Abstract

We discuss two types of optical processing using vortex-producing angular phase plates. In the most common spatial-filtering operation, an input object is Fourier transformed (either by Fraunhofer diffraction or with a lens system). The Fourier transform is then multiplied by an angular phase pattern, and the product is again Fourier transformed. The output is a space-invariant, edge-enhanced version of the input object. Alternatively we can directly image the object using a lens multiplied by the angular phase. The space-variant image is severely distorted along the optical axis of the system. We encode the phase plates onto a liquid-crystal display and present experimental results on both systems.

© 2004 Optical Society of America

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  1. M. Mansuripur, E. M. Wright, “Linear optical vortices,” Opt. Photon News 9, 40–43 (1999).
    [CrossRef]
  2. J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
    [CrossRef]
  3. P. Coullet, L. Gil, F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
    [CrossRef]
  4. M. Harris, C. A. Hill, J. M. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
    [CrossRef]
  5. V. Yu, Bazhenov, M. V. Yasnetsov, M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 428–429 (1990).
  6. N. R. Heckenberg, R. McDuff, C. P. Smith, A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
    [CrossRef] [PubMed]
  7. Z. Jaroszewicz, A. Kolodziejczyk, “Zone plates performing generalized Hankel transforms and their metrological applications,” Opt. Commun. 102, 391–396 (1993).
    [CrossRef]
  8. J. A. Davis, E. Carcole, D. M. Cottrell, “Intensity and phase measurements of nondiffracting beams generated with a magneto-optic spatial light modulator,” Appl. Opt. 35, 593–598 (1996).
    [CrossRef] [PubMed]
  9. D. Ganic, X. Gan, M. Gu, M. Hain, S. Somalingam, S. Stankivic, T. Tschudi, “Generation of doughnut laser beams by use of a liquid-crystal cell with a conversion efficiency near 100%,” Opt. Lett. 27, 1351–1353 (2002).
    [CrossRef]
  10. A. G. Peele, P. J. McMahon, D. Paterson, C. Q. Tran, A. P. Mancuso, K. A. Nugent, J. P. Hayes, E. Harvey, B. Lai, I. McNulty, “Observation of an x-ray vortex,” Opt. Lett. 27, 1752–1754 (2002).
    [CrossRef]
  11. G. Biener, A. Niv, V. Kleiner, E. Hasman, “Formation of helical beams by use of Pancharatnam-Berry phase optical elements,” Opt. Lett. 27, 1875–1877 (2002).
    [CrossRef]
  12. M. Reicherter, T. Haist, E. U. Wageman, H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. 24, 608–610 (1999).
    [CrossRef]
  13. H. He, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
    [CrossRef]
  14. K. T. Gahagan, G. A. Swartzlander, “Optical vortex trapping of particles,” Opt. Lett. 21, 827–829 (1996).
    [CrossRef] [PubMed]
  15. J. A. Davis, L. L. Haavig, D. M. Cottrell, “Bessel function output from an optical correlator,” Appl. Opt. 36, 2376–2379 (1997).
    [CrossRef] [PubMed]
  16. J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, I. Moreno, “Bessel function output from an optical correlator with a phase-only encoded inverse filter,” Appl. Opt. 38, 6709–6713 (1999).
    [CrossRef]
  17. J. A. Davis, D. E. McNamara, D. M. Cottrell, “Image processing with the radial Hilbert transform: theory and experiments,” Opt. Lett. 25, 99–101 (2000).
    [CrossRef]
  18. G. A. Swartzlander, “Peering into darkness with a vortex spatial filter,” Opt. Lett. 26, 497–499 (2001).
    [CrossRef]
  19. J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38, 1051–1057 (1999).
    [CrossRef]
  20. J. A. Davis, J. Adachi, D. M. Cottrell, “Diffraction efficiency of nonsynchronously sampled diffraction gratings,” Opt. Eng. 41, 2983–2986 (2002).
    [CrossRef]
  21. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5.
  22. T. R. Walsh, J. E. Cravatt, B. A. Kast, M. K. Giles, “A time-sequenced rotation and scale-invariant optical correlator for multiple target recognition,” in Optical Information Processing Systems and Architectures, B. Javidi, ed., Proc. SPIE1151, 203–219 (1990).
    [CrossRef]
  23. W. T. Rhodes, “Simple procedure for the analysis of coherent imaging systems,” Opt. Lett. 19, 1559–1561 (1994).
    [CrossRef] [PubMed]
  24. D. Palacios, D. Rozas, G. A. Swartzlander, “Observed scattering into a dark optical vortex core,” Phys. Rev. Lett. 88, 103902 (2002).
    [CrossRef] [PubMed]
  25. J. A. Davis, E. Carcole, D. M. Cottrell, “Nondiffracting interference patterns generated with programmable spatial light modulators,” Appl. Opt. 35, 599–602 (1996).
    [CrossRef] [PubMed]

2002 (5)

2001 (1)

2000 (1)

1999 (4)

J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, I. Moreno, “Bessel function output from an optical correlator with a phase-only encoded inverse filter,” Appl. Opt. 38, 6709–6713 (1999).
[CrossRef]

M. Reicherter, T. Haist, E. U. Wageman, H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. 24, 608–610 (1999).
[CrossRef]

M. Mansuripur, E. M. Wright, “Linear optical vortices,” Opt. Photon News 9, 40–43 (1999).
[CrossRef]

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38, 1051–1057 (1999).
[CrossRef]

1997 (1)

1996 (3)

1995 (1)

H. He, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

1994 (2)

M. Harris, C. A. Hill, J. M. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

W. T. Rhodes, “Simple procedure for the analysis of coherent imaging systems,” Opt. Lett. 19, 1559–1561 (1994).
[CrossRef] [PubMed]

1993 (1)

Z. Jaroszewicz, A. Kolodziejczyk, “Zone plates performing generalized Hankel transforms and their metrological applications,” Opt. Commun. 102, 391–396 (1993).
[CrossRef]

1992 (1)

1990 (1)

V. Yu, Bazhenov, M. V. Yasnetsov, M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 428–429 (1990).

1989 (1)

P. Coullet, L. Gil, F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

1974 (1)

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Adachi, J.

J. A. Davis, J. Adachi, D. M. Cottrell, “Diffraction efficiency of nonsynchronously sampled diffraction gratings,” Opt. Eng. 41, 2983–2986 (2002).
[CrossRef]

Amako, J.

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38, 1051–1057 (1999).
[CrossRef]

Bazhenov,

V. Yu, Bazhenov, M. V. Yasnetsov, M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 428–429 (1990).

Berry, M. V.

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Biener, G.

Campos, J.

Carcole, E.

Cottrell, D. M.

Coullet, P.

P. Coullet, L. Gil, F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

Cravatt, J. E.

T. R. Walsh, J. E. Cravatt, B. A. Kast, M. K. Giles, “A time-sequenced rotation and scale-invariant optical correlator for multiple target recognition,” in Optical Information Processing Systems and Architectures, B. Javidi, ed., Proc. SPIE1151, 203–219 (1990).
[CrossRef]

Davis, J. A.

Gahagan, K. T.

Gan, X.

Ganic, D.

Gil, L.

P. Coullet, L. Gil, F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

Giles, M. K.

T. R. Walsh, J. E. Cravatt, B. A. Kast, M. K. Giles, “A time-sequenced rotation and scale-invariant optical correlator for multiple target recognition,” in Optical Information Processing Systems and Architectures, B. Javidi, ed., Proc. SPIE1151, 203–219 (1990).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5.

Gu, M.

Haavig, L. L.

Hain, M.

Haist, T.

Harris, M.

M. Harris, C. A. Hill, J. M. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

Harvey, E.

Hasman, E.

Hayes, J. P.

He, H.

H. He, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

Heckenberg, N. R.

H. He, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

N. R. Heckenberg, R. McDuff, C. P. Smith, A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
[CrossRef] [PubMed]

Hill, C. A.

M. Harris, C. A. Hill, J. M. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

Jaroszewicz, Z.

Z. Jaroszewicz, A. Kolodziejczyk, “Zone plates performing generalized Hankel transforms and their metrological applications,” Opt. Commun. 102, 391–396 (1993).
[CrossRef]

Kast, B. A.

T. R. Walsh, J. E. Cravatt, B. A. Kast, M. K. Giles, “A time-sequenced rotation and scale-invariant optical correlator for multiple target recognition,” in Optical Information Processing Systems and Architectures, B. Javidi, ed., Proc. SPIE1151, 203–219 (1990).
[CrossRef]

Kleiner, V.

Kolodziejczyk, A.

Z. Jaroszewicz, A. Kolodziejczyk, “Zone plates performing generalized Hankel transforms and their metrological applications,” Opt. Commun. 102, 391–396 (1993).
[CrossRef]

Lai, B.

Mancuso, A. P.

Mansuripur, M.

M. Mansuripur, E. M. Wright, “Linear optical vortices,” Opt. Photon News 9, 40–43 (1999).
[CrossRef]

McDuff, R.

McMahon, P. J.

McNamara, D. E.

McNulty, I.

Moreno, I.

Niv, A.

Nugent, K. A.

Nye, J. F.

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Palacios, D.

D. Palacios, D. Rozas, G. A. Swartzlander, “Observed scattering into a dark optical vortex core,” Phys. Rev. Lett. 88, 103902 (2002).
[CrossRef] [PubMed]

Paterson, D.

Peele, A. G.

Reicherter, M.

Rhodes, W. T.

Rocca, F.

P. Coullet, L. Gil, F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

Rozas, D.

D. Palacios, D. Rozas, G. A. Swartzlander, “Observed scattering into a dark optical vortex core,” Phys. Rev. Lett. 88, 103902 (2002).
[CrossRef] [PubMed]

Rubinsztein-Dunlop, H.

H. He, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

Smith, C. P.

Somalingam, S.

Sonehara, T.

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38, 1051–1057 (1999).
[CrossRef]

Soskin, M. S.

V. Yu, Bazhenov, M. V. Yasnetsov, M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 428–429 (1990).

Stankivic, S.

Swartzlander, G. A.

Tiziani, H. J.

Tran, C. Q.

Tsai, P.

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38, 1051–1057 (1999).
[CrossRef]

Tschudi, T.

Vaughan, J. M.

M. Harris, C. A. Hill, J. M. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

Wageman, E. U.

Walsh, T. R.

T. R. Walsh, J. E. Cravatt, B. A. Kast, M. K. Giles, “A time-sequenced rotation and scale-invariant optical correlator for multiple target recognition,” in Optical Information Processing Systems and Architectures, B. Javidi, ed., Proc. SPIE1151, 203–219 (1990).
[CrossRef]

White, A. G.

Wright, E. M.

M. Mansuripur, E. M. Wright, “Linear optical vortices,” Opt. Photon News 9, 40–43 (1999).
[CrossRef]

Yasnetsov, M. V.

V. Yu, Bazhenov, M. V. Yasnetsov, M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 428–429 (1990).

Yu, V.

V. Yu, Bazhenov, M. V. Yasnetsov, M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 428–429 (1990).

Yzuel, M. J.

Appl. Opt. (4)

J. Mod. Opt. (1)

H. He, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

JETP Lett. (1)

V. Yu, Bazhenov, M. V. Yasnetsov, M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 428–429 (1990).

Opt. Commun. (3)

P. Coullet, L. Gil, F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

M. Harris, C. A. Hill, J. M. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

Z. Jaroszewicz, A. Kolodziejczyk, “Zone plates performing generalized Hankel transforms and their metrological applications,” Opt. Commun. 102, 391–396 (1993).
[CrossRef]

Opt. Eng. (2)

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38, 1051–1057 (1999).
[CrossRef]

J. A. Davis, J. Adachi, D. M. Cottrell, “Diffraction efficiency of nonsynchronously sampled diffraction gratings,” Opt. Eng. 41, 2983–2986 (2002).
[CrossRef]

Opt. Lett. (9)

W. T. Rhodes, “Simple procedure for the analysis of coherent imaging systems,” Opt. Lett. 19, 1559–1561 (1994).
[CrossRef] [PubMed]

D. Ganic, X. Gan, M. Gu, M. Hain, S. Somalingam, S. Stankivic, T. Tschudi, “Generation of doughnut laser beams by use of a liquid-crystal cell with a conversion efficiency near 100%,” Opt. Lett. 27, 1351–1353 (2002).
[CrossRef]

A. G. Peele, P. J. McMahon, D. Paterson, C. Q. Tran, A. P. Mancuso, K. A. Nugent, J. P. Hayes, E. Harvey, B. Lai, I. McNulty, “Observation of an x-ray vortex,” Opt. Lett. 27, 1752–1754 (2002).
[CrossRef]

G. Biener, A. Niv, V. Kleiner, E. Hasman, “Formation of helical beams by use of Pancharatnam-Berry phase optical elements,” Opt. Lett. 27, 1875–1877 (2002).
[CrossRef]

M. Reicherter, T. Haist, E. U. Wageman, H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. 24, 608–610 (1999).
[CrossRef]

N. R. Heckenberg, R. McDuff, C. P. Smith, A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
[CrossRef] [PubMed]

K. T. Gahagan, G. A. Swartzlander, “Optical vortex trapping of particles,” Opt. Lett. 21, 827–829 (1996).
[CrossRef] [PubMed]

J. A. Davis, D. E. McNamara, D. M. Cottrell, “Image processing with the radial Hilbert transform: theory and experiments,” Opt. Lett. 25, 99–101 (2000).
[CrossRef]

G. A. Swartzlander, “Peering into darkness with a vortex spatial filter,” Opt. Lett. 26, 497–499 (2001).
[CrossRef]

Opt. Photon News (1)

M. Mansuripur, E. M. Wright, “Linear optical vortices,” Opt. Photon News 9, 40–43 (1999).
[CrossRef]

Phys. Rev. Lett. (1)

D. Palacios, D. Rozas, G. A. Swartzlander, “Observed scattering into a dark optical vortex core,” Phys. Rev. Lett. 88, 103902 (2002).
[CrossRef] [PubMed]

Proc. R. Soc. London Ser. A (1)

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Other (2)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5.

T. R. Walsh, J. E. Cravatt, B. A. Kast, M. K. Giles, “A time-sequenced rotation and scale-invariant optical correlator for multiple target recognition,” in Optical Information Processing Systems and Architectures, B. Javidi, ed., Proc. SPIE1151, 203–219 (1990).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Lens patterns for (a) normal focusing lens, (b) lens with a clockwise angular phase where m = 2, (c) lens with a clockwise angular phase where m = 6, (d) lens with a counterclockwise angular phase where m = -6. Here white corresponds to a zero phase and black corresponds to a phase of 2π rad.

Fig. 2
Fig. 2

Optical image processing geometries. The input plane is given by P 1, the Fourier-transform plane is given by P 2, and the output or imaging plane is given by P 3. (a) 4-f system, (b) system with a more compact second Fourier-transform system, (c) compact scaling system for the first Fourier transform and imaging. Here the spatial filter and focusing lens are both written in plane P 2.

Fig. 3
Fig. 3

Gray-level images of the real and imaginary parts of the Fourier transform of the angular mask functions V m (ρ, ϕ) for (a) and (b) m = 0, (c) and (d) m = 0.5, (e) and (f) m = 1, (g) and (h) m = 1.5, (i) and (j) m = 2, (k) and (l) m = 3. The maximum positive value is represented by black, and the maximum negative value is represented by white.

Fig. 4
Fig. 4

Image processing results where the input consists of two circular apertures and the Fourier-transform plane contains angular mask functions V m (ρ, ϕ) with (a) m = 0, (b) m = 0.5, (c) m = 1, (d) m = 1.5, (e) m = 2, (f) m = 4, (g) m = 5, (h) m = 6. Here the maximum intensity is shown in white.

Fig. 5
Fig. 5

Imaging system with vortex mask. The pupil function is located in the lens plane.

Fig. 6
Fig. 6

Output images for the system of Fig. 5 where the input is a 0.75-mm slit and the spatial light modulator lens is encoded with lens function Z m (ρ, ϕ, f) where (a) m = 0, (b) m = 2, (c) m = 6, (d) m = -6. Here the maximum intensity is shown in white.

Fig. 7
Fig. 7

Output images for the system of Fig. 5 where the input is a 4-mm slit and the spatial light modulator lens is encoded with lens function Z m (ρ, ϕ, f) where (a) m = 0, (b) m = 0.5, (c) m = 1, (d) m = 1.5, (e) m = 2, (f) m = 6, (g) sum of m = +1 and m = -1, (h) sum of m = +3 and m = -3. Here the maximum intensity is shown in white.

Equations (6)

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

Vmρ, ϕ=expimϕ.
Zmρ, ϕ, f=Vmρ, ϕZ0ρ, f=expimϕexp-iπρ2λf,
gr, θ=gr, θ*vmr, θ.
gr, θ=gr, θcosmπ2+gr, θ*v1r, θsinmπ2.
gimx, y=gobjξ, ηhsvx, y, ξ, ηdξdη.
hsvx, y, ξ, η=expj k2R1ξ2+η2×Psysξλp±xλq, ηλp±yλq×expj k2R2x2+y2.

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