Abstract

We report a new optical vortex phase-shifting method for digital holography, in which an optical vortex mode is taken as the reference beam for holographic recording, and the required phase shifts are directly generated by rotating the vortex mode. In digital reconstruction, the complex amplitude of the object wave can be retrieved by use of the conventional phase shifting algorithm on condition that the digital illumination beam is replaced by an vortex beam with the same topological charge as the reference used. Both the theoretical analysis and experimental results demonstrate the feasibility of this approach.

© 2004 Optical Society of America

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  1. U. Schnars and W. Juptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994).
    [CrossRef] [PubMed]
  2. J. Pomarico, U. Schnars, H.-J. Hartmann, and W. Juptner, “Digital recording and numerical reconstruction of holograms: a new method for displaying light in flight,” Appl. Opt. 34, 8095–8099 (1995).
    [CrossRef] [PubMed]
  3. C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39, 79–85 (2000).
    [CrossRef]
  4. M. K. Kim, “Tomographic three-dimensional imaging of a biological specimen using wavelength scanning digital interference holography,” Opt. Express 7, 305–310 (2000). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-9-305
    [CrossRef] [PubMed]
  5. S. Grilli, P. Ferraro, M. Paturzo, D. Alfieri, P. D. Natale, M. D. Angelis, S. D. Nicola, A. Finizio, and G. Pierattini, “In-situ visualization, monitoring and analysis of electric field domain reversal process in ferroelectric crystals by digital holography,” Opt. Express 12, 1832–1842 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1832
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  11. L. Z. Cai, Q. Liu, Y. R. Wang, and X. L. Yang, “Simultaneous digital correction of amplitude and phaseerrors of retrieved wave-front in phase-shifting interferometry with arbitrary phase errors,” Opt. Commun. 233, 21–26(2004).
    [CrossRef]
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    [CrossRef] [PubMed]
  13. C. S. Guo, L. Zhang, H. T. Wang, J. Liao, and Y.Y. Zhu, “Phase-Shifting Error and Its Elimination in Phase-Shifting Digital Holography,” Opt. Lett. 27, 1–3 (2002).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  16. C. S. Guo, Z. Y. Rong, H. T. Wang, Y. R. Wang, and L.Z. Cai; “Phase-Shifting with Computer-Generated Holograms Written on a Spatial Light Modulator,” Appl. Opt. 42, 6975–6979(2003).
    [CrossRef] [PubMed]
  17. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beam produced with a spiral phase plate,” Opt. Commun. 112, 321–327 (1994).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  21. J. C. Wyant and P. K. O’Neill, “Computer Generated Hologram: Null Lens Test of Aspheric Wavefronts,” Appl. Opt. 13, 2762–2765(1974).
    [CrossRef] [PubMed]
  22. K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40, 2886–2894 (2001).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]

2004 (5)

L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29, 182–185(2004).
[CrossRef]

L. Z. Cai, Q. Liu, Y. R. Wang, and X. L. Yang, “Simultaneous digital correction of amplitude and phaseerrors of retrieved wave-front in phase-shifting interferometry with arbitrary phase errors,” Opt. Commun. 233, 21–26(2004).
[CrossRef]

Z. Chen, H. Martin, E. D. Eugenieva, J. Xu, and A. Bezryadina, “Anisotropic Enhancement of Discrete Diffraction and Formation of Two-Dimensional Discrete-Soliton Trains,” Phys. Rev. Lett. 92, 143902 (2004).
[CrossRef] [PubMed]

K. Ladavac and D. G. Grier, “Microoptomechanical pumps assembled and driven by holographic optical vortex arrays,” Opt. Express 12, 1144–1149 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1144
[CrossRef] [PubMed]

S. Grilli, P. Ferraro, M. Paturzo, D. Alfieri, P. D. Natale, M. D. Angelis, S. D. Nicola, A. Finizio, and G. Pierattini, “In-situ visualization, monitoring and analysis of electric field domain reversal process in ferroelectric crystals by digital holography,” Opt. Express 12, 1832–1842 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1832
[CrossRef] [PubMed]

2003 (3)

2002 (2)

M. B. North-Morris, J. VanDelden, and J. C. Wyant, “Phaseshifting birefringent scatterplate interferometer,” Appl. Opt. 41, 668–677 (2002).
[CrossRef] [PubMed]

C. S. Guo, L. Zhang, H. T. Wang, J. Liao, and Y.Y. Zhu, “Phase-Shifting Error and Its Elimination in Phase-Shifting Digital Holography,” Opt. Lett. 27, 1–3 (2002).
[CrossRef]

2001 (2)

2000 (3)

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39, 79–85 (2000).
[CrossRef]

S. Lai, Brian King, and M. A. Neifeld, “Wave front reconstruction by means of phase-shifting digital inline holography,” Opt. Commun. 173, 155–160 (2000).
[CrossRef]

M. K. Kim, “Tomographic three-dimensional imaging of a biological specimen using wavelength scanning digital interference holography,” Opt. Express 7, 305–310 (2000). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-9-305
[CrossRef] [PubMed]

1999 (1)

L. Allen, M. P. Padgett, and M. Babiker, “The orbital angular momentum of light,” Progress in Optics 39, 291–372 (1999).
[CrossRef]

1998 (2)

T. Zhang and I. Yamaguchi, “Three-dimensional microscopy with phase-shifting digital holography,” Opt. Lett. 23, 1221–1223 (1998).
[CrossRef]

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modesby computer-generated holograms,” J. Modern Opt. 45, 1231–1237 (1998).
[CrossRef]

1997 (1)

1995 (1)

1994 (3)

U. Schnars and W. Juptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994).
[CrossRef] [PubMed]

C. C. Jin, N. K. Bao, and P. S. Chung, “Application of a novel phase-shift method using a computer-controlled polarization mechanism,” Opt. Eng. 33, 2733–2737 (1994).
[CrossRef]

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beam produced with a spiral phase plate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

1992 (1)

1985 (1)

Y. Y. Cheng and James C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl.Opt. 24, 3049–3052 (1985).
[CrossRef] [PubMed]

1974 (1)

1972 (1)

J. C. Wyant and V. P. Bennett, “Using Computer Generated Holograms to Test Aspheric Wavefronts,” Appl. Opt. 9, 2833–2839(1972).
[CrossRef]

Alfieri, D.

Allen, L.

L. Allen, M. P. Padgett, and M. Babiker, “The orbital angular momentum of light,” Progress in Optics 39, 291–372 (1999).
[CrossRef]

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modesby computer-generated holograms,” J. Modern Opt. 45, 1231–1237 (1998).
[CrossRef]

Angelis, M. D.

Arlt, J.

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modesby computer-generated holograms,” J. Modern Opt. 45, 1231–1237 (1998).
[CrossRef]

Babiker, M.

L. Allen, M. P. Padgett, and M. Babiker, “The orbital angular momentum of light,” Progress in Optics 39, 291–372 (1999).
[CrossRef]

Bao, N. K.

C. C. Jin, N. K. Bao, and P. S. Chung, “Application of a novel phase-shift method using a computer-controlled polarization mechanism,” Opt. Eng. 33, 2733–2737 (1994).
[CrossRef]

Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beam produced with a spiral phase plate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Bennett, V. P.

J. C. Wyant and V. P. Bennett, “Using Computer Generated Holograms to Test Aspheric Wavefronts,” Appl. Opt. 9, 2833–2839(1972).
[CrossRef]

Bezryadina, A.

Z. Chen, H. Martin, E. D. Eugenieva, J. Xu, and A. Bezryadina, “Anisotropic Enhancement of Discrete Diffraction and Formation of Two-Dimensional Discrete-Soliton Trains,” Phys. Rev. Lett. 92, 143902 (2004).
[CrossRef] [PubMed]

Bokor, J.

Cai, L. Z.

L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29, 182–185(2004).
[CrossRef]

L. Z. Cai, Q. Liu, Y. R. Wang, and X. L. Yang, “Simultaneous digital correction of amplitude and phaseerrors of retrieved wave-front in phase-shifting interferometry with arbitrary phase errors,” Opt. Commun. 233, 21–26(2004).
[CrossRef]

C. S. Guo, Z. Y. Rong, J. L. He, L. Z. Cai, and Y. R. Wang, “Determination of global phase shifts betweeninterferograms by use of an energy-minimum algorithm,” Appl. Opt. 42, 6514–6519(2003).
[CrossRef] [PubMed]

Cai, L.Z.

Chen, Z.

Z. Chen, H. Martin, E. D. Eugenieva, J. Xu, and A. Bezryadina, “Anisotropic Enhancement of Discrete Diffraction and Formation of Two-Dimensional Discrete-Soliton Trains,” Phys. Rev. Lett. 92, 143902 (2004).
[CrossRef] [PubMed]

Cheng, Y. Y.

Y. Y. Cheng and James C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl.Opt. 24, 3049–3052 (1985).
[CrossRef] [PubMed]

Chung, P. S.

C. C. Jin, N. K. Bao, and P. S. Chung, “Application of a novel phase-shift method using a computer-controlled polarization mechanism,” Opt. Eng. 33, 2733–2737 (1994).
[CrossRef]

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beam produced with a spiral phase plate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Dholakia, K.

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modesby computer-generated holograms,” J. Modern Opt. 45, 1231–1237 (1998).
[CrossRef]

Eugenieva, E. D.

Z. Chen, H. Martin, E. D. Eugenieva, J. Xu, and A. Bezryadina, “Anisotropic Enhancement of Discrete Diffraction and Formation of Two-Dimensional Discrete-Soliton Trains,” Phys. Rev. Lett. 92, 143902 (2004).
[CrossRef] [PubMed]

Ferraro, P.

Finizio, A.

Goldberg, K. A.

Grier, D. G.

Grilli, S.

Guo, C. S.

Hartmann, H.-J.

He, J. L.

Heckenberg, N. R.

Jin, C. C.

C. C. Jin, N. K. Bao, and P. S. Chung, “Application of a novel phase-shift method using a computer-controlled polarization mechanism,” Opt. Eng. 33, 2733–2737 (1994).
[CrossRef]

Juptner, W.

Kato, J.

Kim, M. K.

King, Brian

S. Lai, Brian King, and M. A. Neifeld, “Wave front reconstruction by means of phase-shifting digital inline holography,” Opt. Commun. 173, 155–160 (2000).
[CrossRef]

Kristensen, M.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beam produced with a spiral phase plate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Ladavac, K.

Lai, S.

S. Lai, Brian King, and M. A. Neifeld, “Wave front reconstruction by means of phase-shifting digital inline holography,” Opt. Commun. 173, 155–160 (2000).
[CrossRef]

Liao, J.

C. S. Guo, L. Zhang, H. T. Wang, J. Liao, and Y.Y. Zhu, “Phase-Shifting Error and Its Elimination in Phase-Shifting Digital Holography,” Opt. Lett. 27, 1–3 (2002).
[CrossRef]

Liu, Q.

L. Z. Cai, Q. Liu, Y. R. Wang, and X. L. Yang, “Simultaneous digital correction of amplitude and phaseerrors of retrieved wave-front in phase-shifting interferometry with arbitrary phase errors,” Opt. Commun. 233, 21–26(2004).
[CrossRef]

L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29, 182–185(2004).
[CrossRef]

Martin, H.

Z. Chen, H. Martin, E. D. Eugenieva, J. Xu, and A. Bezryadina, “Anisotropic Enhancement of Discrete Diffraction and Formation of Two-Dimensional Discrete-Soliton Trains,” Phys. Rev. Lett. 92, 143902 (2004).
[CrossRef] [PubMed]

McDuff, R.

Mizuno, J.

Musslimani, Z. H.

Natale, P. D.

Neifeld, M. A.

S. Lai, Brian King, and M. A. Neifeld, “Wave front reconstruction by means of phase-shifting digital inline holography,” Opt. Commun. 173, 155–160 (2000).
[CrossRef]

Nicola, S. D.

North-Morris, M. B.

O’Neill, P. K.

Ohta, S.

Osten, W.

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39, 79–85 (2000).
[CrossRef]

Padgett, M. J.

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modesby computer-generated holograms,” J. Modern Opt. 45, 1231–1237 (1998).
[CrossRef]

Padgett, M. P.

L. Allen, M. P. Padgett, and M. Babiker, “The orbital angular momentum of light,” Progress in Optics 39, 291–372 (1999).
[CrossRef]

Paturzo, M.

Pierattini, G.

Pomarico, J.

Rong, Z. Y.

Schnars, U.

Seebacher, S.

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39, 79–85 (2000).
[CrossRef]

Smith, C. P.

VanDelden, J.

Wagner, C.

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39, 79–85 (2000).
[CrossRef]

Wang, H. T.

C. S. Guo, Z. Y. Rong, H. T. Wang, Y. R. Wang, and L.Z. Cai; “Phase-Shifting with Computer-Generated Holograms Written on a Spatial Light Modulator,” Appl. Opt. 42, 6975–6979(2003).
[CrossRef] [PubMed]

C. S. Guo, L. Zhang, H. T. Wang, J. Liao, and Y.Y. Zhu, “Phase-Shifting Error and Its Elimination in Phase-Shifting Digital Holography,” Opt. Lett. 27, 1–3 (2002).
[CrossRef]

Wang, Y. R.

White, A. G.

Woerdman, J. P.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beam produced with a spiral phase plate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Wyant, J. C.

Wyant, James C.

Y. Y. Cheng and James C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl.Opt. 24, 3049–3052 (1985).
[CrossRef] [PubMed]

Xu, J.

Z. Chen, H. Martin, E. D. Eugenieva, J. Xu, and A. Bezryadina, “Anisotropic Enhancement of Discrete Diffraction and Formation of Two-Dimensional Discrete-Soliton Trains,” Phys. Rev. Lett. 92, 143902 (2004).
[CrossRef] [PubMed]

Yamaguchi, I.

Yang, J. K.

Yang, X. L.

L. Z. Cai, Q. Liu, Y. R. Wang, and X. L. Yang, “Simultaneous digital correction of amplitude and phaseerrors of retrieved wave-front in phase-shifting interferometry with arbitrary phase errors,” Opt. Commun. 233, 21–26(2004).
[CrossRef]

L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29, 182–185(2004).
[CrossRef]

Zhang, L.

C. S. Guo, L. Zhang, H. T. Wang, J. Liao, and Y.Y. Zhu, “Phase-Shifting Error and Its Elimination in Phase-Shifting Digital Holography,” Opt. Lett. 27, 1–3 (2002).
[CrossRef]

Zhang, T.

Zhu, Y.Y.

C. S. Guo, L. Zhang, H. T. Wang, J. Liao, and Y.Y. Zhu, “Phase-Shifting Error and Its Elimination in Phase-Shifting Digital Holography,” Opt. Lett. 27, 1–3 (2002).
[CrossRef]

Appl. Opt. (9)

U. Schnars and W. Juptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994).
[CrossRef] [PubMed]

J. Pomarico, U. Schnars, H.-J. Hartmann, and W. Juptner, “Digital recording and numerical reconstruction of holograms: a new method for displaying light in flight,” Appl. Opt. 34, 8095–8099 (1995).
[CrossRef] [PubMed]

I. Yamaguchi, J. Kato, S. Ohta, and J. Mizuno, “Image Formation in Phase-Shifting Digital Holography and Applications to Microscopy,” Appl. Opt. 40, 6177–6186 (2001).
[CrossRef]

M. B. North-Morris, J. VanDelden, and J. C. Wyant, “Phaseshifting birefringent scatterplate interferometer,” Appl. Opt. 41, 668–677 (2002).
[CrossRef] [PubMed]

C. S. Guo, Z. Y. Rong, H. T. Wang, Y. R. Wang, and L.Z. Cai; “Phase-Shifting with Computer-Generated Holograms Written on a Spatial Light Modulator,” Appl. Opt. 42, 6975–6979(2003).
[CrossRef] [PubMed]

J. C. Wyant and V. P. Bennett, “Using Computer Generated Holograms to Test Aspheric Wavefronts,” Appl. Opt. 9, 2833–2839(1972).
[CrossRef]

J. C. Wyant and P. K. O’Neill, “Computer Generated Hologram: Null Lens Test of Aspheric Wavefronts,” Appl. Opt. 13, 2762–2765(1974).
[CrossRef] [PubMed]

K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40, 2886–2894 (2001).
[CrossRef]

C. S. Guo, Z. Y. Rong, J. L. He, L. Z. Cai, and Y. R. Wang, “Determination of global phase shifts betweeninterferograms by use of an energy-minimum algorithm,” Appl. Opt. 42, 6514–6519(2003).
[CrossRef] [PubMed]

Appl.Opt. (1)

Y. Y. Cheng and James C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl.Opt. 24, 3049–3052 (1985).
[CrossRef] [PubMed]

J. Modern Opt. (1)

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modesby computer-generated holograms,” J. Modern Opt. 45, 1231–1237 (1998).
[CrossRef]

Opt. Commun. (3)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beam produced with a spiral phase plate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

S. Lai, Brian King, and M. A. Neifeld, “Wave front reconstruction by means of phase-shifting digital inline holography,” Opt. Commun. 173, 155–160 (2000).
[CrossRef]

L. Z. Cai, Q. Liu, Y. R. Wang, and X. L. Yang, “Simultaneous digital correction of amplitude and phaseerrors of retrieved wave-front in phase-shifting interferometry with arbitrary phase errors,” Opt. Commun. 233, 21–26(2004).
[CrossRef]

Opt. Eng. (2)

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39, 79–85 (2000).
[CrossRef]

C. C. Jin, N. K. Bao, and P. S. Chung, “Application of a novel phase-shift method using a computer-controlled polarization mechanism,” Opt. Eng. 33, 2733–2737 (1994).
[CrossRef]

Opt. Express (3)

Opt. Lett. (6)

Phys. Rev. Lett. (1)

Z. Chen, H. Martin, E. D. Eugenieva, J. Xu, and A. Bezryadina, “Anisotropic Enhancement of Discrete Diffraction and Formation of Two-Dimensional Discrete-Soliton Trains,” Phys. Rev. Lett. 92, 143902 (2004).
[CrossRef] [PubMed]

Progress in Optics (1)

L. Allen, M. P. Padgett, and M. Babiker, “The orbital angular momentum of light,” Progress in Optics 39, 291–372 (1999).
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

The experimental geometry for in-line phase-shifting digital holography with optical vortices as the reference beam.

Fig. 2.
Fig. 2.

Phase distributions of the optical vortices. (a)–(d) are four optical vortices with topological charge of l = 1; each rotates 90 degree with respect to the former, respectively. (e)–(h) are the respective interferograms with a plane wave.

Fig. 3.
Fig. 3.

Dependence of the phase shift on the rotation angle of optical vortex with l = 1.

Fig. 4.
Fig. 4.

(a) Photo of the object recorded; (b) one of four phase-shifting digital holograms recorded by the CCD camera; (c) the image reconstructed directly by the hologram shown in Fig. 4(b); (d) the final reconstructed image after performing four-step phase-shifting algorithm.

Fig. 5.
Fig. 5.

(a)–(d) are the reconstructed images of the holograms involved in Fig. 4 when the location errors are set on purpose to 8 pixels, 12 pixels, 16 pixels and 32 pixels, respectively.

Equations (3)

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U R = A ( r ) exp [ il ( ϕ + θ ) ] = A ( r ) exp ( ilϕ ) exp ( ilθ )
U h x y = exp ( ) 4 A R ( r ) { I 1 x y 0 I 3 x y π + i [ I 2 x y π 2 I 4 x y 3 π 2 ] }
U o ( x , y , z o ) = exp [ π λ z o ( x 2 + y 2 ) ] U h x y .

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