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. Matter 34(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. Express 14(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. A 336(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,” Nature 467(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. E 63(3), 037602 (2001).
    [CrossRef] [PubMed]
  25. I. Freund and N. Shvartsman, “Wave-field phase singularities: The sign principle,” Phys. Rev. A 50(6), 5164–5172 (1994).
    [CrossRef] [PubMed]

2012 (1)

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

2011 (1)

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 (1)

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature 467(7313), 301–304 (2010).
[CrossRef] [PubMed]

2009 (1)

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 (1)

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

2007 (4)

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 (1)

2005 (1)

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 (1)

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

2003 (1)

2001 (2)

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. E 63(3), 037602 (2001).
[CrossRef] [PubMed]

1994 (2)

I. Freund and N. Shvartsman, “Wave-field phase singularities: The sign principle,” Phys. Rev. A 50(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 (1)

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 (1)

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. Matter 34(5), 3397–3402 (1986).
[CrossRef] [PubMed]

1982 (1)

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A 336(1605), 165–190 (1974).
[CrossRef]

1966 (1)

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

1965 (1)

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. E 63(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. Matter 34(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. A 336(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.

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]

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]

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. Matter 34(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. E 63(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. A 50(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. Matter 34(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. Matter 34(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. E 63(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. A 336(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. E 63(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. E 63(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,” Nature 467(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. A 50(6), 5164–5172 (1994).
[CrossRef] [PubMed]

Takeda, M.

Tian, H.

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature 467(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. Matter 34(5), 3397–3402 (1986).
[CrossRef] [PubMed]

Verbeeck, J.

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature 467(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. (2)

Appl. Phys. Lett. (1)

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

Int. J. Opt. (1)

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

J. Appl. Phys. (1)

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. (1)

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. (1)

J. Phys. Math. Gen. (1)

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. (1)

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

Nature (1)

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature 467(7313), 301–304 (2010).
[CrossRef] [PubMed]

Opt. Commun. (3)

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. (2)

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 (1)

Opt. Laser Technol. (1)

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. (1)

Phys. Rev. A (1)

I. Freund and N. Shvartsman, “Wave-field phase singularities: The sign principle,” Phys. Rev. A 50(6), 5164–5172 (1994).
[CrossRef] [PubMed]

Phys. Rev. B Condens. Matter (1)

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. Matter 34(5), 3397–3402 (1986).
[CrossRef] [PubMed]

Phys. Rev. E (2)

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

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. E 63(3), 037602 (2001).
[CrossRef] [PubMed]

Proc. R. Soc. Lond. A (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A 336(1605), 165–190 (1974).
[CrossRef]

Other (2)

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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Figures (5)

Fig. 1
Fig. 1

(a) Phasor geometry for three waves of unit intensity at a vortex point. (b) Phasor geometry for a vortex when the exit wave is transmitted through an absorbing object.

Fig. 2
Fig. 2

Experimental setup of the three-beam vortex interferometer. A He-Ne laser (λ = 633nm) is spatially filtered by the pinhole and a single polarization direction is selected as it passes through the polarizing beam splitter. Two beamsplitters are used to create the three arms of the interferometer. The central beam is transmitted through an object after which two lenses are used to focus the exit surface of the object into the camera. Each reference arm contains λ/2 and λ/4 wave plates; which are used for phase stepping. Neutral density filters (ND) ensure that the two reference waves have the same intensity as the object wave.

Fig. 3
Fig. 3

Comparison of the experimental three-beam interference pattern with a numerical simulation. (a) Experimental three-beam interference pattern. (b) Simulated three-beam interference pattern. The plus signs and crosses in the top right insert indicate the locations of vortices of positive and negative topological charge, respectively.

Fig. 4
Fig. 4

Experimental results for phase reconstruction of a spherical lens using the three-beam vortex interferometer. (a) The unwrapped reconstructed phase of the lens. The greyscale in this image ranges from 0 (black) to 2π (white). (b) Unwrapped phase profile of the lens. The solid curve corresponds to the experimental data whilst the dashed curve is the fitted curve. The radius of curvature of the fitted profile is 519 ± 1 mm.

Fig. 5
Fig. 5

Experimental results for the wing of a fly. (a) Experimental three-beam interferogram of the fly's wing. (b) Recovered phase of the fly wing using Eq. (7). (c) Recovered phase using the Takeda method [5]. (d) The unwrapped phase in (b). The greyscale in (b) and (c) is [π/2,π/2] from black to white.

Equations (9)

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Ψ(x,y)= ψ obj (x,y)+ ψ A (x,y)+ ψ B (x,y)  =exp[iϕ(x,y)]× {A(x,y)+exp[i2π( k x A x+ k y A y)iϕ(x,y)]+exp[i2π( k x B x+ k y B y)iϕ(x,y)]}.
Γ ϕ·tds=2πp,
2π( k x A x ˜ + k y A y ˜ )ϕ( x ˜ , y ˜ )= 2π 3 +2πn,
2π( k x B x ˜ + k y B y ˜ )ϕ( x ˜ , y ˜ )= 4π 3 +2πm,
4π( k x A x ˜ + k y A y ˜ )2ϕ( x ˜ , y ˜ )=2π( k x B x ˜ + k y B y ˜ )ϕ( x ˜ , y ˜ )+2π(m2n).
ϕ( x ˜ , y ˜ )=2π(2 k x A k x B ) x ˜ +2π(2 k y A k y B ) y ˜ +2πl,
2π( k x A x ˜ + k y A y ˜ )ϕ( x ˜ , y ˜ )=2π(m+n+1)2π( k x B x ˜ + k y B y ˜ )+ϕ( x ˜ , y ˜ ).
ϕ( x ˜ , y ˜ )=π( k x A + k x B ) x ˜ +π( k y A + k y B ) y ˜ +πs,
ϕ(x,y)= tan 1 { sin[ϕ( x ˜ , y ˜ )] cos[ϕ( x ˜ , y ˜ )] }.

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