Abstract

The Cornu spiral is, in essence, the image resulting from an Argand-plane map associated with monochromatic complex scalar plane waves diffracting from an infinite edge. Argand-plane maps can be useful in the analysis of more general optical fields. We experimentally study particular features of Argand-plane mappings known as “vorticity singularities” that are associated with mapping continuous single-valued complex scalar speckle fields to the Argand plane. Vorticity singularities possess a hierarchy of Argand-plane catastrophes including the fold, cusp and elliptic umbilic. We also confirm their connection to vortices in two-dimensional complex scalar waves. The study of vorticity singularities may also have implications for higher-dimensional fields such as coherence functions and multi-component fields such as vector and spinor fields.

© 2014 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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  6. K. O’Holleran, F. Flossman, M. R. Dennis, M. J. Padgett, “Methodology for imaging the 3D structure of singularities in scalar and vector optical fields,” J. Opt. A Pure Appl. Opt. 11, 094020 (2009).
    [CrossRef]
  7. M. V. Berry, “Optical currents,” J. Opt. A Pure Appl. Opt. 11, 094001 (2009).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  20. C. Raman, J. R. Abo-Shaeer, J. M. Vogels, K. Xu, W. Ketterle, “Vortex nucleation in a stirred Bose–Einstein condensate,” Phys. Rev. Lett. 87, 210402 (2001).
    [CrossRef]
  21. I. Freund, “Critical point explosions in two-dimensional wave fields,” Opt. Commun. 159, 99–117 (1999).
    [CrossRef]
  22. G. Gbur, T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003).
    [CrossRef]
  23. M. L. Marasinghe, D. M. Paganin, M. Premaratne, “Coherence-vortex lattice formed via Mie scattering of partially coherent light by several dielectric nanospheres,” Opt. Lett. 36, 936–938 (2011).
    [CrossRef] [PubMed]
  24. P. Liu, H. Yang, J. Rong, G. Wang, Y. Yan, “Coherence vortex evolution of partially coherent vortex beams in the focal region,” Opt. Laser Technol. 42, 99–104 (2010).
    [CrossRef]
  25. W. Wang, M. Takeda, “Coherence current, coherence vortex and the conservation law of coherence,” Phys. Rev. Lett. 69, 223904 (2006).
    [CrossRef]

2012

F. Rothschild, M. J. Kitchen, H. M. L. Faulkner, D. M. Paganin, “Duality between phase vortices and Argand-plane caustics,” Opt. Commun. 285, 4141–4151 (2012).
[CrossRef]

2011

2010

P. Liu, H. Yang, J. Rong, G. Wang, Y. Yan, “Coherence vortex evolution of partially coherent vortex beams in the focal region,” Opt. Laser Technol. 42, 99–104 (2010).
[CrossRef]

K. S. Morgan, K. K. W. Siu, D. M. Paganin, “The projection approximation and edge contrast for x-ray propagation-based phase contrast imaging of a cylindrical edge,” Opt. Express 18, 9865–9878 (2010).
[CrossRef] [PubMed]

2009

M. R. Dennis, K. O’Holleran, M. J. Padgett, “Singular optics: Optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[CrossRef]

K. O’Holleran, F. Flossman, M. R. Dennis, M. J. Padgett, “Methodology for imaging the 3D structure of singularities in scalar and vector optical fields,” J. Opt. A Pure Appl. Opt. 11, 094020 (2009).
[CrossRef]

M. V. Berry, “Optical currents,” J. Opt. A Pure Appl. Opt. 11, 094001 (2009).
[CrossRef]

E. A. L. Henn, J. A. Seman, E. R. F. Ramos, M. Caracanhas, P. Castilho, E. P. Olimpio, G. Roati, D. V. Magalhaes, K. M. F. Magalhaes, V. S. Bagnato, “Observation of vortex formation in an oscillation trapped Bose–Einstein condensate,” Phys. Rev. A 79, 043618 (2009).
[CrossRef]

2008

K. O’Holleran, M. R. Dennis, F. Flossman, M. J. Padgett, “Fractality of light’s darkness,” Phys. Rev. Lett. 100, 053902 (2008).
[CrossRef]

2007

M. V. Berry, M. R. Dennis, “Topological events on wave dislocation lines: Birth and death of loops, and reconnection,” J. Phys. A Math. Theor. 40, 65–74 (2007).
[CrossRef]

2006

W. Wang, M. Takeda, “Coherence current, coherence vortex and the conservation law of coherence,” Phys. Rev. Lett. 69, 223904 (2006).
[CrossRef]

2004

M. J. Kitchen, D. M. Paganin, R. A. Lewis, N. Yagi, K. Uesugi, S. T. Mudie, “On the origin of speckle in x-ray phase contrast images of lung issue,” Phys. Med. Biol. 49, 4335–4348 (2004).
[CrossRef] [PubMed]

2003

2001

L. J. Allen, H. M. L. Faulkner, M. P. Oxley, D. M. Paganin, “Phase retrieval and aberration correction in the presence of vortices in high-resolution transmission electron microscopy,” Ultramicroscopy 88, 85–97 (2001).
[CrossRef] [PubMed]

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, D. M. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
[CrossRef]

C. Raman, J. R. Abo-Shaeer, J. M. Vogels, K. Xu, W. Ketterle, “Vortex nucleation in a stirred Bose–Einstein condensate,” Phys. Rev. Lett. 87, 210402 (2001).
[CrossRef]

1999

I. Freund, “Critical point explosions in two-dimensional wave fields,” Opt. Commun. 159, 99–117 (1999).
[CrossRef]

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, E. A. Cornell, “Vortices in a Bose–Einstein condensate,” Phys. Rev. Lett. 83, 2498–2501 (1999).
[CrossRef]

1962

1953

H. S. Green, E. Wolf, “A scalar representation of electromagnetic fields,” Proc. Phys. Soc. A 66, 1129–1137 (1953).
[CrossRef]

Abo-Shaeer, J. R.

C. Raman, J. R. Abo-Shaeer, J. M. Vogels, K. Xu, W. Ketterle, “Vortex nucleation in a stirred Bose–Einstein condensate,” Phys. Rev. Lett. 87, 210402 (2001).
[CrossRef]

Allen, L. J.

L. J. Allen, H. M. L. Faulkner, M. P. Oxley, D. M. Paganin, “Phase retrieval and aberration correction in the presence of vortices in high-resolution transmission electron microscopy,” Ultramicroscopy 88, 85–97 (2001).
[CrossRef] [PubMed]

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, D. M. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
[CrossRef]

Anderson, B. P.

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, E. A. Cornell, “Vortices in a Bose–Einstein condensate,” Phys. Rev. Lett. 83, 2498–2501 (1999).
[CrossRef]

Bagnato, V. S.

E. A. L. Henn, J. A. Seman, E. R. F. Ramos, M. Caracanhas, P. Castilho, E. P. Olimpio, G. Roati, D. V. Magalhaes, K. M. F. Magalhaes, V. S. Bagnato, “Observation of vortex formation in an oscillation trapped Bose–Einstein condensate,” Phys. Rev. A 79, 043618 (2009).
[CrossRef]

Berry, M. V.

M. V. Berry, “Optical currents,” J. Opt. A Pure Appl. Opt. 11, 094001 (2009).
[CrossRef]

M. V. Berry, M. R. Dennis, “Topological events on wave dislocation lines: Birth and death of loops, and reconnection,” J. Phys. A Math. Theor. 40, 65–74 (2007).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 7 (Cambridge University, 1999).
[CrossRef]

Bruning, J. H.

H. Schreiber, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1992).

Caracanhas, M.

E. A. L. Henn, J. A. Seman, E. R. F. Ramos, M. Caracanhas, P. Castilho, E. P. Olimpio, G. Roati, D. V. Magalhaes, K. M. F. Magalhaes, V. S. Bagnato, “Observation of vortex formation in an oscillation trapped Bose–Einstein condensate,” Phys. Rev. A 79, 043618 (2009).
[CrossRef]

Castilho, P.

E. A. L. Henn, J. A. Seman, E. R. F. Ramos, M. Caracanhas, P. Castilho, E. P. Olimpio, G. Roati, D. V. Magalhaes, K. M. F. Magalhaes, V. S. Bagnato, “Observation of vortex formation in an oscillation trapped Bose–Einstein condensate,” Phys. Rev. A 79, 043618 (2009).
[CrossRef]

Cornell, E. A.

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, E. A. Cornell, “Vortices in a Bose–Einstein condensate,” Phys. Rev. Lett. 83, 2498–2501 (1999).
[CrossRef]

Dennis, M. R.

M. R. Dennis, K. O’Holleran, M. J. Padgett, “Singular optics: Optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[CrossRef]

K. O’Holleran, F. Flossman, M. R. Dennis, M. J. Padgett, “Methodology for imaging the 3D structure of singularities in scalar and vector optical fields,” J. Opt. A Pure Appl. Opt. 11, 094020 (2009).
[CrossRef]

K. O’Holleran, M. R. Dennis, F. Flossman, M. J. Padgett, “Fractality of light’s darkness,” Phys. Rev. Lett. 100, 053902 (2008).
[CrossRef]

M. V. Berry, M. R. Dennis, “Topological events on wave dislocation lines: Birth and death of loops, and reconnection,” J. Phys. A Math. Theor. 40, 65–74 (2007).
[CrossRef]

Faulkner, H. M. L.

F. Rothschild, M. J. Kitchen, H. M. L. Faulkner, D. M. Paganin, “Duality between phase vortices and Argand-plane caustics,” Opt. Commun. 285, 4141–4151 (2012).
[CrossRef]

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, D. M. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
[CrossRef]

L. J. Allen, H. M. L. Faulkner, M. P. Oxley, D. M. Paganin, “Phase retrieval and aberration correction in the presence of vortices in high-resolution transmission electron microscopy,” Ultramicroscopy 88, 85–97 (2001).
[CrossRef] [PubMed]

Flossman, F.

K. O’Holleran, F. Flossman, M. R. Dennis, M. J. Padgett, “Methodology for imaging the 3D structure of singularities in scalar and vector optical fields,” J. Opt. A Pure Appl. Opt. 11, 094020 (2009).
[CrossRef]

K. O’Holleran, M. R. Dennis, F. Flossman, M. J. Padgett, “Fractality of light’s darkness,” Phys. Rev. Lett. 100, 053902 (2008).
[CrossRef]

Freund, I.

I. Freund, “Critical point explosions in two-dimensional wave fields,” Opt. Commun. 159, 99–117 (1999).
[CrossRef]

Gbur, G.

Green, H. S.

H. S. Green, E. Wolf, “A scalar representation of electromagnetic fields,” Proc. Phys. Soc. A 66, 1129–1137 (1953).
[CrossRef]

Haljan, P. C.

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, E. A. Cornell, “Vortices in a Bose–Einstein condensate,” Phys. Rev. Lett. 83, 2498–2501 (1999).
[CrossRef]

Hall, D. S.

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, E. A. Cornell, “Vortices in a Bose–Einstein condensate,” Phys. Rev. Lett. 83, 2498–2501 (1999).
[CrossRef]

Henn, E. A. L.

E. A. L. Henn, J. A. Seman, E. R. F. Ramos, M. Caracanhas, P. Castilho, E. P. Olimpio, G. Roati, D. V. Magalhaes, K. M. F. Magalhaes, V. S. Bagnato, “Observation of vortex formation in an oscillation trapped Bose–Einstein condensate,” Phys. Rev. A 79, 043618 (2009).
[CrossRef]

Keller, J. B.

Ketterle, W.

C. Raman, J. R. Abo-Shaeer, J. M. Vogels, K. Xu, W. Ketterle, “Vortex nucleation in a stirred Bose–Einstein condensate,” Phys. Rev. Lett. 87, 210402 (2001).
[CrossRef]

Kitchen, M. J.

F. Rothschild, M. J. Kitchen, H. M. L. Faulkner, D. M. Paganin, “Duality between phase vortices and Argand-plane caustics,” Opt. Commun. 285, 4141–4151 (2012).
[CrossRef]

M. J. Kitchen, D. M. Paganin, R. A. Lewis, N. Yagi, K. Uesugi, S. T. Mudie, “On the origin of speckle in x-ray phase contrast images of lung issue,” Phys. Med. Biol. 49, 4335–4348 (2004).
[CrossRef] [PubMed]

Kolar, I.

I. Kolar, J. Slovak, P. W. Michor, Natural Operations in Differential Geometry (Springer, 1993).
[CrossRef]

Lewis, R. A.

M. J. Kitchen, D. M. Paganin, R. A. Lewis, N. Yagi, K. Uesugi, S. T. Mudie, “On the origin of speckle in x-ray phase contrast images of lung issue,” Phys. Med. Biol. 49, 4335–4348 (2004).
[CrossRef] [PubMed]

Liu, P.

P. Liu, H. Yang, J. Rong, G. Wang, Y. Yan, “Coherence vortex evolution of partially coherent vortex beams in the focal region,” Opt. Laser Technol. 42, 99–104 (2010).
[CrossRef]

Magalhaes, D. V.

E. A. L. Henn, J. A. Seman, E. R. F. Ramos, M. Caracanhas, P. Castilho, E. P. Olimpio, G. Roati, D. V. Magalhaes, K. M. F. Magalhaes, V. S. Bagnato, “Observation of vortex formation in an oscillation trapped Bose–Einstein condensate,” Phys. Rev. A 79, 043618 (2009).
[CrossRef]

Magalhaes, K. M. F.

E. A. L. Henn, J. A. Seman, E. R. F. Ramos, M. Caracanhas, P. Castilho, E. P. Olimpio, G. Roati, D. V. Magalhaes, K. M. F. Magalhaes, V. S. Bagnato, “Observation of vortex formation in an oscillation trapped Bose–Einstein condensate,” Phys. Rev. A 79, 043618 (2009).
[CrossRef]

Marasinghe, M. L.

Matthews, M. R.

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, E. A. Cornell, “Vortices in a Bose–Einstein condensate,” Phys. Rev. Lett. 83, 2498–2501 (1999).
[CrossRef]

Michor, P. W.

I. Kolar, J. Slovak, P. W. Michor, Natural Operations in Differential Geometry (Springer, 1993).
[CrossRef]

Morgan, K. S.

Mudie, S. T.

M. J. Kitchen, D. M. Paganin, R. A. Lewis, N. Yagi, K. Uesugi, S. T. Mudie, “On the origin of speckle in x-ray phase contrast images of lung issue,” Phys. Med. Biol. 49, 4335–4348 (2004).
[CrossRef] [PubMed]

Nugent, K. A.

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, D. M. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
[CrossRef]

Nye, J. F.

J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics, 1999).

O’Holleran, K.

M. R. Dennis, K. O’Holleran, M. J. Padgett, “Singular optics: Optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[CrossRef]

K. O’Holleran, F. Flossman, M. R. Dennis, M. J. Padgett, “Methodology for imaging the 3D structure of singularities in scalar and vector optical fields,” J. Opt. A Pure Appl. Opt. 11, 094020 (2009).
[CrossRef]

K. O’Holleran, M. R. Dennis, F. Flossman, M. J. Padgett, “Fractality of light’s darkness,” Phys. Rev. Lett. 100, 053902 (2008).
[CrossRef]

Olimpio, E. P.

E. A. L. Henn, J. A. Seman, E. R. F. Ramos, M. Caracanhas, P. Castilho, E. P. Olimpio, G. Roati, D. V. Magalhaes, K. M. F. Magalhaes, V. S. Bagnato, “Observation of vortex formation in an oscillation trapped Bose–Einstein condensate,” Phys. Rev. A 79, 043618 (2009).
[CrossRef]

Oxley, M. P.

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, D. M. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
[CrossRef]

L. J. Allen, H. M. L. Faulkner, M. P. Oxley, D. M. Paganin, “Phase retrieval and aberration correction in the presence of vortices in high-resolution transmission electron microscopy,” Ultramicroscopy 88, 85–97 (2001).
[CrossRef] [PubMed]

Padgett, M. J.

K. O’Holleran, F. Flossman, M. R. Dennis, M. J. Padgett, “Methodology for imaging the 3D structure of singularities in scalar and vector optical fields,” J. Opt. A Pure Appl. Opt. 11, 094020 (2009).
[CrossRef]

M. R. Dennis, K. O’Holleran, M. J. Padgett, “Singular optics: Optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[CrossRef]

K. O’Holleran, M. R. Dennis, F. Flossman, M. J. Padgett, “Fractality of light’s darkness,” Phys. Rev. Lett. 100, 053902 (2008).
[CrossRef]

Paganin, D. M.

F. Rothschild, M. J. Kitchen, H. M. L. Faulkner, D. M. Paganin, “Duality between phase vortices and Argand-plane caustics,” Opt. Commun. 285, 4141–4151 (2012).
[CrossRef]

M. L. Marasinghe, D. M. Paganin, M. Premaratne, “Coherence-vortex lattice formed via Mie scattering of partially coherent light by several dielectric nanospheres,” Opt. Lett. 36, 936–938 (2011).
[CrossRef] [PubMed]

K. S. Morgan, K. K. W. Siu, D. M. Paganin, “The projection approximation and edge contrast for x-ray propagation-based phase contrast imaging of a cylindrical edge,” Opt. Express 18, 9865–9878 (2010).
[CrossRef] [PubMed]

M. J. Kitchen, D. M. Paganin, R. A. Lewis, N. Yagi, K. Uesugi, S. T. Mudie, “On the origin of speckle in x-ray phase contrast images of lung issue,” Phys. Med. Biol. 49, 4335–4348 (2004).
[CrossRef] [PubMed]

L. J. Allen, H. M. L. Faulkner, M. P. Oxley, D. M. Paganin, “Phase retrieval and aberration correction in the presence of vortices in high-resolution transmission electron microscopy,” Ultramicroscopy 88, 85–97 (2001).
[CrossRef] [PubMed]

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, D. M. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
[CrossRef]

Premaratne, M.

Raman, C.

C. Raman, J. R. Abo-Shaeer, J. M. Vogels, K. Xu, W. Ketterle, “Vortex nucleation in a stirred Bose–Einstein condensate,” Phys. Rev. Lett. 87, 210402 (2001).
[CrossRef]

Ramos, E. R. F.

E. A. L. Henn, J. A. Seman, E. R. F. Ramos, M. Caracanhas, P. Castilho, E. P. Olimpio, G. Roati, D. V. Magalhaes, K. M. F. Magalhaes, V. S. Bagnato, “Observation of vortex formation in an oscillation trapped Bose–Einstein condensate,” Phys. Rev. A 79, 043618 (2009).
[CrossRef]

Roati, G.

E. A. L. Henn, J. A. Seman, E. R. F. Ramos, M. Caracanhas, P. Castilho, E. P. Olimpio, G. Roati, D. V. Magalhaes, K. M. F. Magalhaes, V. S. Bagnato, “Observation of vortex formation in an oscillation trapped Bose–Einstein condensate,” Phys. Rev. A 79, 043618 (2009).
[CrossRef]

Rong, J.

P. Liu, H. Yang, J. Rong, G. Wang, Y. Yan, “Coherence vortex evolution of partially coherent vortex beams in the focal region,” Opt. Laser Technol. 42, 99–104 (2010).
[CrossRef]

Rothschild, F.

F. Rothschild, M. J. Kitchen, H. M. L. Faulkner, D. M. Paganin, “Duality between phase vortices and Argand-plane caustics,” Opt. Commun. 285, 4141–4151 (2012).
[CrossRef]

Schouten, H. F.

Schreiber, H.

H. Schreiber, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1992).

Seman, J. A.

E. A. L. Henn, J. A. Seman, E. R. F. Ramos, M. Caracanhas, P. Castilho, E. P. Olimpio, G. Roati, D. V. Magalhaes, K. M. F. Magalhaes, V. S. Bagnato, “Observation of vortex formation in an oscillation trapped Bose–Einstein condensate,” Phys. Rev. A 79, 043618 (2009).
[CrossRef]

Siu, K. K. W.

Slovak, J.

I. Kolar, J. Slovak, P. W. Michor, Natural Operations in Differential Geometry (Springer, 1993).
[CrossRef]

Takeda, M.

W. Wang, M. Takeda, “Coherence current, coherence vortex and the conservation law of coherence,” Phys. Rev. Lett. 69, 223904 (2006).
[CrossRef]

Uesugi, K.

M. J. Kitchen, D. M. Paganin, R. A. Lewis, N. Yagi, K. Uesugi, S. T. Mudie, “On the origin of speckle in x-ray phase contrast images of lung issue,” Phys. Med. Biol. 49, 4335–4348 (2004).
[CrossRef] [PubMed]

Visser, T. D.

Vogels, J. M.

C. Raman, J. R. Abo-Shaeer, J. M. Vogels, K. Xu, W. Ketterle, “Vortex nucleation in a stirred Bose–Einstein condensate,” Phys. Rev. Lett. 87, 210402 (2001).
[CrossRef]

Wang, G.

P. Liu, H. Yang, J. Rong, G. Wang, Y. Yan, “Coherence vortex evolution of partially coherent vortex beams in the focal region,” Opt. Laser Technol. 42, 99–104 (2010).
[CrossRef]

Wang, W.

W. Wang, M. Takeda, “Coherence current, coherence vortex and the conservation law of coherence,” Phys. Rev. Lett. 69, 223904 (2006).
[CrossRef]

Wieman, C. E.

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, E. A. Cornell, “Vortices in a Bose–Einstein condensate,” Phys. Rev. Lett. 83, 2498–2501 (1999).
[CrossRef]

Wolf, E.

H. F. Schouten, G. Gbur, T. D. Visser, E. Wolf, “Phase singularities of the coherence function in Young’s interference pattern,” Opt. Lett. 28, 968–970 (2003).
[CrossRef] [PubMed]

H. S. Green, E. Wolf, “A scalar representation of electromagnetic fields,” Proc. Phys. Soc. A 66, 1129–1137 (1953).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 7 (Cambridge University, 1999).
[CrossRef]

Xu, K.

C. Raman, J. R. Abo-Shaeer, J. M. Vogels, K. Xu, W. Ketterle, “Vortex nucleation in a stirred Bose–Einstein condensate,” Phys. Rev. Lett. 87, 210402 (2001).
[CrossRef]

Yagi, N.

M. J. Kitchen, D. M. Paganin, R. A. Lewis, N. Yagi, K. Uesugi, S. T. Mudie, “On the origin of speckle in x-ray phase contrast images of lung issue,” Phys. Med. Biol. 49, 4335–4348 (2004).
[CrossRef] [PubMed]

Yan, Y.

P. Liu, H. Yang, J. Rong, G. Wang, Y. Yan, “Coherence vortex evolution of partially coherent vortex beams in the focal region,” Opt. Laser Technol. 42, 99–104 (2010).
[CrossRef]

Yang, H.

P. Liu, H. Yang, J. Rong, G. Wang, Y. Yan, “Coherence vortex evolution of partially coherent vortex beams in the focal region,” Opt. Laser Technol. 42, 99–104 (2010).
[CrossRef]

J. Opt. A Pure Appl. Opt.

K. O’Holleran, F. Flossman, M. R. Dennis, M. J. Padgett, “Methodology for imaging the 3D structure of singularities in scalar and vector optical fields,” J. Opt. A Pure Appl. Opt. 11, 094020 (2009).
[CrossRef]

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[CrossRef]

J. Opt. Soc. Am.

J. Phys. A Math. Theor.

M. V. Berry, M. R. Dennis, “Topological events on wave dislocation lines: Birth and death of loops, and reconnection,” J. Phys. A Math. Theor. 40, 65–74 (2007).
[CrossRef]

Opt. Commun.

F. Rothschild, M. J. Kitchen, H. M. L. Faulkner, D. M. Paganin, “Duality between phase vortices and Argand-plane caustics,” Opt. Commun. 285, 4141–4151 (2012).
[CrossRef]

I. Freund, “Critical point explosions in two-dimensional wave fields,” Opt. Commun. 159, 99–117 (1999).
[CrossRef]

G. Gbur, T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003).
[CrossRef]

Opt. Express

Opt. Laser Technol.

P. Liu, H. Yang, J. Rong, G. Wang, Y. Yan, “Coherence vortex evolution of partially coherent vortex beams in the focal region,” Opt. Laser Technol. 42, 99–104 (2010).
[CrossRef]

Opt. Lett.

Phys. Med. Biol.

M. J. Kitchen, D. M. Paganin, R. A. Lewis, N. Yagi, K. Uesugi, S. T. Mudie, “On the origin of speckle in x-ray phase contrast images of lung issue,” Phys. Med. Biol. 49, 4335–4348 (2004).
[CrossRef] [PubMed]

Phys. Rev. A

E. A. L. Henn, J. A. Seman, E. R. F. Ramos, M. Caracanhas, P. Castilho, E. P. Olimpio, G. Roati, D. V. Magalhaes, K. M. F. Magalhaes, V. S. Bagnato, “Observation of vortex formation in an oscillation trapped Bose–Einstein condensate,” Phys. Rev. A 79, 043618 (2009).
[CrossRef]

Phys. Rev. E

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, D. M. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
[CrossRef]

Phys. Rev. Lett.

W. Wang, M. Takeda, “Coherence current, coherence vortex and the conservation law of coherence,” Phys. Rev. Lett. 69, 223904 (2006).
[CrossRef]

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, E. A. Cornell, “Vortices in a Bose–Einstein condensate,” Phys. Rev. Lett. 83, 2498–2501 (1999).
[CrossRef]

C. Raman, J. R. Abo-Shaeer, J. M. Vogels, K. Xu, W. Ketterle, “Vortex nucleation in a stirred Bose–Einstein condensate,” Phys. Rev. Lett. 87, 210402 (2001).
[CrossRef]

K. O’Holleran, M. R. Dennis, F. Flossman, M. J. Padgett, “Fractality of light’s darkness,” Phys. Rev. Lett. 100, 053902 (2008).
[CrossRef]

Proc. Phys. Soc. A

H. S. Green, E. Wolf, “A scalar representation of electromagnetic fields,” Proc. Phys. Soc. A 66, 1129–1137 (1953).
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Prog. Opt.

M. R. Dennis, K. O’Holleran, M. J. Padgett, “Singular optics: Optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
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Ultramicroscopy

L. J. Allen, H. M. L. Faulkner, M. P. Oxley, D. M. Paganin, “Phase retrieval and aberration correction in the presence of vortices in high-resolution transmission electron microscopy,” Ultramicroscopy 88, 85–97 (2001).
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[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

(a) A Cornu spiral generated by taking the Argand-plane image of the complex field resulting when a plane wave diffracts from an infinite opaque edge upon which it is incident. (b) From Morgan et al. [3], the simulated Argand-plane plot for coherent x-ray scalar waves diffracted by a uniform infinitely-long dielectric cylinder. One lobe of the Cornu spiral evolves into a hypocycloid in the geometrical shadow of the cylinder. The dotted circle corresponds to the unscattered plane-wave and has an intensity of 1.

Fig. 2
Fig. 2

The Argand mapping of a complex number Ψ(x, y) for a given point (x, y). The axis in the Argand plane are parameterized by ΨR and ΨI. The magnitude and phase of the wavefunction Ψ(x, y) are represented by |Ψ| and ϕ, respectively.

Fig. 3
Fig. 3

Various visualizations of Ψ(x, y) in Eqn. 11 evaluated over −2 ≤ x, yz. (a) |Ψ|; (b) phase ϕ; (c) a plot of |Ωz|, which falls to zero over a unit circle, as predicted by Eqn. 12; (d) a plot of |Lz| and (e), the parametric plot of Eqn. 14, which is the image of Ψ(x, y) in the Argand plane and takes the form of the cross-section of an elliptic umbilic catastrophe [13].

Fig. 4
Fig. 4

Schematic of apparatus for creating and identifying optical vortices. The apparatus is a form of Mach-Zehnder interferometer and the waveplates allow the recording of interferograms using a CCD camera with different phase shifts of the reference beam, for analysis using phase-step interferometery. Optical vortices are created by the speckle interference from the ground glass screen and the distribution of vortices is controlled by the size of the downstream iris.

Fig. 5
Fig. 5

(a) The intensity and (b) the phase, which were reconstructed using a phase-stepping method applied to the speckle field created by the apparatus in Fig. 4. The broken rectangles indicate the ROI. These areas are enlarged to reveal a greater level of detail.

Fig. 6
Fig. 6

The Argand-plane mapping of the ROI of the data shown in Fig. 5. The mapping is polluted by a high frequency ripple, an example of which is enlarged.

Fig. 7
Fig. 7

The application of Fourier-space Gaussian filter to smooth the data in Fig. 5 and eliminate the caustic ripple in Fig. 6 so that the underlying structure of the Argand-mapping of the data can be revealed. (a) The power spectrum of the data in Fig. 5, as a function of the spatial frequencies (kx, ky) dual to (x, y). The diagonal streaking originates from parasitic scattering. This is related to the ripple behaviour seen in Fig. 6; (b) The Fourier-space Gaussian filter and (c) the result of the application of the Gaussian filter on the power spectrum, from where the diagonal streaking has been removed.

Fig. 8
Fig. 8

The result of applying the Gaussian filter of Fig. 7 and expanding the density of points in the region of interest by a multiple of 15 using cubic interpolation. The intensity (a) and phase (b) appear smoother. Minimum values are represented by black, maximum by white. Several features are indicated in the phase: A vortex α, antivortex β and branch cut γ. The vorticity Ω and angular momentum L are also calculated. |Ωz| < 0.5 × 10−7 is shown in (c) and |Lz| < 0.035 is shown in (d). The locations of vortices are indicated by circles: dark fill for vortices and light fill for antivortices. The field-of-view measures 1.48 mm×1.11 mm. The Argand mapping is shown in e), now with much finer sampling, allowing a clear image of vorticity singularities.

Fig. 9
Fig. 9

(a) The zeroes (|Ωz| < 0.5 × 10−7) of the vorticity, calculated by evaluating the Jacobian of the mapping , as seen in Fig. 8(a). The image measures 1.48 mm×1.11 mm. The locations of vortices and anti-vortices are indicated by filled and unfilled circles, respectively. The regions where the Jacobian is sufficiently close to zero are mapped to the Argand plane (b) to reveal an image containing singularities only. A vortex-antivortex dipole separated by one ‘Jacobian line’ is indicated by the red loop in a) and mapped to the Argand plane in (c) to form a fold caustic. Two antivortices that are not separated by any Jacobian lines are indicated by the green loop and mapped to the Argand plane in (d) to reveal a patch of space that loops around but does not fold. Two vortices separated by two Jacobian lines are indicated by the blue loop and are mapped to the Argand plane in (e) to reveal a fold caustic and a cusp.

Fig. 10
Fig. 10

These data were obtained by shifting the ground glass screen by 10 μm. The region of study measures 1.48 mm×1.11 mm. (a) The zeros of its vorticity, where a “Jacobian ellipse” is indicated by a rectangular loop; (b) the zeroes of the orbital angular momentum, where the area that enclosed the Jacobian ellipse in (a) is shown to be lacking in any equally interesting features of the angular momentum and (c) a vorticity singularity closely resembling the cross-section of an elliptic umbilic catastrophe, induced by the mapping of the Jacobian ellipse to the Argand plane.

Equations (15)

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Ψ ( s ) = C ( s ) + i S ( s )
C = b { [ 1 2 + 𝒞 ( s ) ] [ 1 2 + 𝒮 ( s ) ] }
S = b { [ 1 2 + 𝒞 ( s ) ] + [ 1 2 + 𝒮 ( s ) ] }
( Ψ ( x , y ) ) { Ψ R , Ψ I } .
Ψ ( x , y ) = A + B x + C y + D x y + E x 2 + F y 2 ,
J ( x , y ) = | x Ψ R y Ψ R x Ψ I y Ψ I | = ( B R + D R Y + 2 E R x ) ( C I + D I x + 2 F I y ) ( C R + D R x + 2 f R y ) ( B I + D I y + 2 E I x ) ,
Ω = × j = Im ( Ψ * × Ψ ) = Ψ R × Ψ I ,
Ω z = Ψ R x Ψ I y Ψ I x Ψ R y ,
L z = Im [ Ψ * ( x y Ψ y x Ψ ) ] .
Ψ ± = ( x x 0 ) ± i ( y y 0 ) ,
Ψ ( x , y ) = A + x + i y + x y + i 2 ( x 2 y 2 ) , A
J ( x , y ) = 1 x 2 y 2 = 0 .
{ x ( θ ) = cos θ , y ( θ ) = sin θ , 0 θ 2 π .
{ Ψ R ( θ ) = A R + cos θ + 1 2 sin 2 θ , Ψ I ( θ ) = A I + sin θ + 1 2 cos 2 θ , A , θ [ 0 , 2 π ] .
ϕ ( x , y ) = tan 1 [ I 4 I 2 I 1 I 3 ] ,

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