Abstract

The Cornu spiral is a graphical aid that has been used historically to evaluate Fresnel integrals. It is also the Argand-plane mapping of a monochromatic complex scalar plane wave diffracted by a hard edge. We have successfully reconstructed a Cornu spiral due to diffraction of hard x-rays from a piece of Kapton tape. Additionally, we have explored the generalisation of the Cornu spiral by observing the Argand-plane mapping of complex scalar electromagnetic fields diffracted by a cylinder and a sphere embedded within a cylinder.

© 2016 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  4. F. Rothschild, A. I. Bishop, M. J. Kitchen, and D. M. Paganin, “Argand-plane vorticity singularities in complex scalar optical fields: an experimental study using optical speckle,” Opt. Exp. 22, 6495–6510 (2014).
    [Crossref]
  5. H. S. Green and E. Wolf, “A scalar representation of electromagnetic fields,” Proc. Phys. Soc. A 66, 1129–1137 (1953).
    [Crossref]
  6. M. V. Berry and 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]
  7. M. V. Berry, “Optical currents,” J. Opt. A: Pure Appl. Opt. 11, 094001 (2009).
    [Crossref]
  8. M. J. Kitchen, D. M. Paganin, R. A. Lewis, N. Yagi, K. Uesugi, and 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]
  9. S. Goto, K. Takeshita, Y. Suzuki, H. Ohashi, Y. Asano, H. Kimura, T. Matsushita, N. Yagi, M. Isshiki, H. Yamazaki, Y. Yoneda, K. Umetani, and T. Ishikawa, “Construction and commissioning of a 215-m-long beamline at SPring-8,” Nucl. Instr. Meth. Phys. Res. A 467–468, 682–685 (2001).
    [Crossref]
  10. A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase contrast x-ray computed tomography for observing biological soft tissues,” Nat. Med. 2, 473–475 (1996).
    [Crossref] [PubMed]
  11. E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Krist. Tech. 15, 133–136 (1980).
    [Crossref]
  12. M. J. Kitchen, D. M. Paganin, K. Uesugi, B. J. Allison, R. A. Lewis, S. B. Hooper, and K. M. Pavlov, “Phase contrast image segmentation using a Laue analyser crystal,” Phys. Med. Biol. 56, 515–534 (2011).
    [Crossref] [PubMed]
  13. F. Pfeiffer, O. Bunk, C. David, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52, 6923 (2007).
    [Crossref] [PubMed]
  14. D. M. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
    [Crossref] [PubMed]
  15. F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9, 686–693 (1942).
    [Crossref]
  16. D. M. Paganin, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, Ya. I. Nesterets, and S. W. Wilkins, “X-ray omni microscopy,” J. Microsc. 214, 315–327 (2004).
    [Crossref] [PubMed]
  17. E. Gullikson, “X-ray interaction with matter,” http://henke.lbl.gov/optical_constants/
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    [Crossref]
  20. M. R. Dennis and J. B. Götte, “Beam shifts for pairs of plane waves,” J. Opt. 15, 104015 (2013).
    [Crossref]
  21. T. C. Petersen, A. I. Bishop, S. A. Eastwood, D. M. Paganin, K. S. Morgan, and M. J. Morgan, “Singularimetry of local phase gradients using vortex lattices and in-line holography,” Opt. Exp. 24, 2259–2272 (2016).
    [Crossref]
  22. O. V. Angelsky, M. P. Gorsky, P. P. Maksimyak, A. P. Maksimyak, S. G. Hanson, and C. Yu. Zenkova, “Investigation of optical currents in coherent and partially coherent vector fields,” Opt. Express 19, 660–672 (2011).
    [Crossref] [PubMed]
  23. K. M. Pavlov, D. M. Paganin, D. J. Vine, J. A. Schmalz, Y. Suzuki, K. Uesugi, A. Takeuchi, N. Yagi, A. Kharchenko, G. Blaj, J. Jakubek, M. Altissimo, and J. N. Clark, “Quantized hard x-ray phase vortices nucleated by aberrated nanolenses,” Phys. Rev. A 83, 013813 (2011).
    [Crossref]

2016 (1)

T. C. Petersen, A. I. Bishop, S. A. Eastwood, D. M. Paganin, K. S. Morgan, and M. J. Morgan, “Singularimetry of local phase gradients using vortex lattices and in-line holography,” Opt. Exp. 24, 2259–2272 (2016).
[Crossref]

2014 (1)

F. Rothschild, A. I. Bishop, M. J. Kitchen, and D. M. Paganin, “Argand-plane vorticity singularities in complex scalar optical fields: an experimental study using optical speckle,” Opt. Exp. 22, 6495–6510 (2014).
[Crossref]

2013 (1)

M. R. Dennis and J. B. Götte, “Beam shifts for pairs of plane waves,” J. Opt. 15, 104015 (2013).
[Crossref]

2012 (2)

M. R. Dennis and J. B. Götte, “Topological abberation of optical vortex beams: Determining dielectric interfaces by optical singularity shifts,” Phys. Rev. Lett. 109, 183903 (2012).
[Crossref]

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

2011 (3)

M. J. Kitchen, D. M. Paganin, K. Uesugi, B. J. Allison, R. A. Lewis, S. B. Hooper, and K. M. Pavlov, “Phase contrast image segmentation using a Laue analyser crystal,” Phys. Med. Biol. 56, 515–534 (2011).
[Crossref] [PubMed]

K. M. Pavlov, D. M. Paganin, D. J. Vine, J. A. Schmalz, Y. Suzuki, K. Uesugi, A. Takeuchi, N. Yagi, A. Kharchenko, G. Blaj, J. Jakubek, M. Altissimo, and J. N. Clark, “Quantized hard x-ray phase vortices nucleated by aberrated nanolenses,” Phys. Rev. A 83, 013813 (2011).
[Crossref]

O. V. Angelsky, M. P. Gorsky, P. P. Maksimyak, A. P. Maksimyak, S. G. Hanson, and C. Yu. Zenkova, “Investigation of optical currents in coherent and partially coherent vector fields,” Opt. Express 19, 660–672 (2011).
[Crossref] [PubMed]

2010 (1)

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

2009 (1)

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

2007 (2)

F. Pfeiffer, O. Bunk, C. David, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52, 6923 (2007).
[Crossref] [PubMed]

M. V. Berry and 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]

2004 (2)

D. M. Paganin, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, Ya. I. Nesterets, and S. W. Wilkins, “X-ray omni microscopy,” J. Microsc. 214, 315–327 (2004).
[Crossref] [PubMed]

M. J. Kitchen, D. M. Paganin, R. A. Lewis, N. Yagi, K. Uesugi, and 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]

2002 (1)

D. M. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
[Crossref] [PubMed]

2001 (1)

S. Goto, K. Takeshita, Y. Suzuki, H. Ohashi, Y. Asano, H. Kimura, T. Matsushita, N. Yagi, M. Isshiki, H. Yamazaki, Y. Yoneda, K. Umetani, and T. Ishikawa, “Construction and commissioning of a 215-m-long beamline at SPring-8,” Nucl. Instr. Meth. Phys. Res. A 467–468, 682–685 (2001).
[Crossref]

1996 (1)

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase contrast x-ray computed tomography for observing biological soft tissues,” Nat. Med. 2, 473–475 (1996).
[Crossref] [PubMed]

1980 (1)

E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Krist. Tech. 15, 133–136 (1980).
[Crossref]

1962 (1)

1953 (1)

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

1942 (1)

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9, 686–693 (1942).
[Crossref]

Allison, B. J.

M. J. Kitchen, D. M. Paganin, K. Uesugi, B. J. Allison, R. A. Lewis, S. B. Hooper, and K. M. Pavlov, “Phase contrast image segmentation using a Laue analyser crystal,” Phys. Med. Biol. 56, 515–534 (2011).
[Crossref] [PubMed]

Altissimo, M.

K. M. Pavlov, D. M. Paganin, D. J. Vine, J. A. Schmalz, Y. Suzuki, K. Uesugi, A. Takeuchi, N. Yagi, A. Kharchenko, G. Blaj, J. Jakubek, M. Altissimo, and J. N. Clark, “Quantized hard x-ray phase vortices nucleated by aberrated nanolenses,” Phys. Rev. A 83, 013813 (2011).
[Crossref]

Angelsky, O. V.

Asano, Y.

S. Goto, K. Takeshita, Y. Suzuki, H. Ohashi, Y. Asano, H. Kimura, T. Matsushita, N. Yagi, M. Isshiki, H. Yamazaki, Y. Yoneda, K. Umetani, and T. Ishikawa, “Construction and commissioning of a 215-m-long beamline at SPring-8,” Nucl. Instr. Meth. Phys. Res. A 467–468, 682–685 (2001).
[Crossref]

Bech, M.

F. Pfeiffer, O. Bunk, C. David, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52, 6923 (2007).
[Crossref] [PubMed]

Berry, M. V.

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

M. V. Berry and 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]

Bishop, A. I.

T. C. Petersen, A. I. Bishop, S. A. Eastwood, D. M. Paganin, K. S. Morgan, and M. J. Morgan, “Singularimetry of local phase gradients using vortex lattices and in-line holography,” Opt. Exp. 24, 2259–2272 (2016).
[Crossref]

F. Rothschild, A. I. Bishop, M. J. Kitchen, and D. M. Paganin, “Argand-plane vorticity singularities in complex scalar optical fields: an experimental study using optical speckle,” Opt. Exp. 22, 6495–6510 (2014).
[Crossref]

Blaj, G.

K. M. Pavlov, D. M. Paganin, D. J. Vine, J. A. Schmalz, Y. Suzuki, K. Uesugi, A. Takeuchi, N. Yagi, A. Kharchenko, G. Blaj, J. Jakubek, M. Altissimo, and J. N. Clark, “Quantized hard x-ray phase vortices nucleated by aberrated nanolenses,” Phys. Rev. A 83, 013813 (2011).
[Crossref]

Bravin, A.

F. Pfeiffer, O. Bunk, C. David, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52, 6923 (2007).
[Crossref] [PubMed]

Bunk, O.

F. Pfeiffer, O. Bunk, C. David, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52, 6923 (2007).
[Crossref] [PubMed]

Clark, J. N.

K. M. Pavlov, D. M. Paganin, D. J. Vine, J. A. Schmalz, Y. Suzuki, K. Uesugi, A. Takeuchi, N. Yagi, A. Kharchenko, G. Blaj, J. Jakubek, M. Altissimo, and J. N. Clark, “Quantized hard x-ray phase vortices nucleated by aberrated nanolenses,” Phys. Rev. A 83, 013813 (2011).
[Crossref]

Cloetens, P.

F. Pfeiffer, O. Bunk, C. David, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52, 6923 (2007).
[Crossref] [PubMed]

David, C.

F. Pfeiffer, O. Bunk, C. David, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52, 6923 (2007).
[Crossref] [PubMed]

Dennis, M. R.

M. R. Dennis and J. B. Götte, “Beam shifts for pairs of plane waves,” J. Opt. 15, 104015 (2013).
[Crossref]

M. R. Dennis and J. B. Götte, “Topological abberation of optical vortex beams: Determining dielectric interfaces by optical singularity shifts,” Phys. Rev. Lett. 109, 183903 (2012).
[Crossref]

M. V. Berry and 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]

Eastwood, S. A.

T. C. Petersen, A. I. Bishop, S. A. Eastwood, D. M. Paganin, K. S. Morgan, and M. J. Morgan, “Singularimetry of local phase gradients using vortex lattices and in-line holography,” Opt. Exp. 24, 2259–2272 (2016).
[Crossref]

Faulkner, H. M. L.

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

Förster, E.

E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Krist. Tech. 15, 133–136 (1980).
[Crossref]

Goetz, K.

E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Krist. Tech. 15, 133–136 (1980).
[Crossref]

Gorsky, M. P.

Goto, S.

S. Goto, K. Takeshita, Y. Suzuki, H. Ohashi, Y. Asano, H. Kimura, T. Matsushita, N. Yagi, M. Isshiki, H. Yamazaki, Y. Yoneda, K. Umetani, and T. Ishikawa, “Construction and commissioning of a 215-m-long beamline at SPring-8,” Nucl. Instr. Meth. Phys. Res. A 467–468, 682–685 (2001).
[Crossref]

Götte, J. B.

M. R. Dennis and J. B. Götte, “Beam shifts for pairs of plane waves,” J. Opt. 15, 104015 (2013).
[Crossref]

M. R. Dennis and J. B. Götte, “Topological abberation of optical vortex beams: Determining dielectric interfaces by optical singularity shifts,” Phys. Rev. Lett. 109, 183903 (2012).
[Crossref]

Green, H. S.

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

Gureyev, T. E.

D. M. Paganin, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, Ya. I. Nesterets, and S. W. Wilkins, “X-ray omni microscopy,” J. Microsc. 214, 315–327 (2004).
[Crossref] [PubMed]

D. M. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
[Crossref] [PubMed]

Hanson, S. G.

Hirano, K.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase contrast x-ray computed tomography for observing biological soft tissues,” Nat. Med. 2, 473–475 (1996).
[Crossref] [PubMed]

Hooper, S. B.

M. J. Kitchen, D. M. Paganin, K. Uesugi, B. J. Allison, R. A. Lewis, S. B. Hooper, and K. M. Pavlov, “Phase contrast image segmentation using a Laue analyser crystal,” Phys. Med. Biol. 56, 515–534 (2011).
[Crossref] [PubMed]

Ishikawa, T.

S. Goto, K. Takeshita, Y. Suzuki, H. Ohashi, Y. Asano, H. Kimura, T. Matsushita, N. Yagi, M. Isshiki, H. Yamazaki, Y. Yoneda, K. Umetani, and T. Ishikawa, “Construction and commissioning of a 215-m-long beamline at SPring-8,” Nucl. Instr. Meth. Phys. Res. A 467–468, 682–685 (2001).
[Crossref]

Isshiki, M.

S. Goto, K. Takeshita, Y. Suzuki, H. Ohashi, Y. Asano, H. Kimura, T. Matsushita, N. Yagi, M. Isshiki, H. Yamazaki, Y. Yoneda, K. Umetani, and T. Ishikawa, “Construction and commissioning of a 215-m-long beamline at SPring-8,” Nucl. Instr. Meth. Phys. Res. A 467–468, 682–685 (2001).
[Crossref]

Itai, Y.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase contrast x-ray computed tomography for observing biological soft tissues,” Nat. Med. 2, 473–475 (1996).
[Crossref] [PubMed]

Jakubek, J.

K. M. Pavlov, D. M. Paganin, D. J. Vine, J. A. Schmalz, Y. Suzuki, K. Uesugi, A. Takeuchi, N. Yagi, A. Kharchenko, G. Blaj, J. Jakubek, M. Altissimo, and J. N. Clark, “Quantized hard x-ray phase vortices nucleated by aberrated nanolenses,” Phys. Rev. A 83, 013813 (2011).
[Crossref]

Keller, J. B.

Kharchenko, A.

K. M. Pavlov, D. M. Paganin, D. J. Vine, J. A. Schmalz, Y. Suzuki, K. Uesugi, A. Takeuchi, N. Yagi, A. Kharchenko, G. Blaj, J. Jakubek, M. Altissimo, and J. N. Clark, “Quantized hard x-ray phase vortices nucleated by aberrated nanolenses,” Phys. Rev. A 83, 013813 (2011).
[Crossref]

Kimura, H.

S. Goto, K. Takeshita, Y. Suzuki, H. Ohashi, Y. Asano, H. Kimura, T. Matsushita, N. Yagi, M. Isshiki, H. Yamazaki, Y. Yoneda, K. Umetani, and T. Ishikawa, “Construction and commissioning of a 215-m-long beamline at SPring-8,” Nucl. Instr. Meth. Phys. Res. A 467–468, 682–685 (2001).
[Crossref]

Kitchen, M. J.

F. Rothschild, A. I. Bishop, M. J. Kitchen, and D. M. Paganin, “Argand-plane vorticity singularities in complex scalar optical fields: an experimental study using optical speckle,” Opt. Exp. 22, 6495–6510 (2014).
[Crossref]

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

M. J. Kitchen, D. M. Paganin, K. Uesugi, B. J. Allison, R. A. Lewis, S. B. Hooper, and K. M. Pavlov, “Phase contrast image segmentation using a Laue analyser crystal,” Phys. Med. Biol. 56, 515–534 (2011).
[Crossref] [PubMed]

M. J. Kitchen, D. M. Paganin, R. A. Lewis, N. Yagi, K. Uesugi, and 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]

Le Duc, G.

F. Pfeiffer, O. Bunk, C. David, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52, 6923 (2007).
[Crossref] [PubMed]

Lewis, R. A.

M. J. Kitchen, D. M. Paganin, K. Uesugi, B. J. Allison, R. A. Lewis, S. B. Hooper, and K. M. Pavlov, “Phase contrast image segmentation using a Laue analyser crystal,” Phys. Med. Biol. 56, 515–534 (2011).
[Crossref] [PubMed]

M. J. Kitchen, D. M. Paganin, R. A. Lewis, N. Yagi, K. Uesugi, and 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]

Maksimyak, A. P.

Maksimyak, P. P.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

Matsushita, T.

S. Goto, K. Takeshita, Y. Suzuki, H. Ohashi, Y. Asano, H. Kimura, T. Matsushita, N. Yagi, M. Isshiki, H. Yamazaki, Y. Yoneda, K. Umetani, and T. Ishikawa, “Construction and commissioning of a 215-m-long beamline at SPring-8,” Nucl. Instr. Meth. Phys. Res. A 467–468, 682–685 (2001).
[Crossref]

Mayo, S. C.

D. M. Paganin, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, Ya. I. Nesterets, and S. W. Wilkins, “X-ray omni microscopy,” J. Microsc. 214, 315–327 (2004).
[Crossref] [PubMed]

D. M. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
[Crossref] [PubMed]

Miller, P. R.

D. M. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
[Crossref] [PubMed]

Momose, A.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase contrast x-ray computed tomography for observing biological soft tissues,” Nat. Med. 2, 473–475 (1996).
[Crossref] [PubMed]

Morgan, K. S.

T. C. Petersen, A. I. Bishop, S. A. Eastwood, D. M. Paganin, K. S. Morgan, and M. J. Morgan, “Singularimetry of local phase gradients using vortex lattices and in-line holography,” Opt. Exp. 24, 2259–2272 (2016).
[Crossref]

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

Morgan, M. J.

T. C. Petersen, A. I. Bishop, S. A. Eastwood, D. M. Paganin, K. S. Morgan, and M. J. Morgan, “Singularimetry of local phase gradients using vortex lattices and in-line holography,” Opt. Exp. 24, 2259–2272 (2016).
[Crossref]

Mudie, S. T.

M. J. Kitchen, D. M. Paganin, R. A. Lewis, N. Yagi, K. Uesugi, and 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]

Nesterets, Ya. I.

D. M. Paganin, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, Ya. I. Nesterets, and S. W. Wilkins, “X-ray omni microscopy,” J. Microsc. 214, 315–327 (2004).
[Crossref] [PubMed]

Ohashi, H.

S. Goto, K. Takeshita, Y. Suzuki, H. Ohashi, Y. Asano, H. Kimura, T. Matsushita, N. Yagi, M. Isshiki, H. Yamazaki, Y. Yoneda, K. Umetani, and T. Ishikawa, “Construction and commissioning of a 215-m-long beamline at SPring-8,” Nucl. Instr. Meth. Phys. Res. A 467–468, 682–685 (2001).
[Crossref]

Paganin, D. M.

T. C. Petersen, A. I. Bishop, S. A. Eastwood, D. M. Paganin, K. S. Morgan, and M. J. Morgan, “Singularimetry of local phase gradients using vortex lattices and in-line holography,” Opt. Exp. 24, 2259–2272 (2016).
[Crossref]

F. Rothschild, A. I. Bishop, M. J. Kitchen, and D. M. Paganin, “Argand-plane vorticity singularities in complex scalar optical fields: an experimental study using optical speckle,” Opt. Exp. 22, 6495–6510 (2014).
[Crossref]

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

M. J. Kitchen, D. M. Paganin, K. Uesugi, B. J. Allison, R. A. Lewis, S. B. Hooper, and K. M. Pavlov, “Phase contrast image segmentation using a Laue analyser crystal,” Phys. Med. Biol. 56, 515–534 (2011).
[Crossref] [PubMed]

K. M. Pavlov, D. M. Paganin, D. J. Vine, J. A. Schmalz, Y. Suzuki, K. Uesugi, A. Takeuchi, N. Yagi, A. Kharchenko, G. Blaj, J. Jakubek, M. Altissimo, and J. N. Clark, “Quantized hard x-ray phase vortices nucleated by aberrated nanolenses,” Phys. Rev. A 83, 013813 (2011).
[Crossref]

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

M. J. Kitchen, D. M. Paganin, R. A. Lewis, N. Yagi, K. Uesugi, and 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]

D. M. Paganin, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, Ya. I. Nesterets, and S. W. Wilkins, “X-ray omni microscopy,” J. Microsc. 214, 315–327 (2004).
[Crossref] [PubMed]

D. M. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
[Crossref] [PubMed]

Pavlov, K. M.

M. J. Kitchen, D. M. Paganin, K. Uesugi, B. J. Allison, R. A. Lewis, S. B. Hooper, and K. M. Pavlov, “Phase contrast image segmentation using a Laue analyser crystal,” Phys. Med. Biol. 56, 515–534 (2011).
[Crossref] [PubMed]

K. M. Pavlov, D. M. Paganin, D. J. Vine, J. A. Schmalz, Y. Suzuki, K. Uesugi, A. Takeuchi, N. Yagi, A. Kharchenko, G. Blaj, J. Jakubek, M. Altissimo, and J. N. Clark, “Quantized hard x-ray phase vortices nucleated by aberrated nanolenses,” Phys. Rev. A 83, 013813 (2011).
[Crossref]

Petersen, T. C.

T. C. Petersen, A. I. Bishop, S. A. Eastwood, D. M. Paganin, K. S. Morgan, and M. J. Morgan, “Singularimetry of local phase gradients using vortex lattices and in-line holography,” Opt. Exp. 24, 2259–2272 (2016).
[Crossref]

Pfeiffer, F.

F. Pfeiffer, O. Bunk, C. David, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52, 6923 (2007).
[Crossref] [PubMed]

Rothschild, F.

F. Rothschild, A. I. Bishop, M. J. Kitchen, and D. M. Paganin, “Argand-plane vorticity singularities in complex scalar optical fields: an experimental study using optical speckle,” Opt. Exp. 22, 6495–6510 (2014).
[Crossref]

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

Schmalz, J. A.

K. M. Pavlov, D. M. Paganin, D. J. Vine, J. A. Schmalz, Y. Suzuki, K. Uesugi, A. Takeuchi, N. Yagi, A. Kharchenko, G. Blaj, J. Jakubek, M. Altissimo, and J. N. Clark, “Quantized hard x-ray phase vortices nucleated by aberrated nanolenses,” Phys. Rev. A 83, 013813 (2011).
[Crossref]

Siu, K. K. W.

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

Stevenson, A. W.

D. M. Paganin, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, Ya. I. Nesterets, and S. W. Wilkins, “X-ray omni microscopy,” J. Microsc. 214, 315–327 (2004).
[Crossref] [PubMed]

Suzuki, Y.

K. M. Pavlov, D. M. Paganin, D. J. Vine, J. A. Schmalz, Y. Suzuki, K. Uesugi, A. Takeuchi, N. Yagi, A. Kharchenko, G. Blaj, J. Jakubek, M. Altissimo, and J. N. Clark, “Quantized hard x-ray phase vortices nucleated by aberrated nanolenses,” Phys. Rev. A 83, 013813 (2011).
[Crossref]

S. Goto, K. Takeshita, Y. Suzuki, H. Ohashi, Y. Asano, H. Kimura, T. Matsushita, N. Yagi, M. Isshiki, H. Yamazaki, Y. Yoneda, K. Umetani, and T. Ishikawa, “Construction and commissioning of a 215-m-long beamline at SPring-8,” Nucl. Instr. Meth. Phys. Res. A 467–468, 682–685 (2001).
[Crossref]

Takeda, T.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase contrast x-ray computed tomography for observing biological soft tissues,” Nat. Med. 2, 473–475 (1996).
[Crossref] [PubMed]

Takeshita, K.

S. Goto, K. Takeshita, Y. Suzuki, H. Ohashi, Y. Asano, H. Kimura, T. Matsushita, N. Yagi, M. Isshiki, H. Yamazaki, Y. Yoneda, K. Umetani, and T. Ishikawa, “Construction and commissioning of a 215-m-long beamline at SPring-8,” Nucl. Instr. Meth. Phys. Res. A 467–468, 682–685 (2001).
[Crossref]

Takeuchi, A.

K. M. Pavlov, D. M. Paganin, D. J. Vine, J. A. Schmalz, Y. Suzuki, K. Uesugi, A. Takeuchi, N. Yagi, A. Kharchenko, G. Blaj, J. Jakubek, M. Altissimo, and J. N. Clark, “Quantized hard x-ray phase vortices nucleated by aberrated nanolenses,” Phys. Rev. A 83, 013813 (2011).
[Crossref]

Uesugi, K.

K. M. Pavlov, D. M. Paganin, D. J. Vine, J. A. Schmalz, Y. Suzuki, K. Uesugi, A. Takeuchi, N. Yagi, A. Kharchenko, G. Blaj, J. Jakubek, M. Altissimo, and J. N. Clark, “Quantized hard x-ray phase vortices nucleated by aberrated nanolenses,” Phys. Rev. A 83, 013813 (2011).
[Crossref]

M. J. Kitchen, D. M. Paganin, K. Uesugi, B. J. Allison, R. A. Lewis, S. B. Hooper, and K. M. Pavlov, “Phase contrast image segmentation using a Laue analyser crystal,” Phys. Med. Biol. 56, 515–534 (2011).
[Crossref] [PubMed]

M. J. Kitchen, D. M. Paganin, R. A. Lewis, N. Yagi, K. Uesugi, and 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]

Umetani, K.

S. Goto, K. Takeshita, Y. Suzuki, H. Ohashi, Y. Asano, H. Kimura, T. Matsushita, N. Yagi, M. Isshiki, H. Yamazaki, Y. Yoneda, K. Umetani, and T. Ishikawa, “Construction and commissioning of a 215-m-long beamline at SPring-8,” Nucl. Instr. Meth. Phys. Res. A 467–468, 682–685 (2001).
[Crossref]

Vine, D. J.

K. M. Pavlov, D. M. Paganin, D. J. Vine, J. A. Schmalz, Y. Suzuki, K. Uesugi, A. Takeuchi, N. Yagi, A. Kharchenko, G. Blaj, J. Jakubek, M. Altissimo, and J. N. Clark, “Quantized hard x-ray phase vortices nucleated by aberrated nanolenses,” Phys. Rev. A 83, 013813 (2011).
[Crossref]

Wilkins, S. W.

D. M. Paganin, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, Ya. I. Nesterets, and S. W. Wilkins, “X-ray omni microscopy,” J. Microsc. 214, 315–327 (2004).
[Crossref] [PubMed]

D. M. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
[Crossref] [PubMed]

Wolf, E.

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

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

Yagi, N.

K. M. Pavlov, D. M. Paganin, D. J. Vine, J. A. Schmalz, Y. Suzuki, K. Uesugi, A. Takeuchi, N. Yagi, A. Kharchenko, G. Blaj, J. Jakubek, M. Altissimo, and J. N. Clark, “Quantized hard x-ray phase vortices nucleated by aberrated nanolenses,” Phys. Rev. A 83, 013813 (2011).
[Crossref]

M. J. Kitchen, D. M. Paganin, R. A. Lewis, N. Yagi, K. Uesugi, and 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]

S. Goto, K. Takeshita, Y. Suzuki, H. Ohashi, Y. Asano, H. Kimura, T. Matsushita, N. Yagi, M. Isshiki, H. Yamazaki, Y. Yoneda, K. Umetani, and T. Ishikawa, “Construction and commissioning of a 215-m-long beamline at SPring-8,” Nucl. Instr. Meth. Phys. Res. A 467–468, 682–685 (2001).
[Crossref]

Yamazaki, H.

S. Goto, K. Takeshita, Y. Suzuki, H. Ohashi, Y. Asano, H. Kimura, T. Matsushita, N. Yagi, M. Isshiki, H. Yamazaki, Y. Yoneda, K. Umetani, and T. Ishikawa, “Construction and commissioning of a 215-m-long beamline at SPring-8,” Nucl. Instr. Meth. Phys. Res. A 467–468, 682–685 (2001).
[Crossref]

Yoneda, Y.

S. Goto, K. Takeshita, Y. Suzuki, H. Ohashi, Y. Asano, H. Kimura, T. Matsushita, N. Yagi, M. Isshiki, H. Yamazaki, Y. Yoneda, K. Umetani, and T. Ishikawa, “Construction and commissioning of a 215-m-long beamline at SPring-8,” Nucl. Instr. Meth. Phys. Res. A 467–468, 682–685 (2001).
[Crossref]

Zaumseil, P.

E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Krist. Tech. 15, 133–136 (1980).
[Crossref]

Zenkova, C. Yu.

Zernike, F.

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9, 686–693 (1942).
[Crossref]

J. Microsc. (2)

D. M. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
[Crossref] [PubMed]

D. M. Paganin, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, Ya. I. Nesterets, and S. W. Wilkins, “X-ray omni microscopy,” J. Microsc. 214, 315–327 (2004).
[Crossref] [PubMed]

J. Opt. (1)

M. R. Dennis and J. B. Götte, “Beam shifts for pairs of plane waves,” J. Opt. 15, 104015 (2013).
[Crossref]

J. Opt. A: Pure Appl. Opt. (1)

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

J. Opt. Soc. Am. (1)

J. Phys. A: Math. Theor. (1)

M. V. Berry and 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]

Krist. Tech. (1)

E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Krist. Tech. 15, 133–136 (1980).
[Crossref]

Nat. Med. (1)

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase contrast x-ray computed tomography for observing biological soft tissues,” Nat. Med. 2, 473–475 (1996).
[Crossref] [PubMed]

Nucl. Instr. Meth. Phys. Res. A (1)

S. Goto, K. Takeshita, Y. Suzuki, H. Ohashi, Y. Asano, H. Kimura, T. Matsushita, N. Yagi, M. Isshiki, H. Yamazaki, Y. Yoneda, K. Umetani, and T. Ishikawa, “Construction and commissioning of a 215-m-long beamline at SPring-8,” Nucl. Instr. Meth. Phys. Res. A 467–468, 682–685 (2001).
[Crossref]

Opt. Commun. (1)

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

Opt. Exp. (3)

F. Rothschild, A. I. Bishop, M. J. Kitchen, and D. M. Paganin, “Argand-plane vorticity singularities in complex scalar optical fields: an experimental study using optical speckle,” Opt. Exp. 22, 6495–6510 (2014).
[Crossref]

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

T. C. Petersen, A. I. Bishop, S. A. Eastwood, D. M. Paganin, K. S. Morgan, and M. J. Morgan, “Singularimetry of local phase gradients using vortex lattices and in-line holography,” Opt. Exp. 24, 2259–2272 (2016).
[Crossref]

Opt. Express (1)

Phys. Med. Biol. (3)

M. J. Kitchen, D. M. Paganin, R. A. Lewis, N. Yagi, K. Uesugi, and 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]

M. J. Kitchen, D. M. Paganin, K. Uesugi, B. J. Allison, R. A. Lewis, S. B. Hooper, and K. M. Pavlov, “Phase contrast image segmentation using a Laue analyser crystal,” Phys. Med. Biol. 56, 515–534 (2011).
[Crossref] [PubMed]

F. Pfeiffer, O. Bunk, C. David, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52, 6923 (2007).
[Crossref] [PubMed]

Phys. Rev. A (1)

K. M. Pavlov, D. M. Paganin, D. J. Vine, J. A. Schmalz, Y. Suzuki, K. Uesugi, A. Takeuchi, N. Yagi, A. Kharchenko, G. Blaj, J. Jakubek, M. Altissimo, and J. N. Clark, “Quantized hard x-ray phase vortices nucleated by aberrated nanolenses,” Phys. Rev. A 83, 013813 (2011).
[Crossref]

Phys. Rev. Lett. (1)

M. R. Dennis and J. B. Götte, “Topological abberation of optical vortex beams: Determining dielectric interfaces by optical singularity shifts,” Phys. Rev. Lett. 109, 183903 (2012).
[Crossref]

Physica (1)

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9, 686–693 (1942).
[Crossref]

Proc. Phys. Soc. A (1)

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

Other (2)

E. Gullikson, “X-ray interaction with matter,” http://henke.lbl.gov/optical_constants/

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

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Figures (11)

Fig. 1
Fig. 1 (a) A Cornu spiral resulting from the Argand-plane mapping of a monochromatic plane wave diffracted by a partially absorbing aluminium half-plane screen (the dotted line is a unit circle that represents the unscattered plane wave), and (b) a hypocycloid resulting from a monochromatic plane wave diffracted by a partially absorbing cylinder, described by Morgan et al. [2]. The diffracted light that emerges outside the geometric shadow of the cylinder is indicated in blue.
Fig. 2
Fig. 2 For a sphere lying in the object plane, the source of the diffracted rays is a ring around the outermost edge (red). To find the contribution of the diffracted rays through point P lying at a distance of z = Δ from the object plane, we find the shortest distance from P in the image plane to the ring source in the object plane, defined by R3. The diffracted rays through P are a result of a toroidal wavefront emanating from the point where R3 meets the ring.
Fig. 3
Fig. 3 The general set up of the experiments used to obtain results shown in Secs 3.2 – 3.4. The object to be imaged lies downstream of the x-ray source. The partially coherent light is then allowed to propagate further in order to observe edge enhancement via interference.
Fig. 4
Fig. 4 (a) A 1.10 cm × 0.24 cm image 2.0 m from a sample containing a piece of Kapton (polyimide) tape, taken using 24 keV x-rays. A profile plot of the raw image is shown in (b), wherein lies a single bright phase contrast fringe pair. Note as well the highly transmissive quality of the tape; (c) An image of the projected thickness of the tape, with a profile plot shown in (d), highlighting the low signal-to-noise ratio; (e) For comparison, a simulated image of the tape at 2.0 m is shown, along with a profile plot (f) showing the previously noted single bright fringe. The fringe in (f) appears brighter than the one in (d) as the pixel size was halved for the purpose of the simulation.
Fig. 5
Fig. 5 (a) Intensity of the field forward propagated to 8.0 m, where several fringes are visible; (b) Argand mapping of the field propagated to 8.0 m, where the spiral is obscured by noise; (c) Argand mapping of the field after being summed and averaged along the vertical axis, showing a clean Cornu spiral.
Fig. 6
Fig. 6 The evolution of the Cornu spiral with propagation distance: (a) 0.0 m, (b) 2.0 m and (c) 6.0 m. The spiral clings to the unit circle (dashed line) in each instance due to the low absorption of the Kapton tape. The field in (a) traces out an arc due to its smooth and continuous nature, rather than two discrete points representing either side of the tape.
Fig. 7
Fig. 7 (a) A 9.1 mm × 1.6 mm image taken 35 cm downstream of an aluminium rod illuminated by 24 keV x-rays with a plot profile (b) showing a single bright fringe at the edge of the highly absorbent cylinder; (c) a reconstruction of the projected thickness of the rod and profile in (d); (e) forward propagation to 35 cm for comparison (the intensity is shown) and profile in (f) showing a brighter central fringe due to the interpolation as part of the simulation.
Fig. 8
Fig. 8 Evolution of the Argand-plane mapping of the aluminium cylinder with propagation distance: (a) 35 cm, (b) 1.0 m and (c) 4.0 m. The spiral in each image represents the outside of the cylinder, where the intensity and phase oscillate against the unscattered plane wave. Within the cylinder, the varying phase causes the trace to move around the Argand plane as the thickness increases and the intensity falls to a minimum. The hypocycloid, as described in Fig. 1(b), is noticable in (c).
Fig. 9
Fig. 9 The projected thickness is forward propagated using x-rays with reduced energy. (a) 18 keV x-rays propagated to 4.0 m; (b) 16 keV x-rays at 4.0 m; (c) 15 keV x-rays at 6.0 m. The retrograde motion of the Argand trace can be seen in (c). The lower energy rays have a higher absorption coefficient and diffract more heavily.
Fig. 10
Fig. 10 (a) A 1 cm × 1 cm image taken 2.0 m from a bubble trapped in a perspex cylinder filled with agar illuminated by 24 keV x-rays. A plot profile (b) taken along the red line in (a) shows the three distinct regions: air, perspex and agar. The reconstruction uses μ and δ for the latter; (c) The projected thickness is shown, with plot profile (d) taken along the red line in (c), within the boundary of the cylinder. Noise and fringes are suppressed.
Fig. 11
Fig. 11 (a) Image of the cylinder propagated to a distance of 50 m; (b) profile across the mid-section of the bubble, which has a strong signal at this distance; (c) Argand mapping of the bubble; (d) Close-up image of (c), with behaviour seen in fully-realized two-dimensional vorticity singularities, such as the fold singularity indicated by α; (e) With energy halved to 12 keV, the x-rays diffract more strongly and are absorbed heavily, resulting in some hypocylodic behaviour seen on the left side of the image.

Equations (7)

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( Ψ ( x , y ) ) ( Ψ R , Ψ I ) ,
J ( x , y ) = | x Ψ R y Ψ R x Ψ I y Ψ I | .
Ω = × j = Im ( Ψ × Ψ ) = Ψ R × Ψ I ,
Ω z = Ψ R x Ψ I y Ψ I x Ψ R y ,
Ψ ( x , y ) = e i k z e μ T / 2 e i k δ T + A e i k R R ,
Ψ ( x , y ) = e i k z e μ T ( x ) / 2 e i k δ T ( x ) + A e i k R 1 R 1 + e i k R 2 R 2 ,
Ψ ( x , y ) = e i k z e μ T ( x , y ) / 2 e i k δ T ( x , y ) + A e i k R 1 R 1 + e i k R 2 R 2 + e i k R ˜ 3 R ˜ 3 ,

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