Abstract

A new phase-measurement technique is proposed, which utilizes a three-beam interferometer. Three-wave interference in the interferometer generates a uniform lattice of optical vortices, which is distorted by the presence of an object inserted in one arm of the interferometer. The transverse displacement of the vortices is proportional to the phase shift in the object wave. Tracking the vortices permits the phase of the object to be reconstructed. We demonstrate the method experimentally using a simple lens and a more complex object, namely the wing of a common house fly. Since the technique is implemented in real space, it is capable of reconstructing the phase locally.

© 2012 OSA

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  1. L. O. Heflinger, R. F. Wuerker, and R. E. Brooks, “Holographic interferometry,” J. Appl. Phys.37(2), 642–649 (1966).
    [CrossRef]
  2. M. Zhan, K. Li, P. Wang, L. Kong, X. Wang, R. Li, X. Tu, L. He, J. Wang, and B. Lu, “Cold atom interferometry,” J. Phys.: Conf. Ser. 80, 012047 (2007).
  3. A. Tonomura, T. Matsuda, J. Endo, T. Arii, and K. Mihama, “Holographic interference electron microscopy for determining specimen magnetic structure and thickness distribution,” Phys. Rev. B Condens. Matter34(5), 3397–3402 (1986).
    [CrossRef] [PubMed]
  4. U. Bonse and M. Hart, “An X-ray interferometer,” Appl. Phys. Lett.6(8), 155–156 (1965).
    [CrossRef]
  5. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am.72(1), 156–160 (1982).
    [CrossRef]
  6. P. Senthilkumaran, J. Masajada, and S. Sato, “Interferometry with vortices,” Int. J. Opt.2012, 517591 (2012).
    [CrossRef]
  7. A. Popiołek-Masajada, M. Borwinska, and B. Dubik, “Reconstruction of a plane wave’s tilt and orientation using an optical vortex interferometer,” Opt. Eng.46(7), 073604 (2007).
    [CrossRef]
  8. J. Masajada, “Small-angle rotations measurement using optical vortex interferometer,” Opt. Commun.239(4-6), 373–381 (2004).
    [CrossRef]
  9. M. Borwińska, A. Popiołek-Masajada, and P. Kurzynowski, “Measurements of birefringent media properties using optical vortex birefringence compensator,” Appl. Opt.46(25), 6419–6426 (2007).
    [CrossRef] [PubMed]
  10. W. A. Woźniak and M. Banach, “Measurements of linearly birefringent media parameters using the optical vortex interferometer with the Wollaston compensator,” J. Opt. A, Pure Appl. Opt.11(9), 094024 (2009).
    [CrossRef]
  11. W. Fraczek and J. Mroczka, “Optical vortices as phase markers to wave-front deformation measurement,” Metrol. Meas. Syst.15, 433–440 (2008).
  12. D. M. Paganin, Coherent X-Ray Optics (Clarendon Press, 2006).
  13. K. W. Nicholls and J. F. Nye, “Three-beam model for studying dislocations in wave pulses,” J. Phys. Math. Gen.20(14), 4673–4696 (1987).
    [CrossRef]
  14. J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun.198(1-3), 21–27 (2001).
    [CrossRef]
  15. K. O’Holleran, M. J. Padgett, and M. R. Dennis, “Topology of optical vortex lines formed by the interference of three, four, and five plane waves,” Opt. Express14(7), 3039–3044 (2006).
    [CrossRef] [PubMed]
  16. G. Ruben and D. M. Paganin, “Phase vortices from a Young’s three-pinhole interferometer,” Phys. Rev. E 75(6), 066613 (2007).
    [CrossRef] [PubMed]
  17. S. Vyas and P. Senthilkumaran, “Interferometric optical vortex array generator,” Appl. Opt.46(15), 2893–2898 (2007).
    [CrossRef] [PubMed]
  18. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A336(1605), 165–190 (1974).
    [CrossRef]
  19. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun.112(5-6), 321–327 (1994).
    [CrossRef]
  20. J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature467(7313), 301–304 (2010).
    [CrossRef] [PubMed]
  21. A. Popiołek-Masajada and W. Frączek, “Evaluation of a phase shifting method for vortex localization in optical vortex interferometery,” Opt. Laser Technol.43(7), 1219–1224 (2011).
    [CrossRef]
  22. E. Frączek, W. Fraczek, and J. Mroczka, “Experimental method for topological charge determination of optical vortices in a regular net,” Opt. Eng.44(2), 025601 (2005).
    [CrossRef]
  23. V. V. Volkov and Y. Zhu, “Deterministic phase unwrapping in the presence of noise,” Opt. Lett.28(22), 2156–2158 (2003).
    [CrossRef] [PubMed]
  24. L. J. Allen, H. M. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E63(3), 037602 (2001).
    [CrossRef] [PubMed]
  25. I. Freund and N. Shvartsman, “Wave-field phase singularities: The sign principle,” Phys. Rev. A50(6), 5164–5172 (1994).
    [CrossRef] [PubMed]

2012

P. Senthilkumaran, J. Masajada, and S. Sato, “Interferometry with vortices,” Int. J. Opt.2012, 517591 (2012).
[CrossRef]

2011

A. Popiołek-Masajada and W. Frączek, “Evaluation of a phase shifting method for vortex localization in optical vortex interferometery,” Opt. Laser Technol.43(7), 1219–1224 (2011).
[CrossRef]

2010

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature467(7313), 301–304 (2010).
[CrossRef] [PubMed]

2009

W. A. Woźniak and M. Banach, “Measurements of linearly birefringent media parameters using the optical vortex interferometer with the Wollaston compensator,” J. Opt. A, Pure Appl. Opt.11(9), 094024 (2009).
[CrossRef]

2008

W. Fraczek and J. Mroczka, “Optical vortices as phase markers to wave-front deformation measurement,” Metrol. Meas. Syst.15, 433–440 (2008).

2007

A. Popiołek-Masajada, M. Borwinska, and B. Dubik, “Reconstruction of a plane wave’s tilt and orientation using an optical vortex interferometer,” Opt. Eng.46(7), 073604 (2007).
[CrossRef]

G. Ruben and D. M. Paganin, “Phase vortices from a Young’s three-pinhole interferometer,” Phys. Rev. E 75(6), 066613 (2007).
[CrossRef] [PubMed]

S. Vyas and P. Senthilkumaran, “Interferometric optical vortex array generator,” Appl. Opt.46(15), 2893–2898 (2007).
[CrossRef] [PubMed]

M. Borwińska, A. Popiołek-Masajada, and P. Kurzynowski, “Measurements of birefringent media properties using optical vortex birefringence compensator,” Appl. Opt.46(25), 6419–6426 (2007).
[CrossRef] [PubMed]

2006

2005

E. Frączek, W. Fraczek, and J. Mroczka, “Experimental method for topological charge determination of optical vortices in a regular net,” Opt. Eng.44(2), 025601 (2005).
[CrossRef]

2004

J. Masajada, “Small-angle rotations measurement using optical vortex interferometer,” Opt. Commun.239(4-6), 373–381 (2004).
[CrossRef]

2003

2001

J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun.198(1-3), 21–27 (2001).
[CrossRef]

L. J. Allen, H. M. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E63(3), 037602 (2001).
[CrossRef] [PubMed]

1994

I. Freund and N. Shvartsman, “Wave-field phase singularities: The sign principle,” Phys. Rev. A50(6), 5164–5172 (1994).
[CrossRef] [PubMed]

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun.112(5-6), 321–327 (1994).
[CrossRef]

1987

K. W. Nicholls and J. F. Nye, “Three-beam model for studying dislocations in wave pulses,” J. Phys. Math. Gen.20(14), 4673–4696 (1987).
[CrossRef]

1986

A. Tonomura, T. Matsuda, J. Endo, T. Arii, and K. Mihama, “Holographic interference electron microscopy for determining specimen magnetic structure and thickness distribution,” Phys. Rev. B Condens. Matter34(5), 3397–3402 (1986).
[CrossRef] [PubMed]

1982

1974

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A336(1605), 165–190 (1974).
[CrossRef]

1966

L. O. Heflinger, R. F. Wuerker, and R. E. Brooks, “Holographic interferometry,” J. Appl. Phys.37(2), 642–649 (1966).
[CrossRef]

1965

U. Bonse and M. Hart, “An X-ray interferometer,” Appl. Phys. Lett.6(8), 155–156 (1965).
[CrossRef]

Allen, L. J.

L. J. Allen, H. M. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E63(3), 037602 (2001).
[CrossRef] [PubMed]

Arii, T.

A. Tonomura, T. Matsuda, J. Endo, T. Arii, and K. Mihama, “Holographic interference electron microscopy for determining specimen magnetic structure and thickness distribution,” Phys. Rev. B Condens. Matter34(5), 3397–3402 (1986).
[CrossRef] [PubMed]

Banach, M.

W. A. Woźniak and M. Banach, “Measurements of linearly birefringent media parameters using the optical vortex interferometer with the Wollaston compensator,” J. Opt. A, Pure Appl. Opt.11(9), 094024 (2009).
[CrossRef]

Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun.112(5-6), 321–327 (1994).
[CrossRef]

Berry, M. V.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A336(1605), 165–190 (1974).
[CrossRef]

Bonse, U.

U. Bonse and M. Hart, “An X-ray interferometer,” Appl. Phys. Lett.6(8), 155–156 (1965).
[CrossRef]

Borwinska, M.

A. Popiołek-Masajada, M. Borwinska, and B. Dubik, “Reconstruction of a plane wave’s tilt and orientation using an optical vortex interferometer,” Opt. Eng.46(7), 073604 (2007).
[CrossRef]

M. Borwińska, A. Popiołek-Masajada, and P. Kurzynowski, “Measurements of birefringent media properties using optical vortex birefringence compensator,” Appl. Opt.46(25), 6419–6426 (2007).
[CrossRef] [PubMed]

Brooks, R. E.

L. O. Heflinger, R. F. Wuerker, and R. E. Brooks, “Holographic interferometry,” J. Appl. Phys.37(2), 642–649 (1966).
[CrossRef]

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun.112(5-6), 321–327 (1994).
[CrossRef]

Dennis, M. R.

Dubik, B.

A. Popiołek-Masajada, M. Borwinska, and B. Dubik, “Reconstruction of a plane wave’s tilt and orientation using an optical vortex interferometer,” Opt. Eng.46(7), 073604 (2007).
[CrossRef]

J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun.198(1-3), 21–27 (2001).
[CrossRef]

Endo, J.

A. Tonomura, T. Matsuda, J. Endo, T. Arii, and K. Mihama, “Holographic interference electron microscopy for determining specimen magnetic structure and thickness distribution,” Phys. Rev. B Condens. Matter34(5), 3397–3402 (1986).
[CrossRef] [PubMed]

Faulkner, H. M.

L. J. Allen, H. M. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E63(3), 037602 (2001).
[CrossRef] [PubMed]

Fraczek, E.

E. Frączek, W. Fraczek, and J. Mroczka, “Experimental method for topological charge determination of optical vortices in a regular net,” Opt. Eng.44(2), 025601 (2005).
[CrossRef]

Fraczek, W.

A. Popiołek-Masajada and W. Frączek, “Evaluation of a phase shifting method for vortex localization in optical vortex interferometery,” Opt. Laser Technol.43(7), 1219–1224 (2011).
[CrossRef]

W. Fraczek and J. Mroczka, “Optical vortices as phase markers to wave-front deformation measurement,” Metrol. Meas. Syst.15, 433–440 (2008).

E. Frączek, W. Fraczek, and J. Mroczka, “Experimental method for topological charge determination of optical vortices in a regular net,” Opt. Eng.44(2), 025601 (2005).
[CrossRef]

Freund, I.

I. Freund and N. Shvartsman, “Wave-field phase singularities: The sign principle,” Phys. Rev. A50(6), 5164–5172 (1994).
[CrossRef] [PubMed]

Hart, M.

U. Bonse and M. Hart, “An X-ray interferometer,” Appl. Phys. Lett.6(8), 155–156 (1965).
[CrossRef]

Heflinger, L. O.

L. O. Heflinger, R. F. Wuerker, and R. E. Brooks, “Holographic interferometry,” J. Appl. Phys.37(2), 642–649 (1966).
[CrossRef]

Ina, H.

Kobayashi, S.

Kristensen, M.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun.112(5-6), 321–327 (1994).
[CrossRef]

Kurzynowski, P.

Masajada, J.

P. Senthilkumaran, J. Masajada, and S. Sato, “Interferometry with vortices,” Int. J. Opt.2012, 517591 (2012).
[CrossRef]

J. Masajada, “Small-angle rotations measurement using optical vortex interferometer,” Opt. Commun.239(4-6), 373–381 (2004).
[CrossRef]

J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun.198(1-3), 21–27 (2001).
[CrossRef]

Matsuda, T.

A. Tonomura, T. Matsuda, J. Endo, T. Arii, and K. Mihama, “Holographic interference electron microscopy for determining specimen magnetic structure and thickness distribution,” Phys. Rev. B Condens. Matter34(5), 3397–3402 (1986).
[CrossRef] [PubMed]

Mihama, K.

A. Tonomura, T. Matsuda, J. Endo, T. Arii, and K. Mihama, “Holographic interference electron microscopy for determining specimen magnetic structure and thickness distribution,” Phys. Rev. B Condens. Matter34(5), 3397–3402 (1986).
[CrossRef] [PubMed]

Mroczka, J.

W. Fraczek and J. Mroczka, “Optical vortices as phase markers to wave-front deformation measurement,” Metrol. Meas. Syst.15, 433–440 (2008).

E. Frączek, W. Fraczek, and J. Mroczka, “Experimental method for topological charge determination of optical vortices in a regular net,” Opt. Eng.44(2), 025601 (2005).
[CrossRef]

Nicholls, K. W.

K. W. Nicholls and J. F. Nye, “Three-beam model for studying dislocations in wave pulses,” J. Phys. Math. Gen.20(14), 4673–4696 (1987).
[CrossRef]

Nugent, K. A.

L. J. Allen, H. M. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E63(3), 037602 (2001).
[CrossRef] [PubMed]

Nye, J. F.

K. W. Nicholls and J. F. Nye, “Three-beam model for studying dislocations in wave pulses,” J. Phys. Math. Gen.20(14), 4673–4696 (1987).
[CrossRef]

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A336(1605), 165–190 (1974).
[CrossRef]

O’Holleran, K.

Oxley, M. P.

L. J. Allen, H. M. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E63(3), 037602 (2001).
[CrossRef] [PubMed]

Padgett, M. J.

Paganin, D.

L. J. Allen, H. M. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E63(3), 037602 (2001).
[CrossRef] [PubMed]

Paganin, D. M.

G. Ruben and D. M. Paganin, “Phase vortices from a Young’s three-pinhole interferometer,” Phys. Rev. E 75(6), 066613 (2007).
[CrossRef] [PubMed]

Popiolek-Masajada, A.

A. Popiołek-Masajada and W. Frączek, “Evaluation of a phase shifting method for vortex localization in optical vortex interferometery,” Opt. Laser Technol.43(7), 1219–1224 (2011).
[CrossRef]

A. Popiołek-Masajada, M. Borwinska, and B. Dubik, “Reconstruction of a plane wave’s tilt and orientation using an optical vortex interferometer,” Opt. Eng.46(7), 073604 (2007).
[CrossRef]

M. Borwińska, A. Popiołek-Masajada, and P. Kurzynowski, “Measurements of birefringent media properties using optical vortex birefringence compensator,” Appl. Opt.46(25), 6419–6426 (2007).
[CrossRef] [PubMed]

Ruben, G.

G. Ruben and D. M. Paganin, “Phase vortices from a Young’s three-pinhole interferometer,” Phys. Rev. E 75(6), 066613 (2007).
[CrossRef] [PubMed]

Sato, S.

P. Senthilkumaran, J. Masajada, and S. Sato, “Interferometry with vortices,” Int. J. Opt.2012, 517591 (2012).
[CrossRef]

Schattschneider, P.

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature467(7313), 301–304 (2010).
[CrossRef] [PubMed]

Senthilkumaran, P.

P. Senthilkumaran, J. Masajada, and S. Sato, “Interferometry with vortices,” Int. J. Opt.2012, 517591 (2012).
[CrossRef]

S. Vyas and P. Senthilkumaran, “Interferometric optical vortex array generator,” Appl. Opt.46(15), 2893–2898 (2007).
[CrossRef] [PubMed]

Shvartsman, N.

I. Freund and N. Shvartsman, “Wave-field phase singularities: The sign principle,” Phys. Rev. A50(6), 5164–5172 (1994).
[CrossRef] [PubMed]

Takeda, M.

Tian, H.

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature467(7313), 301–304 (2010).
[CrossRef] [PubMed]

Tonomura, A.

A. Tonomura, T. Matsuda, J. Endo, T. Arii, and K. Mihama, “Holographic interference electron microscopy for determining specimen magnetic structure and thickness distribution,” Phys. Rev. B Condens. Matter34(5), 3397–3402 (1986).
[CrossRef] [PubMed]

Verbeeck, J.

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature467(7313), 301–304 (2010).
[CrossRef] [PubMed]

Volkov, V. V.

Vyas, S.

Woerdman, J. P.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun.112(5-6), 321–327 (1994).
[CrossRef]

Wozniak, W. A.

W. A. Woźniak and M. Banach, “Measurements of linearly birefringent media parameters using the optical vortex interferometer with the Wollaston compensator,” J. Opt. A, Pure Appl. Opt.11(9), 094024 (2009).
[CrossRef]

Wuerker, R. F.

L. O. Heflinger, R. F. Wuerker, and R. E. Brooks, “Holographic interferometry,” J. Appl. Phys.37(2), 642–649 (1966).
[CrossRef]

Zhu, Y.

Appl. Opt.

Appl. Phys. Lett.

U. Bonse and M. Hart, “An X-ray interferometer,” Appl. Phys. Lett.6(8), 155–156 (1965).
[CrossRef]

Int. J. Opt.

P. Senthilkumaran, J. Masajada, and S. Sato, “Interferometry with vortices,” Int. J. Opt.2012, 517591 (2012).
[CrossRef]

J. Appl. Phys.

L. O. Heflinger, R. F. Wuerker, and R. E. Brooks, “Holographic interferometry,” J. Appl. Phys.37(2), 642–649 (1966).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

W. A. Woźniak and M. Banach, “Measurements of linearly birefringent media parameters using the optical vortex interferometer with the Wollaston compensator,” J. Opt. A, Pure Appl. Opt.11(9), 094024 (2009).
[CrossRef]

J. Opt. Soc. Am.

J. Phys. Math. Gen.

K. W. Nicholls and J. F. Nye, “Three-beam model for studying dislocations in wave pulses,” J. Phys. Math. Gen.20(14), 4673–4696 (1987).
[CrossRef]

Metrol. Meas. Syst.

W. Fraczek and J. Mroczka, “Optical vortices as phase markers to wave-front deformation measurement,” Metrol. Meas. Syst.15, 433–440 (2008).

Nature

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature467(7313), 301–304 (2010).
[CrossRef] [PubMed]

Opt. Commun.

J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun.198(1-3), 21–27 (2001).
[CrossRef]

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun.112(5-6), 321–327 (1994).
[CrossRef]

J. Masajada, “Small-angle rotations measurement using optical vortex interferometer,” Opt. Commun.239(4-6), 373–381 (2004).
[CrossRef]

Opt. Eng.

A. Popiołek-Masajada, M. Borwinska, and B. Dubik, “Reconstruction of a plane wave’s tilt and orientation using an optical vortex interferometer,” Opt. Eng.46(7), 073604 (2007).
[CrossRef]

E. Frączek, W. Fraczek, and J. Mroczka, “Experimental method for topological charge determination of optical vortices in a regular net,” Opt. Eng.44(2), 025601 (2005).
[CrossRef]

Opt. Express

Opt. Laser Technol.

A. Popiołek-Masajada and W. Frączek, “Evaluation of a phase shifting method for vortex localization in optical vortex interferometery,” Opt. Laser Technol.43(7), 1219–1224 (2011).
[CrossRef]

Opt. Lett.

Phys. Rev. A

I. Freund and N. Shvartsman, “Wave-field phase singularities: The sign principle,” Phys. Rev. A50(6), 5164–5172 (1994).
[CrossRef] [PubMed]

Phys. Rev. B Condens. Matter

A. Tonomura, T. Matsuda, J. Endo, T. Arii, and K. Mihama, “Holographic interference electron microscopy for determining specimen magnetic structure and thickness distribution,” Phys. Rev. B Condens. Matter34(5), 3397–3402 (1986).
[CrossRef] [PubMed]

Phys. Rev. E

G. Ruben and D. M. Paganin, “Phase vortices from a Young’s three-pinhole interferometer,” Phys. Rev. E 75(6), 066613 (2007).
[CrossRef] [PubMed]

Phys. Rev. E

L. J. Allen, H. M. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E63(3), 037602 (2001).
[CrossRef] [PubMed]

Proc. R. Soc. Lond. A

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A336(1605), 165–190 (1974).
[CrossRef]

Other

D. M. Paganin, Coherent X-Ray Optics (Clarendon Press, 2006).

M. Zhan, K. Li, P. Wang, L. Kong, X. Wang, R. Li, X. Tu, L. He, J. Wang, and B. Lu, “Cold atom interferometry,” J. Phys.: Conf. Ser. 80, 012047 (2007).

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