Abstract

We propose a novel phase retrieval algorithm in Hankel (or called Fourier-Bessel) transform domains by using Monte-Carlo method. Based on the proposed algorithm, we investigate the generation of Gaussian-like beams, such as hollow Gaussian beam, Bessel-Gaussian beam and Laguerre-Gaussian beam, with double phase filtering operations. The phase distributions of filters are determined by employing the proposed phase retrieval algorithm. The advantage of the method is that the total energy of the beam is conservative. Numerical simulations are shown to demonstrate the validity of the scheme.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
    [CrossRef]
  2. H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, "Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers," Phys. Rev. Lett. 76, 4500 (1996).
    [CrossRef] [PubMed]
  3. Y. K. Wu, J. Li, and J. Wu, "Anomalous hollow electron beams in a storage ring," Phys. Rev. Lett. 94, 134802 (2005).
    [CrossRef] [PubMed]
  4. Y. Cai, X. Lu, and Q. Lin, "Hollow Gaussian beams and their propagation properties," Opt. Lett. 28, 1084 (2003).
    [CrossRef] [PubMed]
  5. Z. Liu, H. Zhao, J. Liu, J. Lin, M. A. Ahmad, and S. Liu, "Generation of hollow Gaussian beams by spatial filtering," Opt. Lett. 32, 2076 (2007).
    [CrossRef] [PubMed]
  6. Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, "Spin-to-orbital angular momentum conversion in a strongly focused optical beam," Phys. Rev. Lett. 99, 073901 (2007).
    [CrossRef] [PubMed]
  7. R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).
  8. Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, "Gerchberg-Saxton algorithm applied in the fractional Fourieror the Fresnel domain," Opt. Lett. 21, 842-844 (1996).
    [CrossRef] [PubMed]
  9. D. F. McAlister, M. Beck, L. Clarke, A. Mayer, and M. G. Raymer, "Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms," Opt. Lett. 20, 1181-1183 (1995).
    [CrossRef] [PubMed]
  10. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barena, "Quantitative Phase Imaging using Hard X Rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
    [CrossRef] [PubMed]
  11. J. Miao T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, "High resolution 3D X-ray diffraction microscopy," Phys. Rev. Lett. 89, 088303 (2002).
    [CrossRef] [PubMed]
  12. F. Meng, H. Liu, and X. Wu, "An iterative phase retrieval algorithm for in-line x-ray phase imaging," Opt. Express 15, 8383-8390 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-13-8383.
    [CrossRef] [PubMed]
  13. J. M. Zuo I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, "Atomic resolution imaging of a carbon nanotube from diffraction intensities," Science 300, 1419 (2003).
    [CrossRef] [PubMed]
  14. R. Borghi, F. Gori, and M. Santarsiero, "Phase and amplitude retrieval in ghost diffraction from field-correlation measurements," Phys. Rev. Lett. 96, 183901 (2006).
    [CrossRef] [PubMed]
  15. A. Anand, G. Pedrini, W. Osten, and P. Almoro, "Wavefront sensing with random amplitude mask and phase retrieval," Opt. Lett. 32, 1584-1586 (2007).
    [CrossRef] [PubMed]
  16. P. Almoro, G. Pedrini, and W. Osten, "Wavefront sensing with random amplitude mask and phase retrieval," Opt. Lett. 32, 733-735 (2007).
    [CrossRef] [PubMed]
  17. A. K. Ellerbee and J. A. Izatt, "Phase retrieval in low-coherence interferometric microscopy," Opt. Lett. 32, 388-390 (2007).
    [CrossRef] [PubMed]
  18. J. Zhong and J. Weng, "Phase retrieval of optical fringe patterns from the ridge of a wavelet transform," Opt. Lett. 30, 2560-2562 (2005).
    [CrossRef] [PubMed]
  19. J. S. Wu, U. Weierstall, J. C. H. Spence, and C. T. Koch, "Iterative phase retrieval without support," Opt. Lett. 29, 2737 (2004).
    [CrossRef] [PubMed]
  20. N. Nakajima, "Phase and amplitude retrieval in ghost diffraction from field-correlation measurements," Phys. Rev. Lett. 98, 223901 (2007).
    [CrossRef] [PubMed]
  21. F. Zhang, G. Pedrini, and W. Osten, "Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation," Phys. Rev. A 75, 043805 (2007).
    [CrossRef]
  22. T. Latychevskaia and H.-W. Fink, "Solution to the twin image problem in holography," Phys. Rev. Lett. 98, 233901 (2007).
    [CrossRef] [PubMed]
  23. Y. Zhang, G. Pedrini, W. Osten, and H. Tiziani, "Whole optical wave field reconstruction from double or multi in-line holograms by phase retrieval algorithm," Opt. Express 11, 3234-3241 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-24-3234.
    [CrossRef] [PubMed]
  24. Z. Liu and S. Liu, "Double image encryption based on iterative fractional Fourier transform," Opt. Commun. 272, 324-329 (2007).
    [CrossRef]
  25. G.-Z. Yang, B.-Z. Dong, B.-Y. Gu, J.-Y. Zhuang, and O. K. Ersoy, "Gerchberg-Saxton and Yang-Gu algorithms for phase retrieval in a nonunitary transform system: a comparison," Appl. Opt. 33, 209 (1994).
    [CrossRef] [PubMed]
  26. B. Dong, Y. Zhang, B. Gu, and G. Yang, "Numerical investigation of phase retrieval in a fractional Fourier transform," J. Opt. Soc. Am. A 14, 2709-2714 (1997).
    [CrossRef]

2007 (10)

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, "Spin-to-orbital angular momentum conversion in a strongly focused optical beam," Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

N. Nakajima, "Phase and amplitude retrieval in ghost diffraction from field-correlation measurements," Phys. Rev. Lett. 98, 223901 (2007).
[CrossRef] [PubMed]

F. Zhang, G. Pedrini, and W. Osten, "Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation," Phys. Rev. A 75, 043805 (2007).
[CrossRef]

T. Latychevskaia and H.-W. Fink, "Solution to the twin image problem in holography," Phys. Rev. Lett. 98, 233901 (2007).
[CrossRef] [PubMed]

Z. Liu and S. Liu, "Double image encryption based on iterative fractional Fourier transform," Opt. Commun. 272, 324-329 (2007).
[CrossRef]

A. K. Ellerbee and J. A. Izatt, "Phase retrieval in low-coherence interferometric microscopy," Opt. Lett. 32, 388-390 (2007).
[CrossRef] [PubMed]

P. Almoro, G. Pedrini, and W. Osten, "Wavefront sensing with random amplitude mask and phase retrieval," Opt. Lett. 32, 733-735 (2007).
[CrossRef] [PubMed]

A. Anand, G. Pedrini, W. Osten, and P. Almoro, "Wavefront sensing with random amplitude mask and phase retrieval," Opt. Lett. 32, 1584-1586 (2007).
[CrossRef] [PubMed]

F. Meng, H. Liu, and X. Wu, "An iterative phase retrieval algorithm for in-line x-ray phase imaging," Opt. Express 15, 8383-8390 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-13-8383.
[CrossRef] [PubMed]

Z. Liu, H. Zhao, J. Liu, J. Lin, M. A. Ahmad, and S. Liu, "Generation of hollow Gaussian beams by spatial filtering," Opt. Lett. 32, 2076 (2007).
[CrossRef] [PubMed]

2006 (1)

R. Borghi, F. Gori, and M. Santarsiero, "Phase and amplitude retrieval in ghost diffraction from field-correlation measurements," Phys. Rev. Lett. 96, 183901 (2006).
[CrossRef] [PubMed]

2005 (2)

Y. K. Wu, J. Li, and J. Wu, "Anomalous hollow electron beams in a storage ring," Phys. Rev. Lett. 94, 134802 (2005).
[CrossRef] [PubMed]

J. Zhong and J. Weng, "Phase retrieval of optical fringe patterns from the ridge of a wavelet transform," Opt. Lett. 30, 2560-2562 (2005).
[CrossRef] [PubMed]

2004 (1)

2003 (3)

2002 (1)

J. Miao T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, "High resolution 3D X-ray diffraction microscopy," Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

1997 (2)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

B. Dong, Y. Zhang, B. Gu, and G. Yang, "Numerical investigation of phase retrieval in a fractional Fourier transform," J. Opt. Soc. Am. A 14, 2709-2714 (1997).
[CrossRef]

1996 (3)

Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, "Gerchberg-Saxton algorithm applied in the fractional Fourieror the Fresnel domain," Opt. Lett. 21, 842-844 (1996).
[CrossRef] [PubMed]

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, "Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers," Phys. Rev. Lett. 76, 4500 (1996).
[CrossRef] [PubMed]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barena, "Quantitative Phase Imaging using Hard X Rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

1995 (1)

1994 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).

Ahmad, M. A.

Almoro, P.

Anand, A.

Barena, Z.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barena, "Quantitative Phase Imaging using Hard X Rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Beck, M.

Borghi, R.

R. Borghi, F. Gori, and M. Santarsiero, "Phase and amplitude retrieval in ghost diffraction from field-correlation measurements," Phys. Rev. Lett. 96, 183901 (2006).
[CrossRef] [PubMed]

Cai, Y.

Chiu, D. T.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, "Spin-to-orbital angular momentum conversion in a strongly focused optical beam," Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

Clarke, L.

Cookson, D. F.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barena, "Quantitative Phase Imaging using Hard X Rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Dong, B.

Dong, B.-Z.

Dorsch, R. G.

Edgar, J. S.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, "Spin-to-orbital angular momentum conversion in a strongly focused optical beam," Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

Ellerbee, A. K.

Ersoy, O. K.

Fink, H.-W.

T. Latychevskaia and H.-W. Fink, "Solution to the twin image problem in holography," Phys. Rev. Lett. 98, 233901 (2007).
[CrossRef] [PubMed]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).

Gori, F.

R. Borghi, F. Gori, and M. Santarsiero, "Phase and amplitude retrieval in ghost diffraction from field-correlation measurements," Phys. Rev. Lett. 96, 183901 (2006).
[CrossRef] [PubMed]

Gu, B.

Gu, B.-Y.

Gureyev, T. E.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barena, "Quantitative Phase Imaging using Hard X Rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Ito, H.

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, "Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers," Phys. Rev. Lett. 76, 4500 (1996).
[CrossRef] [PubMed]

Izatt, J. A.

Jeffries, G. D. M.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, "Spin-to-orbital angular momentum conversion in a strongly focused optical beam," Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

Koch, C. T.

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Latychevskaia, T.

T. Latychevskaia and H.-W. Fink, "Solution to the twin image problem in holography," Phys. Rev. Lett. 98, 233901 (2007).
[CrossRef] [PubMed]

Lee, K. I.

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, "Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers," Phys. Rev. Lett. 76, 4500 (1996).
[CrossRef] [PubMed]

Li, J.

Y. K. Wu, J. Li, and J. Wu, "Anomalous hollow electron beams in a storage ring," Phys. Rev. Lett. 94, 134802 (2005).
[CrossRef] [PubMed]

Lin, J.

Lin, Q.

Liu, H.

Liu, J.

Liu, S.

Z. Liu, H. Zhao, J. Liu, J. Lin, M. A. Ahmad, and S. Liu, "Generation of hollow Gaussian beams by spatial filtering," Opt. Lett. 32, 2076 (2007).
[CrossRef] [PubMed]

Z. Liu and S. Liu, "Double image encryption based on iterative fractional Fourier transform," Opt. Commun. 272, 324-329 (2007).
[CrossRef]

Liu, Z.

Z. Liu and S. Liu, "Double image encryption based on iterative fractional Fourier transform," Opt. Commun. 272, 324-329 (2007).
[CrossRef]

Z. Liu, H. Zhao, J. Liu, J. Lin, M. A. Ahmad, and S. Liu, "Generation of hollow Gaussian beams by spatial filtering," Opt. Lett. 32, 2076 (2007).
[CrossRef] [PubMed]

Lu, X.

Mayer, A.

McAlister, D. F.

McGloin, D.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, "Spin-to-orbital angular momentum conversion in a strongly focused optical beam," Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

Mendlovic, D.

Meng, F.

Nakajima, N.

N. Nakajima, "Phase and amplitude retrieval in ghost diffraction from field-correlation measurements," Phys. Rev. Lett. 98, 223901 (2007).
[CrossRef] [PubMed]

Nakata, T.

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, "Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers," Phys. Rev. Lett. 76, 4500 (1996).
[CrossRef] [PubMed]

Nugent, K. A.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barena, "Quantitative Phase Imaging using Hard X Rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Ohtsu, M.

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, "Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers," Phys. Rev. Lett. 76, 4500 (1996).
[CrossRef] [PubMed]

Osten, W.

Paganin, D.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barena, "Quantitative Phase Imaging using Hard X Rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Pedrini, G.

Raymer, M. G.

Sakaki, K.

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, "Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers," Phys. Rev. Lett. 76, 4500 (1996).
[CrossRef] [PubMed]

Santarsiero, M.

R. Borghi, F. Gori, and M. Santarsiero, "Phase and amplitude retrieval in ghost diffraction from field-correlation measurements," Phys. Rev. Lett. 96, 183901 (2006).
[CrossRef] [PubMed]

Sasada, H.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).

Shimizu, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Spence, J. C. H.

Tiziani, H.

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Weierstall, U.

Weng, J.

Wu, J.

Y. K. Wu, J. Li, and J. Wu, "Anomalous hollow electron beams in a storage ring," Phys. Rev. Lett. 94, 134802 (2005).
[CrossRef] [PubMed]

Wu, J. S.

Wu, X.

Wu, Y. K.

Y. K. Wu, J. Li, and J. Wu, "Anomalous hollow electron beams in a storage ring," Phys. Rev. Lett. 94, 134802 (2005).
[CrossRef] [PubMed]

Yang, G.

Yang, G.-Z.

Zalevsky, Z.

Zhang, F.

F. Zhang, G. Pedrini, and W. Osten, "Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation," Phys. Rev. A 75, 043805 (2007).
[CrossRef]

Zhang, Y.

Zhao, H.

Zhao, Y.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, "Spin-to-orbital angular momentum conversion in a strongly focused optical beam," Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

Zhong, J.

Zhuang, J.-Y.

Appl. Opt. (1)

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

Opt. Commun. (1)

Z. Liu and S. Liu, "Double image encryption based on iterative fractional Fourier transform," Opt. Commun. 272, 324-329 (2007).
[CrossRef]

Opt. Express (2)

Opt. Lett. (9)

J. S. Wu, U. Weierstall, J. C. H. Spence, and C. T. Koch, "Iterative phase retrieval without support," Opt. Lett. 29, 2737 (2004).
[CrossRef] [PubMed]

J. Zhong and J. Weng, "Phase retrieval of optical fringe patterns from the ridge of a wavelet transform," Opt. Lett. 30, 2560-2562 (2005).
[CrossRef] [PubMed]

A. K. Ellerbee and J. A. Izatt, "Phase retrieval in low-coherence interferometric microscopy," Opt. Lett. 32, 388-390 (2007).
[CrossRef] [PubMed]

P. Almoro, G. Pedrini, and W. Osten, "Wavefront sensing with random amplitude mask and phase retrieval," Opt. Lett. 32, 733-735 (2007).
[CrossRef] [PubMed]

A. Anand, G. Pedrini, W. Osten, and P. Almoro, "Wavefront sensing with random amplitude mask and phase retrieval," Opt. Lett. 32, 1584-1586 (2007).
[CrossRef] [PubMed]

Z. Liu, H. Zhao, J. Liu, J. Lin, M. A. Ahmad, and S. Liu, "Generation of hollow Gaussian beams by spatial filtering," Opt. Lett. 32, 2076 (2007).
[CrossRef] [PubMed]

D. F. McAlister, M. Beck, L. Clarke, A. Mayer, and M. G. Raymer, "Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms," Opt. Lett. 20, 1181-1183 (1995).
[CrossRef] [PubMed]

Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, "Gerchberg-Saxton algorithm applied in the fractional Fourieror the Fresnel domain," Opt. Lett. 21, 842-844 (1996).
[CrossRef] [PubMed]

Y. Cai, X. Lu, and Q. Lin, "Hollow Gaussian beams and their propagation properties," Opt. Lett. 28, 1084 (2003).
[CrossRef] [PubMed]

Optik (1)

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).

Phys. Rev. A (1)

F. Zhang, G. Pedrini, and W. Osten, "Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation," Phys. Rev. A 75, 043805 (2007).
[CrossRef]

Phys. Rev. Lett. (9)

T. Latychevskaia and H.-W. Fink, "Solution to the twin image problem in holography," Phys. Rev. Lett. 98, 233901 (2007).
[CrossRef] [PubMed]

R. Borghi, F. Gori, and M. Santarsiero, "Phase and amplitude retrieval in ghost diffraction from field-correlation measurements," Phys. Rev. Lett. 96, 183901 (2006).
[CrossRef] [PubMed]

N. Nakajima, "Phase and amplitude retrieval in ghost diffraction from field-correlation measurements," Phys. Rev. Lett. 98, 223901 (2007).
[CrossRef] [PubMed]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barena, "Quantitative Phase Imaging using Hard X Rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

J. Miao T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, "High resolution 3D X-ray diffraction microscopy," Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, "Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers," Phys. Rev. Lett. 76, 4500 (1996).
[CrossRef] [PubMed]

Y. K. Wu, J. Li, and J. Wu, "Anomalous hollow electron beams in a storage ring," Phys. Rev. Lett. 94, 134802 (2005).
[CrossRef] [PubMed]

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, "Spin-to-orbital angular momentum conversion in a strongly focused optical beam," Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

Science (1)

J. M. Zuo I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, "Atomic resolution imaging of a carbon nanotube from diffraction intensities," Science 300, 1419 (2003).
[CrossRef] [PubMed]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1.
Fig. 1.

Schematic Illustration of the optical setup for the generation of hollow Gaussian beams based on phase retrieval algorithm.

Fig. 2.
Fig. 2.

Flow chart of the proposed phase retrieval algorithm in Hankel transform domains.

Fig. 3.
Fig. 3.

The numerical result of the proposed phase retrieval algorithm. Here the output beam is the second-order hollow Gaussian beam.

Fig. 4.
Fig. 4.

The intensity distribution of the hollow Gaussian beam: (a) the theoretical result directly from the analytical expression, (b) the result generated by the phase retrieval algorithm and double phase filtering.

Fig. 5.
Fig. 5.

Convergence speed curves of G-S, Yang-Gu and the proposed algorithms.

Tables (1)

Tables Icon

Table 1. The total iterative number under different parameters c0 and c1 and other parameters is the same in Fig.3.

Equations (12)

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

E ( x , y ) = G ( z ) exp [ x 2 + y 2 ω 2 ( z ) ] exp [ ik x 2 + y 2 2 R ( z ) ] ,
A 1 ( x , y ) = G 10 exp ( x 2 + y 2 ω 0 2 ) ,
A 3 ( x , y ) = G 30 ( x 2 + y 2 ω 0 2 ) p exp ( x 2 + y 2 ω 0 2 ) ,
ϕ 3 ( x , y ) = C 0 ,
A 2 ( ρ ) exp [ i ϕ 2 ( ρ ) ] = H { A 1 ( r ) exp [ i ϕ 1 ( r ) ] } ,
A 3 ( r ) exp [ i ϕ 3 ( r ) ] = H { A 2 ( ρ ) exp [ i φ 2 ( ρ ) ] } ,
H [ f ( ρ ) ] = 2 π 0 f ( r ) J 0 ( 2 π ρ r ) d r ,
A 2 ( ρ ) = H { A 3 ( r ) exp [ i ϕ 3 ( r ) ] } ,
φ 2 ( ρ ) = arg { H { A 3 ( r ) exp [ i ϕ 3 ( r ) ] } A 2 ( ρ ) } ,
A 2 ( ρ ) exp [ i ϕ 2 ( ρ ) ] = H { A 1 ( r ) exp [ i ϕ 1 ( r ) ] } ,
MSE = mse ( A 2 , k , A 2 ) = Σ A 2 , k A 2 2 L ,
Δ ϕ 1 = ϕ 1 r ( r ) φ 1 , Δ ϕ 2 = φ 2 ( ρ ) ϕ 2 r ( ρ ) .

Metrics