattern generated for the value M = 0.5 of the modulation depth parameter, for different values of the period ( 1 / A ) of the linear phase grating:(a) 32 pixels, (b) 16 pixels, and (c) 8 pixels.

Fig. 4
Fig. 4

Interference patterns generated when a constant phase is added to the upper half-plane relative to the lower half-plane. The modulation depth parameter is set at M = 0.5 . The upper half-plane is phase shifted by values of (a) 0 rad , (b) π / 2 rad , and (c) π rad .

Fig. 5
Fig. 5

Interference patterns when spiral phase patterns are encoded onto the LCD where the phase varies azimuthally as exp ( i ϕ ) for (a) = 1 , (b) = 2 , and (c) = 3 . The modulation depth parameter is equal to M = 0.5 in all cases.

Equations (7)

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T ( x ) = exp [ i M { Φ ( x ) + 2 π A x } ] .
T ( x ) = - T n exp [ i n { Φ ( x ) + 2 π A x } ] .
T n = sin [ π ( n - M ) ] π ( n - M ) .
T ( x ) = T 0 + T 1 exp [ - i { Φ ( x ) + 2 π x A } ] .
I ( x ) = T 0 2 + T 1 2 + 2 T 0 T 1 cos [ Φ ( x ) + 2 π x A ] .
I ( x ) = [ T 0 2 + T 1 2 ] [ 1 + C cos { Φ ( x ) + 2 π x A } ] .
C = 2 T 0 T 1 [ T 0 2 + T 1 2 ] .

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