Abstract

We review the current work on x-ray phase vortices. We explain the role of an x-ray vortex in phase recovery and speculate on its possible applications in other fields of x-ray optical research. We present our theoretical understanding of the structure of phase vortices and test these predictions against experiment. We present experimental observations of phase vortices with charge greater than 3 and observe that their propagation appears to be consistent with our theoretical models.

© 2004 Optical Society of America

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  1. A. G. Peele, P. J. McMahon, D. Paterson, C. Q. Tran, A. P. Mancuso, K. A. Nugent, J. P. Hayes, E. Harvey, B. Lai, I. McNulty, “Observation of an x-ray vortex,” Opt. Lett. 27, 1752–1754 (2002).
    [CrossRef]
  2. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
    [CrossRef] [PubMed]
  3. M. V. Vasnetsov, I. G. Marienko, M. S. Soskin, “Self-restoration of an optical vortex,” JETP Lett. 71, 130–133 (2000).
    [CrossRef]
  4. M. Berry, “Paraxial beams of spinning light,” in International Conference on Singular Optics, M. S. Soskin, ed., Proc. SPIE3487, 6–11 (1998).
    [CrossRef]
  5. J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
    [CrossRef]
  6. M. S. Soskin, M. V. Vasnetsov, “Singular optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 2001), Vol. 42, pp. 220–276.
  7. D. Rozas, C. T. Law, G. A. Swartzlander, “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B 14, 3054–3065 (1997).
    [CrossRef]
  8. M. Harris, “Light-field fluctuations in space and time,” Contemp. Phys. 36, 215–233 (1995).
    [CrossRef]
  9. N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamayev, N. F. Pilipetskii, V. V. Shkukov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” Pis'ma Zh. Eksp. Teor. Fiz. 33, 206–210 (1981) [JETP Lett.33, 195–199 (1981)].
  10. Z. S. Sacks, D. Rozas, G. A. Swartzlander, “Holographic formation of optical-vortex filaments,” J. Opt. Soc. Am. B 15, 2226–2234 (1998).
    [CrossRef]
  11. H. He, M. E. J. Friese, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
    [CrossRef] [PubMed]
  12. S. Chávez-Cerda, G. S. McDonald, G. H. C. New, “Nondiffracting beams: travelling, standing, rotating, and spiral waves,” Opt. Commun. 123, 225–233 (1996).
    [CrossRef]
  13. Z. Bouchal, “Resistance of nondiffracting vortex beam against amplitude and phase perturbations,” Opt. Commun. 210, 155–164 (2002).
    [CrossRef]
  14. M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991).
    [CrossRef] [PubMed]
  15. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
    [CrossRef]
  16. N. R. Heckenberg, R. McDuff, C. P. Smith, A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
    [CrossRef] [PubMed]
  17. V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
    [CrossRef]
  18. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
    [CrossRef]
  19. G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, M. J. Padgett, “The generation of free-space Laguerre–Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127, 183–188 (1996).
    [CrossRef]
  20. G.-H. Kim, J.-H. Jeon, K.-H. Ko, H.-J. Moon, J.-H. Lee, J.-S. Chang, “Optical vortices produced with a nonspiral phase plate,” Appl. Opt. 36, 8614–8621 (1997).
    [CrossRef]
  21. I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
    [CrossRef]
  22. J. E. Curtis, B. A. Koss, D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
    [CrossRef]
  23. G. A. Swartzlander, “Peering into darkness with a vortex spatial filter,” Opt. Lett. 26, 497–499 (2001).
    [CrossRef]
  24. M. Harwit, “Photon orbital angular momentum in astrophysics,” Astrophys. J. 597, 1266–1270 (2003).
    [CrossRef]
  25. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
    [CrossRef] [PubMed]
  26. H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yao, J. Courtial, “Simplified measurement of the orbital angular momentum of single photons,” Opt. Commun. 223, 117–122 (2003).
    [CrossRef]
  27. P. J. McMahon, A. G. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480–1482 (2003).
    [CrossRef]
  28. J. W. Miao, P. Charalambous, J. Kirz, D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
    [CrossRef]
  29. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  30. R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: underlying theory,” Optik (Stuttgart) 61, 247–262 (1982).
  31. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [CrossRef] [PubMed]
  32. D. Sayre, H. N. Chapman, “X-ray microscopy,” Acta Crystallogr., Sect. A 51, 237–252 (1995).
    [CrossRef] [PubMed]
  33. J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419–1421 (2003).
    [CrossRef] [PubMed]
  34. I. K. Robinson, Department of Physics, University of Illinois at Urbana–Champaign, Urbana, Ill. 61801 (personal communication, 2003).
  35. K. A. Nugent, A. G. Peele, H. N. Chapman, A. P. Mancuso, “Unique phase recovery for nonperiodic objects,” Phys. Rev. Lett. 91, 203902 (2003).
    [CrossRef] [PubMed]
  36. J. H. Seldin, J. R. Fienup, “Numerical investigation of the uniqueness of phase retrieval,” J. Opt. Soc. Am. A 7, 412–427 (1990).
    [CrossRef]
  37. I. K. Robinson, F. Pfeiffer, I. A. Vartanyants, Y. Sun, Y. Xia, “Enhancement of coherent x-ray diffraction from nanocrystals by introduction of x-ray optics,” Opt. Express11, 2329–2334 (2003), http://www.opticsexpress.org .
    [CrossRef] [PubMed]
  38. D. Paganin, K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
    [CrossRef]
  39. T. E. Gureyev, A. Roberts, K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12, 1942–1946 (1995).
    [CrossRef]
  40. K. A. Nugent, “Wave field determination using three-dimensional intensity information,” Phys. Rev. Lett. 68, 2261–2264 (1992).
    [CrossRef] [PubMed]
  41. M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
    [CrossRef] [PubMed]
  42. C. Kurtsiefer, T. Pfau, J. Mlynek, “Measurement of the Wigner function of an ensemble of helium atoms,” Nature 386, 150–153 (1997).
    [CrossRef]
  43. C. Q. Tran, A. G. Peele, D. Paterson, A. Roberts, I. McNulty, K. A. Nugent, “Correlations in a synchrotron beam measured using phase-space tomography,” manuscript available from C. Q. Tran at tran@physics.unimelb.edu.au.
  44. F. Gori, M. Santarsiero, G. Guattari, “Coherence and the spatial distribution of intensity,” J. Opt. Soc. Am. A 10, 673–679 (1993).
    [CrossRef]
  45. A. G. Peele, K. A. Nugent, “X-ray vortex beams: a the-oretical analysis,” Opt. Express11, 2315–2322 (2003), http://www.opticsexpress.org .
    [CrossRef] [PubMed]
  46. I. V. Basistiy, M. S. Soskin, M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
    [CrossRef]
  47. E. C. Harvey, P. T. Rumsby, “Fabrication techniques and their application to produce novel micromachined structures and devices using excimer laser projection,” in Micromachining and Microfabrication Process Technology III, S.-C. Chang, S. W. Pang, eds., Proc. SPIE3223, 26–33 (1997).
  48. J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. 90, 074801 (2003).
    [PubMed]
  49. J. H. Hubbell, W. J. Veigele, E. A. Briggs, R. T. Brown, D. T. Cromer, R. J. Howerton, “Atomic form factors, incoherent scattering functions, and photon scattering cross sections,” J. Phys. Chem. Ref. Data 4, 471–538 (1975).
  50. A. Tartari, E. Casnati, C. Bonifazzi, C. Baraldi, “Molecular differential cross sections for x-ray coherent scattering in fat and polymethyl methacrylate,” Phys. Med. Biol. 42, 2551–2560 (1997).
  51. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).

2003

M. Harwit, “Photon orbital angular momentum in astrophysics,” Astrophys. J. 597, 1266–1270 (2003).
[CrossRef]

H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yao, J. Courtial, “Simplified measurement of the orbital angular momentum of single photons,” Opt. Commun. 223, 117–122 (2003).
[CrossRef]

P. J. McMahon, A. G. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480–1482 (2003).
[CrossRef]

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419–1421 (2003).
[CrossRef] [PubMed]

K. A. Nugent, A. G. Peele, H. N. Chapman, A. P. Mancuso, “Unique phase recovery for nonperiodic objects,” Phys. Rev. Lett. 91, 203902 (2003).
[CrossRef] [PubMed]

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. 90, 074801 (2003).
[PubMed]

2002

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

J. E. Curtis, B. A. Koss, D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
[CrossRef]

A. G. Peele, P. J. McMahon, D. Paterson, C. Q. Tran, A. P. Mancuso, K. A. Nugent, J. P. Hayes, E. Harvey, B. Lai, I. McNulty, “Observation of an x-ray vortex,” Opt. Lett. 27, 1752–1754 (2002).
[CrossRef]

Z. Bouchal, “Resistance of nondiffracting vortex beam against amplitude and phase perturbations,” Opt. Commun. 210, 155–164 (2002).
[CrossRef]

2001

2000

M. V. Vasnetsov, I. G. Marienko, M. S. Soskin, “Self-restoration of an optical vortex,” JETP Lett. 71, 130–133 (2000).
[CrossRef]

1999

J. W. Miao, P. Charalambous, J. Kirz, D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

1998

D. Paganin, K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[CrossRef]

Z. S. Sacks, D. Rozas, G. A. Swartzlander, “Holographic formation of optical-vortex filaments,” J. Opt. Soc. Am. B 15, 2226–2234 (1998).
[CrossRef]

1997

D. Rozas, C. T. Law, G. A. Swartzlander, “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B 14, 3054–3065 (1997).
[CrossRef]

G.-H. Kim, J.-H. Jeon, K.-H. Ko, H.-J. Moon, J.-H. Lee, J.-S. Chang, “Optical vortices produced with a nonspiral phase plate,” Appl. Opt. 36, 8614–8621 (1997).
[CrossRef]

A. Tartari, E. Casnati, C. Bonifazzi, C. Baraldi, “Molecular differential cross sections for x-ray coherent scattering in fat and polymethyl methacrylate,” Phys. Med. Biol. 42, 2551–2560 (1997).

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).

C. Kurtsiefer, T. Pfau, J. Mlynek, “Measurement of the Wigner function of an ensemble of helium atoms,” Nature 386, 150–153 (1997).
[CrossRef]

1996

S. Chávez-Cerda, G. S. McDonald, G. H. C. New, “Nondiffracting beams: travelling, standing, rotating, and spiral waves,” Opt. Commun. 123, 225–233 (1996).
[CrossRef]

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, M. J. Padgett, “The generation of free-space Laguerre–Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127, 183–188 (1996).
[CrossRef]

1995

D. Sayre, H. N. Chapman, “X-ray microscopy,” Acta Crystallogr., Sect. A 51, 237–252 (1995).
[CrossRef] [PubMed]

T. E. Gureyev, A. Roberts, K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12, 1942–1946 (1995).
[CrossRef]

M. Harris, “Light-field fluctuations in space and time,” Contemp. Phys. 36, 215–233 (1995).
[CrossRef]

H. He, M. E. J. Friese, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

I. V. Basistiy, M. S. Soskin, M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

1994

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

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

1993

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

F. Gori, M. Santarsiero, G. Guattari, “Coherence and the spatial distribution of intensity,” J. Opt. Soc. Am. A 10, 673–679 (1993).
[CrossRef]

1992

N. R. Heckenberg, R. McDuff, C. P. Smith, A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
[CrossRef] [PubMed]

V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

K. A. Nugent, “Wave field determination using three-dimensional intensity information,” Phys. Rev. Lett. 68, 2261–2264 (1992).
[CrossRef] [PubMed]

1991

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

1990

1982

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: underlying theory,” Optik (Stuttgart) 61, 247–262 (1982).

1981

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamayev, N. F. Pilipetskii, V. V. Shkukov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” Pis'ma Zh. Eksp. Teor. Fiz. 33, 206–210 (1981) [JETP Lett.33, 195–199 (1981)].

1978

1975

J. H. Hubbell, W. J. Veigele, E. A. Briggs, R. T. Brown, D. T. Cromer, R. J. Howerton, “Atomic form factors, incoherent scattering functions, and photon scattering cross sections,” J. Phys. Chem. Ref. Data 4, 471–538 (1975).

1974

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

1972

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

Allen, L.

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, M. J. Padgett, “The generation of free-space Laguerre–Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127, 183–188 (1996).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Baraldi, C.

A. Tartari, E. Casnati, C. Bonifazzi, C. Baraldi, “Molecular differential cross sections for x-ray coherent scattering in fat and polymethyl methacrylate,” Phys. Med. Biol. 42, 2551–2560 (1997).

Baranova, N. B.

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamayev, N. F. Pilipetskii, V. V. Shkukov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” Pis'ma Zh. Eksp. Teor. Fiz. 33, 206–210 (1981) [JETP Lett.33, 195–199 (1981)].

Barnett, S. M.

H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yao, J. Courtial, “Simplified measurement of the orbital angular momentum of single photons,” Opt. Commun. 223, 117–122 (2003).
[CrossRef]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Basistiy, I. V.

I. V. Basistiy, M. S. Soskin, M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

Bates, R. H. T.

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: underlying theory,” Optik (Stuttgart) 61, 247–262 (1982).

Battipede, F.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

Bazhenov, V. Yu.

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Beck, M.

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Beijersbergen, M. W.

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

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Berry, M.

M. Berry, “Paraxial beams of spinning light,” in International Conference on Singular Optics, M. S. Soskin, ed., Proc. SPIE3487, 6–11 (1998).
[CrossRef]

Berry, M. V.

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

Bonifazzi, C.

A. Tartari, E. Casnati, C. Bonifazzi, C. Baraldi, “Molecular differential cross sections for x-ray coherent scattering in fat and polymethyl methacrylate,” Phys. Med. Biol. 42, 2551–2560 (1997).

Bouchal, Z.

Z. Bouchal, “Resistance of nondiffracting vortex beam against amplitude and phase perturbations,” Opt. Commun. 210, 155–164 (2002).
[CrossRef]

Brambilla, M.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

Briggs, E. A.

J. H. Hubbell, W. J. Veigele, E. A. Briggs, R. T. Brown, D. T. Cromer, R. J. Howerton, “Atomic form factors, incoherent scattering functions, and photon scattering cross sections,” J. Phys. Chem. Ref. Data 4, 471–538 (1975).

Brown, R. T.

J. H. Hubbell, W. J. Veigele, E. A. Briggs, R. T. Brown, D. T. Cromer, R. J. Howerton, “Atomic form factors, incoherent scattering functions, and photon scattering cross sections,” J. Phys. Chem. Ref. Data 4, 471–538 (1975).

Cai, Z.

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. 90, 074801 (2003).
[PubMed]

Casnati, E.

A. Tartari, E. Casnati, C. Bonifazzi, C. Baraldi, “Molecular differential cross sections for x-ray coherent scattering in fat and polymethyl methacrylate,” Phys. Med. Biol. 42, 2551–2560 (1997).

Chang, J.-S.

Chantler, C. T.

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. 90, 074801 (2003).
[PubMed]

Chapman, H. N.

K. A. Nugent, A. G. Peele, H. N. Chapman, A. P. Mancuso, “Unique phase recovery for nonperiodic objects,” Phys. Rev. Lett. 91, 203902 (2003).
[CrossRef] [PubMed]

D. Sayre, H. N. Chapman, “X-ray microscopy,” Acta Crystallogr., Sect. A 51, 237–252 (1995).
[CrossRef] [PubMed]

Charalambous, P.

J. W. Miao, P. Charalambous, J. Kirz, D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

Chávez-Cerda, S.

S. Chávez-Cerda, G. S. McDonald, G. H. C. New, “Nondiffracting beams: travelling, standing, rotating, and spiral waves,” Opt. Commun. 123, 225–233 (1996).
[CrossRef]

Coerwinkel, R. P. C.

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

Courtial, J.

H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yao, J. Courtial, “Simplified measurement of the orbital angular momentum of single photons,” Opt. Commun. 223, 117–122 (2003).
[CrossRef]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Cromer, D. T.

J. H. Hubbell, W. J. Veigele, E. A. Briggs, R. T. Brown, D. T. Cromer, R. J. Howerton, “Atomic form factors, incoherent scattering functions, and photon scattering cross sections,” J. Phys. Chem. Ref. Data 4, 471–538 (1975).

Curtis, J. E.

J. E. Curtis, B. A. Koss, D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
[CrossRef]

Fienup, J. R.

Franke-Arnold, S.

H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yao, J. Courtial, “Simplified measurement of the orbital angular momentum of single photons,” Opt. Commun. 223, 117–122 (2003).
[CrossRef]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Friese, M. E. J.

H. He, M. E. J. Friese, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

Gao, M.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419–1421 (2003).
[CrossRef] [PubMed]

Gerchberg, R. W.

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

Gori, F.

Gorshkov, V. N.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).

Grier, D. G.

J. E. Curtis, B. A. Koss, D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
[CrossRef]

Guattari, G.

Gureyev, T. E.

Harris, M.

M. Harris, “Light-field fluctuations in space and time,” Contemp. Phys. 36, 215–233 (1995).
[CrossRef]

Harvey, E.

Harvey, E. C.

E. C. Harvey, P. T. Rumsby, “Fabrication techniques and their application to produce novel micromachined structures and devices using excimer laser projection,” in Micromachining and Microfabrication Process Technology III, S.-C. Chang, S. W. Pang, eds., Proc. SPIE3223, 26–33 (1997).

Harwit, M.

M. Harwit, “Photon orbital angular momentum in astrophysics,” Astrophys. J. 597, 1266–1270 (2003).
[CrossRef]

Hayes, J. P.

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

Heckenberg, N. R.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).

H. He, M. E. J. Friese, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

N. R. Heckenberg, R. McDuff, C. P. Smith, A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
[CrossRef] [PubMed]

Howerton, R. J.

J. H. Hubbell, W. J. Veigele, E. A. Briggs, R. T. Brown, D. T. Cromer, R. J. Howerton, “Atomic form factors, incoherent scattering functions, and photon scattering cross sections,” J. Phys. Chem. Ref. Data 4, 471–538 (1975).

Hubbell, J. H.

J. H. Hubbell, W. J. Veigele, E. A. Briggs, R. T. Brown, D. T. Cromer, R. J. Howerton, “Atomic form factors, incoherent scattering functions, and photon scattering cross sections,” J. Phys. Chem. Ref. Data 4, 471–538 (1975).

Jeon, J.-H.

Kim, G.-H.

Kirz, J.

J. W. Miao, P. Charalambous, J. Kirz, D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

Ko, K.-H.

Koss, B. A.

J. E. Curtis, B. A. Koss, D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
[CrossRef]

Kristensen, M.

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

Kurtsiefer, C.

C. Kurtsiefer, T. Pfau, J. Mlynek, “Measurement of the Wigner function of an ensemble of helium atoms,” Nature 386, 150–153 (1997).
[CrossRef]

Lai, B.

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. 90, 074801 (2003).
[PubMed]

A. G. Peele, P. J. McMahon, D. Paterson, C. Q. Tran, A. P. Mancuso, K. A. Nugent, J. P. Hayes, E. Harvey, B. Lai, I. McNulty, “Observation of an x-ray vortex,” Opt. Lett. 27, 1752–1754 (2002).
[CrossRef]

Law, C. T.

Leach, J.

H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yao, J. Courtial, “Simplified measurement of the orbital angular momentum of single photons,” Opt. Commun. 223, 117–122 (2003).
[CrossRef]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Lee, J.-H.

Lin, J. J. A.

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. 90, 074801 (2003).
[PubMed]

Lugiato, L. A.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

Malos, J. T.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).

Mamayev, A. V.

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamayev, N. F. Pilipetskii, V. V. Shkukov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” Pis'ma Zh. Eksp. Teor. Fiz. 33, 206–210 (1981) [JETP Lett.33, 195–199 (1981)].

Mancini, D. C.

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. 90, 074801 (2003).
[PubMed]

Mancuso, A. P.

Marienko, I. G.

M. V. Vasnetsov, I. G. Marienko, M. S. Soskin, “Self-restoration of an optical vortex,” JETP Lett. 71, 130–133 (2000).
[CrossRef]

McAlister, D. F.

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

McDonald, G. S.

S. Chávez-Cerda, G. S. McDonald, G. H. C. New, “Nondiffracting beams: travelling, standing, rotating, and spiral waves,” Opt. Commun. 123, 225–233 (1996).
[CrossRef]

McDuff, R.

McMahon, P. J.

P. J. McMahon, A. G. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480–1482 (2003).
[CrossRef]

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. 90, 074801 (2003).
[PubMed]

A. G. Peele, P. J. McMahon, D. Paterson, C. Q. Tran, A. P. Mancuso, K. A. Nugent, J. P. Hayes, E. Harvey, B. Lai, I. McNulty, “Observation of an x-ray vortex,” Opt. Lett. 27, 1752–1754 (2002).
[CrossRef]

McNulty, I.

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. 90, 074801 (2003).
[PubMed]

A. G. Peele, P. J. McMahon, D. Paterson, C. Q. Tran, A. P. Mancuso, K. A. Nugent, J. P. Hayes, E. Harvey, B. Lai, I. McNulty, “Observation of an x-ray vortex,” Opt. Lett. 27, 1752–1754 (2002).
[CrossRef]

C. Q. Tran, A. G. Peele, D. Paterson, A. Roberts, I. McNulty, K. A. Nugent, “Correlations in a synchrotron beam measured using phase-space tomography,” manuscript available from C. Q. Tran at tran@physics.unimelb.edu.au.

Miao, J. W.

J. W. Miao, P. Charalambous, J. Kirz, D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

Mlynek, J.

C. Kurtsiefer, T. Pfau, J. Mlynek, “Measurement of the Wigner function of an ensemble of helium atoms,” Nature 386, 150–153 (1997).
[CrossRef]

Moldovan, N.

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. 90, 074801 (2003).
[PubMed]

Moon, H.-J.

Nagahara, L. A.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419–1421 (2003).
[CrossRef] [PubMed]

New, G. H. C.

S. Chávez-Cerda, G. S. McDonald, G. H. C. New, “Nondiffracting beams: travelling, standing, rotating, and spiral waves,” Opt. Commun. 123, 225–233 (1996).
[CrossRef]

Nugent, K. A.

K. A. Nugent, A. G. Peele, H. N. Chapman, A. P. Mancuso, “Unique phase recovery for nonperiodic objects,” Phys. Rev. Lett. 91, 203902 (2003).
[CrossRef] [PubMed]

P. J. McMahon, A. G. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480–1482 (2003).
[CrossRef]

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. 90, 074801 (2003).
[PubMed]

A. G. Peele, P. J. McMahon, D. Paterson, C. Q. Tran, A. P. Mancuso, K. A. Nugent, J. P. Hayes, E. Harvey, B. Lai, I. McNulty, “Observation of an x-ray vortex,” Opt. Lett. 27, 1752–1754 (2002).
[CrossRef]

D. Paganin, K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[CrossRef]

T. E. Gureyev, A. Roberts, K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12, 1942–1946 (1995).
[CrossRef]

K. A. Nugent, “Wave field determination using three-dimensional intensity information,” Phys. Rev. Lett. 68, 2261–2264 (1992).
[CrossRef] [PubMed]

C. Q. Tran, A. G. Peele, D. Paterson, A. Roberts, I. McNulty, K. A. Nugent, “Correlations in a synchrotron beam measured using phase-space tomography,” manuscript available from C. Q. Tran at tran@physics.unimelb.edu.au.

Nye, J. F.

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

Padgett, M. J.

H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yao, J. Courtial, “Simplified measurement of the orbital angular momentum of single photons,” Opt. Commun. 223, 117–122 (2003).
[CrossRef]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, M. J. Padgett, “The generation of free-space Laguerre–Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127, 183–188 (1996).
[CrossRef]

Paganin, D.

D. Paganin, K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[CrossRef]

Paterson, D.

P. J. McMahon, A. G. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480–1482 (2003).
[CrossRef]

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. 90, 074801 (2003).
[PubMed]

A. G. Peele, P. J. McMahon, D. Paterson, C. Q. Tran, A. P. Mancuso, K. A. Nugent, J. P. Hayes, E. Harvey, B. Lai, I. McNulty, “Observation of an x-ray vortex,” Opt. Lett. 27, 1752–1754 (2002).
[CrossRef]

C. Q. Tran, A. G. Peele, D. Paterson, A. Roberts, I. McNulty, K. A. Nugent, “Correlations in a synchrotron beam measured using phase-space tomography,” manuscript available from C. Q. Tran at tran@physics.unimelb.edu.au.

Peele, A. G.

K. A. Nugent, A. G. Peele, H. N. Chapman, A. P. Mancuso, “Unique phase recovery for nonperiodic objects,” Phys. Rev. Lett. 91, 203902 (2003).
[CrossRef] [PubMed]

P. J. McMahon, A. G. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480–1482 (2003).
[CrossRef]

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. 90, 074801 (2003).
[PubMed]

A. G. Peele, P. J. McMahon, D. Paterson, C. Q. Tran, A. P. Mancuso, K. A. Nugent, J. P. Hayes, E. Harvey, B. Lai, I. McNulty, “Observation of an x-ray vortex,” Opt. Lett. 27, 1752–1754 (2002).
[CrossRef]

C. Q. Tran, A. G. Peele, D. Paterson, A. Roberts, I. McNulty, K. A. Nugent, “Correlations in a synchrotron beam measured using phase-space tomography,” manuscript available from C. Q. Tran at tran@physics.unimelb.edu.au.

Penna, V.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

Pfau, T.

C. Kurtsiefer, T. Pfau, J. Mlynek, “Measurement of the Wigner function of an ensemble of helium atoms,” Nature 386, 150–153 (1997).
[CrossRef]

Pilipetskii, N. F.

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamayev, N. F. Pilipetskii, V. V. Shkukov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” Pis'ma Zh. Eksp. Teor. Fiz. 33, 206–210 (1981) [JETP Lett.33, 195–199 (1981)].

Prati, F.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

Rau, C.

P. J. McMahon, A. G. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480–1482 (2003).
[CrossRef]

Raymer, M. G.

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Roberts, A.

T. E. Gureyev, A. Roberts, K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12, 1942–1946 (1995).
[CrossRef]

C. Q. Tran, A. G. Peele, D. Paterson, A. Roberts, I. McNulty, K. A. Nugent, “Correlations in a synchrotron beam measured using phase-space tomography,” manuscript available from C. Q. Tran at tran@physics.unimelb.edu.au.

Robertson, D. A.

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, M. J. Padgett, “The generation of free-space Laguerre–Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127, 183–188 (1996).
[CrossRef]

Robinson, I. K.

I. K. Robinson, Department of Physics, University of Illinois at Urbana–Champaign, Urbana, Ill. 61801 (personal communication, 2003).

Rozas, D.

Rubinsztein-Dunlop, H.

H. He, M. E. J. Friese, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

Rumsby, P. T.

E. C. Harvey, P. T. Rumsby, “Fabrication techniques and their application to produce novel micromachined structures and devices using excimer laser projection,” in Micromachining and Microfabrication Process Technology III, S.-C. Chang, S. W. Pang, eds., Proc. SPIE3223, 26–33 (1997).

Sacks, Z. S.

Santarsiero, M.

Saxton, W. O.

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

Sayre, D.

J. W. Miao, P. Charalambous, J. Kirz, D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

D. Sayre, H. N. Chapman, “X-ray microscopy,” Acta Crystallogr., Sect. A 51, 237–252 (1995).
[CrossRef] [PubMed]

Seldin, J. H.

Shkukov, V. V.

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamayev, N. F. Pilipetskii, V. V. Shkukov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” Pis'ma Zh. Eksp. Teor. Fiz. 33, 206–210 (1981) [JETP Lett.33, 195–199 (1981)].

Smith, C. P.

Smith, G. M.

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, M. J. Padgett, “The generation of free-space Laguerre–Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127, 183–188 (1996).
[CrossRef]

Snigirev, A.

P. J. McMahon, A. G. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480–1482 (2003).
[CrossRef]

Soskin, M. S.

M. V. Vasnetsov, I. G. Marienko, M. S. Soskin, “Self-restoration of an optical vortex,” JETP Lett. 71, 130–133 (2000).
[CrossRef]

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).

I. V. Basistiy, M. S. Soskin, M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

M. S. Soskin, M. V. Vasnetsov, “Singular optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 2001), Vol. 42, pp. 220–276.

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Swartzlander, G. A.

Tamm, C.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

Tartari, A.

A. Tartari, E. Casnati, C. Bonifazzi, C. Baraldi, “Molecular differential cross sections for x-ray coherent scattering in fat and polymethyl methacrylate,” Phys. Med. Biol. 42, 2551–2560 (1997).

Tran, C. Q.

A. G. Peele, P. J. McMahon, D. Paterson, C. Q. Tran, A. P. Mancuso, K. A. Nugent, J. P. Hayes, E. Harvey, B. Lai, I. McNulty, “Observation of an x-ray vortex,” Opt. Lett. 27, 1752–1754 (2002).
[CrossRef]

C. Q. Tran, A. G. Peele, D. Paterson, A. Roberts, I. McNulty, K. A. Nugent, “Correlations in a synchrotron beam measured using phase-space tomography,” manuscript available from C. Q. Tran at tran@physics.unimelb.edu.au.

C. Q. Tran, A. G. Peele, D. Paterson, A. Roberts, I. McNulty, K. A. Nugent, “Correlations in a synchrotron beam measured using phase-space tomography,” manuscript available from C. Q. Tran at tran@physics.unimelb.edu.au.

Turnbull, G. A.

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, M. J. Padgett, “The generation of free-space Laguerre–Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127, 183–188 (1996).
[CrossRef]

van der Veen, H. E. L. O.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Vartanyants, I.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419–1421 (2003).
[CrossRef] [PubMed]

Vasnetsov, M. V.

M. V. Vasnetsov, I. G. Marienko, M. S. Soskin, “Self-restoration of an optical vortex,” JETP Lett. 71, 130–133 (2000).
[CrossRef]

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).

I. V. Basistiy, M. S. Soskin, M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

M. S. Soskin, M. V. Vasnetsov, “Singular optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 2001), Vol. 42, pp. 220–276.

Veigele, W. J.

J. H. Hubbell, W. J. Veigele, E. A. Briggs, R. T. Brown, D. T. Cromer, R. J. Howerton, “Atomic form factors, incoherent scattering functions, and photon scattering cross sections,” J. Phys. Chem. Ref. Data 4, 471–538 (1975).

Wei, H.

H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yao, J. Courtial, “Simplified measurement of the orbital angular momentum of single photons,” Opt. Commun. 223, 117–122 (2003).
[CrossRef]

Weiss, C. O.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

Weitkamp, T.

P. J. McMahon, A. G. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480–1482 (2003).
[CrossRef]

White, A. G.

Woerdman, J. P.

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

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Xue, X.

H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yao, J. Courtial, “Simplified measurement of the orbital angular momentum of single photons,” Opt. Commun. 223, 117–122 (2003).
[CrossRef]

Yao, E.

H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yao, J. Courtial, “Simplified measurement of the orbital angular momentum of single photons,” Opt. Commun. 223, 117–122 (2003).
[CrossRef]

Zel’dovich, B. Ya.

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamayev, N. F. Pilipetskii, V. V. Shkukov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” Pis'ma Zh. Eksp. Teor. Fiz. 33, 206–210 (1981) [JETP Lett.33, 195–199 (1981)].

Zhang, R.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419–1421 (2003).
[CrossRef] [PubMed]

Zuo, J. M.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419–1421 (2003).
[CrossRef] [PubMed]

Acta Crystallogr., Sect. A

D. Sayre, H. N. Chapman, “X-ray microscopy,” Acta Crystallogr., Sect. A 51, 237–252 (1995).
[CrossRef] [PubMed]

Appl. Opt.

Appl. Phys. Lett.

P. J. McMahon, A. G. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480–1482 (2003).
[CrossRef]

Astrophys. J.

M. Harwit, “Photon orbital angular momentum in astrophysics,” Astrophys. J. 597, 1266–1270 (2003).
[CrossRef]

Contemp. Phys.

M. Harris, “Light-field fluctuations in space and time,” Contemp. Phys. 36, 215–233 (1995).
[CrossRef]

J. Mod. Opt.

V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Phys. Chem. Ref. Data

J. H. Hubbell, W. J. Veigele, E. A. Briggs, R. T. Brown, D. T. Cromer, R. J. Howerton, “Atomic form factors, incoherent scattering functions, and photon scattering cross sections,” J. Phys. Chem. Ref. Data 4, 471–538 (1975).

JETP Lett.

M. V. Vasnetsov, I. G. Marienko, M. S. Soskin, “Self-restoration of an optical vortex,” JETP Lett. 71, 130–133 (2000).
[CrossRef]

Nature

J. W. Miao, P. Charalambous, J. Kirz, D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

C. Kurtsiefer, T. Pfau, J. Mlynek, “Measurement of the Wigner function of an ensemble of helium atoms,” Nature 386, 150–153 (1997).
[CrossRef]

Opt. Commun.

I. V. Basistiy, M. S. Soskin, M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yao, J. Courtial, “Simplified measurement of the orbital angular momentum of single photons,” Opt. Commun. 223, 117–122 (2003).
[CrossRef]

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

J. E. Curtis, B. A. Koss, D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
[CrossRef]

S. Chávez-Cerda, G. S. McDonald, G. H. C. New, “Nondiffracting beams: travelling, standing, rotating, and spiral waves,” Opt. Commun. 123, 225–233 (1996).
[CrossRef]

Z. Bouchal, “Resistance of nondiffracting vortex beam against amplitude and phase perturbations,” Opt. Commun. 210, 155–164 (2002).
[CrossRef]

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

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, M. J. Padgett, “The generation of free-space Laguerre–Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127, 183–188 (1996).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Opt. Lett.

Optik (Stuttgart)

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

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: underlying theory,” Optik (Stuttgart) 61, 247–262 (1982).

Phys. Med. Biol.

A. Tartari, E. Casnati, C. Bonifazzi, C. Baraldi, “Molecular differential cross sections for x-ray coherent scattering in fat and polymethyl methacrylate,” Phys. Med. Biol. 42, 2551–2560 (1997).

Phys. Rev. A

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

Phys. Rev. Lett.

H. He, M. E. J. Friese, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

K. A. Nugent, A. G. Peele, H. N. Chapman, A. P. Mancuso, “Unique phase recovery for nonperiodic objects,” Phys. Rev. Lett. 91, 203902 (2003).
[CrossRef] [PubMed]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

D. Paganin, K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[CrossRef]

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. 90, 074801 (2003).
[PubMed]

K. A. Nugent, “Wave field determination using three-dimensional intensity information,” Phys. Rev. Lett. 68, 2261–2264 (1992).
[CrossRef] [PubMed]

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Pis'ma Zh. Eksp. Teor. Fiz.

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamayev, N. F. Pilipetskii, V. V. Shkukov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” Pis'ma Zh. Eksp. Teor. Fiz. 33, 206–210 (1981) [JETP Lett.33, 195–199 (1981)].

Proc. R. Soc. London Ser. A

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

Science

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419–1421 (2003).
[CrossRef] [PubMed]

Other

I. K. Robinson, Department of Physics, University of Illinois at Urbana–Champaign, Urbana, Ill. 61801 (personal communication, 2003).

M. S. Soskin, M. V. Vasnetsov, “Singular optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 2001), Vol. 42, pp. 220–276.

M. Berry, “Paraxial beams of spinning light,” in International Conference on Singular Optics, M. S. Soskin, ed., Proc. SPIE3487, 6–11 (1998).
[CrossRef]

I. K. Robinson, F. Pfeiffer, I. A. Vartanyants, Y. Sun, Y. Xia, “Enhancement of coherent x-ray diffraction from nanocrystals by introduction of x-ray optics,” Opt. Express11, 2329–2334 (2003), http://www.opticsexpress.org .
[CrossRef] [PubMed]

C. Q. Tran, A. G. Peele, D. Paterson, A. Roberts, I. McNulty, K. A. Nugent, “Correlations in a synchrotron beam measured using phase-space tomography,” manuscript available from C. Q. Tran at tran@physics.unimelb.edu.au.

A. G. Peele, K. A. Nugent, “X-ray vortex beams: a the-oretical analysis,” Opt. Express11, 2315–2322 (2003), http://www.opticsexpress.org .
[CrossRef] [PubMed]

E. C. Harvey, P. T. Rumsby, “Fabrication techniques and their application to produce novel micromachined structures and devices using excimer laser projection,” in Micromachining and Microfabrication Process Technology III, S.-C. Chang, S. W. Pang, eds., Proc. SPIE3223, 26–33 (1997).

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Figures (10)

Fig. 1
Fig. 1

Calculated intensity for an ideal theoretical vortex calculated by use of Eq. (9). The synchrotron parameters used were z1=41.4 m, ω0=200 μm, λ=0.14 nm, and Z=5.8 m.

Fig. 2
Fig. 2

Modulation to the intensity distribution of Eq. (10) produced by the cosine of θinterf given by (a) Eq. (11): λ=0.14 nm, Z=5.8 m, xoffs=42.5 μm, and zoffs=3 mm and (b) the modified form of Eq. (11) discussed in Subsection 4.B: λ=0.14 nm, Z=5.8 m, xoffs=42.5 μm, zoffs=3 mm, and ν=0.5.

Fig. 3
Fig. 3

Microscope image of one of the 16-step phase plates used in the experimental work. The surface roughness of each step can be seen to increase in a clockwise direction around the phase plate, indicating the increase in depth of each step.

Fig. 4
Fig. 4

Diffracted intensity produced by a 16-step phase plate for Z=5.8 m and λ=0.14 nm measured with a crystal scintillator and a CCD camera. The circular edge of the ∼1-mm-diameter phase plate can be seen. The inset shows the intensity as measured through the core and along the vertical and horizontal lines shown on the image.

Fig. 5
Fig. 5

(a) Magnified portion of another experimental image as reported for Fig. 4. (b) Simulation of the same experiment done with the known experimental parameters as described in the text.

Fig. 6
Fig. 6

(a) Experimental vortex interferogram produced by interfering the wave field produced by a 7.5-μm tungsten wire 3 mm downstream of the phase plate with the vortex wave field produced by the phase plate. The other experimental parameters are as for Fig. 4. (b) Simulation of the division of the wave-front interferogram experimental result shown in (a) done with the known experimental parameters and assuming a maximum phase shift of 1.8π. The result shows excellent agreement. (c) Same as for (a) but with the phase plate removed. The expected Fresnel fringes for the wire are seen with no splitting.

Fig. 7
Fig. 7

Demonstration of the association of the position of the interferogram fork with the central position of the phase plate produced by movement of an edge relative to the phase plate. As indicated by the arrows, the fork moves from the second diffraction fringe of the edge in (a) to the third fringe and in (b) as the edge is moved with respect to the phase plate.

Fig. 8
Fig. 8

Demonstration of negative charge: reversal of the fork direction by reversal of the phase plate.

Fig. 9
Fig. 9

(a) Experimental interferogram for a 32-step polyimide phase plate with a maximum step depth of ∼80 μm measured at 6 keV (λ=0.21 nm); the other experimental parameters are as described previously. The overlaid sketch shows the fringe positions with branching points with the label A indicating the first branch point. The arrows indicate the locations of additional single-charge vortices of the opposite sign to the main vortex. (b) Calculated interferogram for the phase plate at 6 keV (λ=0.21 nm) by use of the modified form of Eq. (11) discussed in Subsection 4.B. The sketch from (a) is overlaid showing the fringe positions with branching points for the experimental data.

Fig. 10
Fig. 10

(a) Experimental interferograms as for Fig. 9(a) for energies of 12–7 keV (λ=0.10.18 nm) from left to right, decreasing in 1-keV steps. (b) Calculated interferograms as for Fig. 9(b) at the same energies as in (a). Note that the size and contrast in the region about the wire position will differ from the experimental data as the calculated images assume an infinitely narrow wire. However, the fringes to the side of the wire position are in good agreement with the data.

Equations (19)

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S(r)=I(r)[ΦS(r)+×Φν(r)],
k I(r)z=S(r),
k I(r)z=[I(r)ΦS(r)]+I(r)×Φν(r),
I(r)×Φν(r)=0.
u(r, θ, 0)=2π ω0exp-r2ω02,
T(r, θ)=exp(imθ)
u(R, Θ, Z)=A(Z)×exp[-A(Z)R2]exp[-i(mΘ-Φ(Z)-Φ(Z)R2)]×RI1/2(m-1)γ28β-I1/2(m+1)γ28β,
A(Z)π2πλZ1λz12π1ω(z1)k8Z×1ω(z1)4+πλ21R(z1)-1Z2-3/4;
A(Z)kZ2181ω(z1)2×1ω(z1)4+πλ21R(z1)-1Z2-1;
Φ(Z)π-m π2+kz1-Ψ(z1)-32arctanπλω(z1)21R(z1)-1Z;
Φ(Z)k2Z+kZ218πλ1R(z1)-1Z1ω(z1)4+πλ21R(z1)-1Z2-1;
β1ω(z1)2+i πλ1R(z1)-1Z;γRk2;
ω(z)ω01+zzR21/2;ω(z)ω01+zzR21/2;
Ψ(z)arctanzzR;R(z)z1+zRz2;
zRπω02λ;k=2πλ.
I(R, Θ, Z)=|u(R, Θ, Z)|2.
I=Av2+Acyl2+2AvAcylcos θinterf,
θinterf=-m arctanyx+kZ+k[(x-xoffs)2+(Z-zoffs)2]1/2+m arctanyxoffs,
rm=ω0AAm(1/|m|)[z2+[kω02/2]2]1/2(kω02/2),

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