Abstract

In this paper we present some quantitative measurements of X-ray phase contrast images and noise evaluation obtained with a recent grating based X-ray phase contrast interferometer. This device is built using a single phase grating and a large broadband X-ray source. It was calibrated using a reference sample and finally used to perform measurements of a biological fossil: a mosquito trapped in amber. As phase images, noise was evaluated from the measured interferograms.

© 2013 OSA

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  1. G. Nomarski, “Nouveau dispositif pour l’observation en contraste de phase differentiel,” J.Phys.Radium 16, S88–S88 (1955).
  2. U. Bonse and M. Hart, “An X-ray interferometer,” Appl. Phys. Lett 6(8), 155–156 (1965).
    [Crossref]
  3. R. Fitzgerald, “Phase sensitive X-ray imaging,” Phys. Today 53, 23–26 (2000).
    [Crossref]
  4. A. Momose, “Phase-sensitive imaging and phase tomography using X-ray interferometers,” Opt. Express 11(19), 2303–2314 (2003).
    [Crossref] [PubMed]
  5. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum 66(12), 5486–5492 (1995).
    [Crossref]
  6. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996).
    [Crossref]
  7. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jap. Jour. Appl. Phys 42(7B), 866–868 (2003).
    [Crossref]
  8. A. Momose, “Recent advances in X-ray phase imaging,” Jap. Jour. Appl. Phys 44(9A), 6355–6367 (2005).
    [Crossref]
  9. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005).
    [Crossref] [PubMed]
  10. H. F. Talbot, “Facts relating to optical science,” Phil. Mag. Series 3 9, 401–407 (1836).
  11. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72(1), 156–160 (1982).
    [Crossref]
  12. Y. Takeda, W. Yashiro, T. Hattori, A. Takeuchi, Y. Suzuki, and A. Momose, “Differential phase X-ray imaging microscopy with Talbot interferometer,” Appl. Phys. Express 1, 117002 (2008).
    [Crossref]
  13. S. Ruthishauer, I. Zanette, T. Weitkamp, T. Donath, and C. David, “At-wavelength charcterization of refractive x-ray lenses using a two-dimensional grating interferometer,” Appl. Phys. Lett 99, 221104 (2011).
    [Crossref]
  14. M. Creath, “Phase-Measurement Interferometry Techniques” (Elsevier Science, 1988) pp. 349–393.
  15. A. Momose, W. Yashiro, H. Maikusa, and Y. Takeda, “High-speed X-ray phase imaging and X-ray phase tomography with Talbot interferometer and white synchrotron radiation,” Opt. Express 17(15), 12540–12545 (2009).
    [Crossref] [PubMed]
  16. J. M. Kim, I. H. Cho, S. Y. Lee, H. C. Kang, R. Conley, C. Liu, A. T. Macrander, and D. Y. Noh, “Observation of the Talbot effect using broadband hard x-ray beam,” Opt. Express 18(24), 24975–24982 (2010).
    [Crossref] [PubMed]
  17. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brillance X-ray sources,” Nature Phys. 2, 258–261 (2006).
    [Crossref]
  18. F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard X-ray phase tomography with low-brillance sources,” Phys. Rev. Lett 98, 108105 (2007).
    [Crossref] [PubMed]
  19. J. Primot, “Three-wave lateral shearing interferometry,” Appl. Opt 32(31), 6242–6249 (1993).
    [Crossref] [PubMed]
  20. J. Primot and L. Sogno, “Achromatic three-wave (or more) lateral shearing interferometer,” J. Opt. Soc. Am. A 12(12), 2679–2685 (1995).
    [Crossref]
  21. A. Momose and S. Kawamoto, “X-ray Talbot interferometry with capillary plates,” Jap. Jour. Appl. Phys 45(1A), 314–316 (2006).
    [Crossref]
  22. C. Kottler, C. David, F. Pfeiffer, and O. Bunk, “A two-directional approach for grating based differential phase contrast imaging using hard x-rays,” Opt. Express 15(3), 1175–1181 (2007).
    [Crossref] [PubMed]
  23. H. Itoh, K. Nagai, G. sato, K. Yamaguchi, T. Nakamura, T. Kondoh, C. Ouchi, T. Teshima, Y. Setomoto, and T. Den, “Two-dimensional grating-based X-ray phase contrast imaging using Fourier transform phase retrieval,” Opt. Express 19(4), 3339–3346 (2011).
    [Crossref] [PubMed]
  24. I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional X-ray grating interferometer,” Phys. Rev. Lett 105, 248102 (2010).
    [Crossref]
  25. H. Wen, E. E. Bennett, R. Kopace, A. F. Stein, and V. Pai, “Single-shot x-ray differential phase-contrast and diffraction imaging using two-dimensional transmission grating,” Opt. Lett 35(12), 1932–1934 (2010).
    [Crossref] [PubMed]
  26. V. Revol, C. Kottler, R. Kaufmann, I. Jerjen, T. lüthi, F. Cardot, P. Niedermann, U. Straumann, U. Sennhauser, and C. Urban, “X-ray interferometer with bent grating: toward larger fields of view,” Nucl. Instr. Meth. Phys. Res. A 648, 302–305 (2011).
    [Crossref]
  27. K. S. Morgan, D. M. Paganin, and K. K. W. Siu, “Quantitative single-exposure x-ray phase contrast imaging using a single attenuation grid,” Opt. Express 19(20), 19781–19789 (2011).
    [Crossref] [PubMed]
  28. J. Durnin, “Continuously self-imaging fields of finite aperture,” J. Opt. Soc. Am. A 2, 110 (1985).
  29. N. Guérineau and J. Primot, “Non diffracting array generation using an N-wave interferometer,” J. Opt. Soc. Am. A 16(2), 293–298 (1999).
    [Crossref]
  30. M. Piponnier, G. Druart, N. Guérineau, J. L. de Bougrenet, and J. Primot, “Optimal conditions for using the binary approximation of continuously self-imaging gratings,” Opt. Express 19(23), 23054–23066 (2011).
    [Crossref] [PubMed]
  31. N. Guérineau, B. Harchaoui, and J. Primot, “Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime,” Opt. Com 180, 199–203 (2000).
    [Crossref]
  32. J. R. Leger and G. J. Swanson, “Efficient array illuminator using binary-optics phase plates at fractional-Talbot planes,” Opt. Lett 15(5), 288–290 (1990).
    [Crossref] [PubMed]
  33. P. Cloetens, J. P. Guigay, C. De Martino, and J. Baruchel, “fractionnal Talbot imaging of phase gratings with hard x rays,” Opt. Lett 22(14), 1059–1061 (1997).
    [Crossref] [PubMed]
  34. J. Primot and N. Guérineau, “Extended Hartmann test based on the pseudo-giuding property of a Hartmann mask completed by a phase chessboard,” Appl. Opt 39(31), 5715–5720 (2000).
    [Crossref]
  35. J. Rizzi, T. Weitkamp, N. Guérineau, M. Idir, P. Mercère, G. Druart, G. Vincent, P. Da Silva, and J. Primot, “Quadriwave lateral shearing interferometry in an achromatic and continuously self-imaging regime for future x-ray phase imaging,” Opt. Lett 36(8), 1398–1400 (2011).
    [Crossref] [PubMed]
  36. T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brillance sources,” Proc. of SPIE 631863180S (2006).
    [Crossref]
  37. X. Ge, Z. Wang, K. Gao, K. Zhang, Y. Hong, D. Wang, Z. Zhu, P. Zhu, and Z. Wu, “Inverstigation of the partially coherent effects in a 2D Talbot interferometer,” Anal. Bioanal. Chem 401, 865–870 (2011).
    [Crossref] [PubMed]
  38. N. Guérineau, B. Harchaoui, K. Heggarty, and J. Primot, “Generation of achromatic and propagation-invariant spot arrays by use of continuously self-imaging gratings,” Opt. lett 26(7), 411–413 (2001).
    [Crossref]
  39. A. A. Michelson, Studies in Optics (University of Chicago Press, Chicago, 1927).
  40. P. Bon, S. Monneret, and B. Wattelier, “Noninterative boundary-artifact-free wavefront reconstruction from its derivatives,” Appl. Opt 51(23), 5698–5704 (2012).
    [Crossref] [PubMed]
  41. D. Ghilia and M. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998) pp. 34.
  42. J. Rizzi, P. Mercre, M. Idir, N. Gurineau, E. Sakat, R. Hadar, G. Vincent, P. Da Silva, and J. Primot, “X-ray phase contrast imaging using a broadband X-ray beam and a single phase grating used in its achromatic and propagation-invariant regime,” J. Phys.: Conf. Ser. 425,192002 (2013).
    [Crossref]
  43. R. C. Jennison, “A phase sensitive interferometer technique for the measurement of the Fourier Transfoms of spatial brightness distributions of small angular extent,” Mon. Not. Roy. Astron. Soc. 118(3), 276–284(1958).
  44. D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. A 67(3), 370–375 (1977).
    [Crossref]
  45. W. H. Southwell, “Wavefront estimation from wavefront slope measurements,” J. Opt. Soc. Am. A 70(8), 998–1006 (1980).
    [Crossref]
  46. K. R. Freischlad and C. L. Koliopoulos, “Wavefront estimation from wavefront slope measurements,” J. Opt. Soc. Am. A 3(11), 1852–1861 (1986).
    [Crossref]
  47. S. Velghe, “Wave-front reconstruction from multidirectional phase derivatives generated by multilateral shearing interferometers,” Opt. Lett 30(3), 245–247 (2005).
    [Crossref] [PubMed]

2013 (1)

J. Rizzi, P. Mercre, M. Idir, N. Gurineau, E. Sakat, R. Hadar, G. Vincent, P. Da Silva, and J. Primot, “X-ray phase contrast imaging using a broadband X-ray beam and a single phase grating used in its achromatic and propagation-invariant regime,” J. Phys.: Conf. Ser. 425,192002 (2013).
[Crossref]

2012 (1)

P. Bon, S. Monneret, and B. Wattelier, “Noninterative boundary-artifact-free wavefront reconstruction from its derivatives,” Appl. Opt 51(23), 5698–5704 (2012).
[Crossref] [PubMed]

2011 (7)

S. Ruthishauer, I. Zanette, T. Weitkamp, T. Donath, and C. David, “At-wavelength charcterization of refractive x-ray lenses using a two-dimensional grating interferometer,” Appl. Phys. Lett 99, 221104 (2011).
[Crossref]

H. Itoh, K. Nagai, G. sato, K. Yamaguchi, T. Nakamura, T. Kondoh, C. Ouchi, T. Teshima, Y. Setomoto, and T. Den, “Two-dimensional grating-based X-ray phase contrast imaging using Fourier transform phase retrieval,” Opt. Express 19(4), 3339–3346 (2011).
[Crossref] [PubMed]

V. Revol, C. Kottler, R. Kaufmann, I. Jerjen, T. lüthi, F. Cardot, P. Niedermann, U. Straumann, U. Sennhauser, and C. Urban, “X-ray interferometer with bent grating: toward larger fields of view,” Nucl. Instr. Meth. Phys. Res. A 648, 302–305 (2011).
[Crossref]

K. S. Morgan, D. M. Paganin, and K. K. W. Siu, “Quantitative single-exposure x-ray phase contrast imaging using a single attenuation grid,” Opt. Express 19(20), 19781–19789 (2011).
[Crossref] [PubMed]

M. Piponnier, G. Druart, N. Guérineau, J. L. de Bougrenet, and J. Primot, “Optimal conditions for using the binary approximation of continuously self-imaging gratings,” Opt. Express 19(23), 23054–23066 (2011).
[Crossref] [PubMed]

J. Rizzi, T. Weitkamp, N. Guérineau, M. Idir, P. Mercère, G. Druart, G. Vincent, P. Da Silva, and J. Primot, “Quadriwave lateral shearing interferometry in an achromatic and continuously self-imaging regime for future x-ray phase imaging,” Opt. Lett 36(8), 1398–1400 (2011).
[Crossref] [PubMed]

X. Ge, Z. Wang, K. Gao, K. Zhang, Y. Hong, D. Wang, Z. Zhu, P. Zhu, and Z. Wu, “Inverstigation of the partially coherent effects in a 2D Talbot interferometer,” Anal. Bioanal. Chem 401, 865–870 (2011).
[Crossref] [PubMed]

2010 (3)

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional X-ray grating interferometer,” Phys. Rev. Lett 105, 248102 (2010).
[Crossref]

H. Wen, E. E. Bennett, R. Kopace, A. F. Stein, and V. Pai, “Single-shot x-ray differential phase-contrast and diffraction imaging using two-dimensional transmission grating,” Opt. Lett 35(12), 1932–1934 (2010).
[Crossref] [PubMed]

J. M. Kim, I. H. Cho, S. Y. Lee, H. C. Kang, R. Conley, C. Liu, A. T. Macrander, and D. Y. Noh, “Observation of the Talbot effect using broadband hard x-ray beam,” Opt. Express 18(24), 24975–24982 (2010).
[Crossref] [PubMed]

2009 (1)

2008 (1)

Y. Takeda, W. Yashiro, T. Hattori, A. Takeuchi, Y. Suzuki, and A. Momose, “Differential phase X-ray imaging microscopy with Talbot interferometer,” Appl. Phys. Express 1, 117002 (2008).
[Crossref]

2007 (2)

2006 (3)

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brillance X-ray sources,” Nature Phys. 2, 258–261 (2006).
[Crossref]

A. Momose and S. Kawamoto, “X-ray Talbot interferometry with capillary plates,” Jap. Jour. Appl. Phys 45(1A), 314–316 (2006).
[Crossref]

T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brillance sources,” Proc. of SPIE 631863180S (2006).
[Crossref]

2005 (3)

A. Momose, “Recent advances in X-ray phase imaging,” Jap. Jour. Appl. Phys 44(9A), 6355–6367 (2005).
[Crossref]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005).
[Crossref] [PubMed]

S. Velghe, “Wave-front reconstruction from multidirectional phase derivatives generated by multilateral shearing interferometers,” Opt. Lett 30(3), 245–247 (2005).
[Crossref] [PubMed]

2003 (2)

A. Momose, “Phase-sensitive imaging and phase tomography using X-ray interferometers,” Opt. Express 11(19), 2303–2314 (2003).
[Crossref] [PubMed]

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jap. Jour. Appl. Phys 42(7B), 866–868 (2003).
[Crossref]

2001 (1)

N. Guérineau, B. Harchaoui, K. Heggarty, and J. Primot, “Generation of achromatic and propagation-invariant spot arrays by use of continuously self-imaging gratings,” Opt. lett 26(7), 411–413 (2001).
[Crossref]

2000 (3)

N. Guérineau, B. Harchaoui, and J. Primot, “Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime,” Opt. Com 180, 199–203 (2000).
[Crossref]

R. Fitzgerald, “Phase sensitive X-ray imaging,” Phys. Today 53, 23–26 (2000).
[Crossref]

J. Primot and N. Guérineau, “Extended Hartmann test based on the pseudo-giuding property of a Hartmann mask completed by a phase chessboard,” Appl. Opt 39(31), 5715–5720 (2000).
[Crossref]

1999 (1)

1997 (1)

P. Cloetens, J. P. Guigay, C. De Martino, and J. Baruchel, “fractionnal Talbot imaging of phase gratings with hard x rays,” Opt. Lett 22(14), 1059–1061 (1997).
[Crossref] [PubMed]

1996 (1)

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996).
[Crossref]

1995 (2)

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum 66(12), 5486–5492 (1995).
[Crossref]

J. Primot and L. Sogno, “Achromatic three-wave (or more) lateral shearing interferometer,” J. Opt. Soc. Am. A 12(12), 2679–2685 (1995).
[Crossref]

1993 (1)

J. Primot, “Three-wave lateral shearing interferometry,” Appl. Opt 32(31), 6242–6249 (1993).
[Crossref] [PubMed]

1990 (1)

J. R. Leger and G. J. Swanson, “Efficient array illuminator using binary-optics phase plates at fractional-Talbot planes,” Opt. Lett 15(5), 288–290 (1990).
[Crossref] [PubMed]

1986 (1)

1985 (1)

J. Durnin, “Continuously self-imaging fields of finite aperture,” J. Opt. Soc. Am. A 2, 110 (1985).

1982 (1)

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72(1), 156–160 (1982).
[Crossref]

1980 (1)

W. H. Southwell, “Wavefront estimation from wavefront slope measurements,” J. Opt. Soc. Am. A 70(8), 998–1006 (1980).
[Crossref]

1977 (1)

D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. A 67(3), 370–375 (1977).
[Crossref]

1965 (1)

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

1958 (1)

R. C. Jennison, “A phase sensitive interferometer technique for the measurement of the Fourier Transfoms of spatial brightness distributions of small angular extent,” Mon. Not. Roy. Astron. Soc. 118(3), 276–284(1958).

1955 (1)

G. Nomarski, “Nouveau dispositif pour l’observation en contraste de phase differentiel,” J.Phys.Radium 16, S88–S88 (1955).

1836 (1)

H. F. Talbot, “Facts relating to optical science,” Phil. Mag. Series 3 9, 401–407 (1836).

Baruchel, J.

P. Cloetens, J. P. Guigay, C. De Martino, and J. Baruchel, “fractionnal Talbot imaging of phase gratings with hard x rays,” Opt. Lett 22(14), 1059–1061 (1997).
[Crossref] [PubMed]

Bennett, E. E.

H. Wen, E. E. Bennett, R. Kopace, A. F. Stein, and V. Pai, “Single-shot x-ray differential phase-contrast and diffraction imaging using two-dimensional transmission grating,” Opt. Lett 35(12), 1932–1934 (2010).
[Crossref] [PubMed]

Bon, P.

P. Bon, S. Monneret, and B. Wattelier, “Noninterative boundary-artifact-free wavefront reconstruction from its derivatives,” Appl. Opt 51(23), 5698–5704 (2012).
[Crossref] [PubMed]

Bonse, U.

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

Bunk, O.

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard X-ray phase tomography with low-brillance sources,” Phys. Rev. Lett 98, 108105 (2007).
[Crossref] [PubMed]

C. Kottler, C. David, F. Pfeiffer, and O. Bunk, “A two-directional approach for grating based differential phase contrast imaging using hard x-rays,” Opt. Express 15(3), 1175–1181 (2007).
[Crossref] [PubMed]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brillance X-ray sources,” Nature Phys. 2, 258–261 (2006).
[Crossref]

T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brillance sources,” Proc. of SPIE 631863180S (2006).
[Crossref]

Cardot, F.

V. Revol, C. Kottler, R. Kaufmann, I. Jerjen, T. lüthi, F. Cardot, P. Niedermann, U. Straumann, U. Sennhauser, and C. Urban, “X-ray interferometer with bent grating: toward larger fields of view,” Nucl. Instr. Meth. Phys. Res. A 648, 302–305 (2011).
[Crossref]

Cho, I. H.

Cloetens, P.

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005).
[Crossref] [PubMed]

P. Cloetens, J. P. Guigay, C. De Martino, and J. Baruchel, “fractionnal Talbot imaging of phase gratings with hard x rays,” Opt. Lett 22(14), 1059–1061 (1997).
[Crossref] [PubMed]

Conley, R.

Creath, M.

M. Creath, “Phase-Measurement Interferometry Techniques” (Elsevier Science, 1988) pp. 349–393.

Da Silva, P.

J. Rizzi, P. Mercre, M. Idir, N. Gurineau, E. Sakat, R. Hadar, G. Vincent, P. Da Silva, and J. Primot, “X-ray phase contrast imaging using a broadband X-ray beam and a single phase grating used in its achromatic and propagation-invariant regime,” J. Phys.: Conf. Ser. 425,192002 (2013).
[Crossref]

J. Rizzi, T. Weitkamp, N. Guérineau, M. Idir, P. Mercère, G. Druart, G. Vincent, P. Da Silva, and J. Primot, “Quadriwave lateral shearing interferometry in an achromatic and continuously self-imaging regime for future x-ray phase imaging,” Opt. Lett 36(8), 1398–1400 (2011).
[Crossref] [PubMed]

David, C.

S. Ruthishauer, I. Zanette, T. Weitkamp, T. Donath, and C. David, “At-wavelength charcterization of refractive x-ray lenses using a two-dimensional grating interferometer,” Appl. Phys. Lett 99, 221104 (2011).
[Crossref]

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional X-ray grating interferometer,” Phys. Rev. Lett 105, 248102 (2010).
[Crossref]

C. Kottler, C. David, F. Pfeiffer, and O. Bunk, “A two-directional approach for grating based differential phase contrast imaging using hard x-rays,” Opt. Express 15(3), 1175–1181 (2007).
[Crossref] [PubMed]

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard X-ray phase tomography with low-brillance sources,” Phys. Rev. Lett 98, 108105 (2007).
[Crossref] [PubMed]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brillance X-ray sources,” Nature Phys. 2, 258–261 (2006).
[Crossref]

T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brillance sources,” Proc. of SPIE 631863180S (2006).
[Crossref]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005).
[Crossref] [PubMed]

de Bougrenet, J. L.

De Martino, C.

P. Cloetens, J. P. Guigay, C. De Martino, and J. Baruchel, “fractionnal Talbot imaging of phase gratings with hard x rays,” Opt. Lett 22(14), 1059–1061 (1997).
[Crossref] [PubMed]

Den, T.

Diaz, A.

Donath, T.

S. Ruthishauer, I. Zanette, T. Weitkamp, T. Donath, and C. David, “At-wavelength charcterization of refractive x-ray lenses using a two-dimensional grating interferometer,” Appl. Phys. Lett 99, 221104 (2011).
[Crossref]

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional X-ray grating interferometer,” Phys. Rev. Lett 105, 248102 (2010).
[Crossref]

Druart, G.

J. Rizzi, T. Weitkamp, N. Guérineau, M. Idir, P. Mercère, G. Druart, G. Vincent, P. Da Silva, and J. Primot, “Quadriwave lateral shearing interferometry in an achromatic and continuously self-imaging regime for future x-ray phase imaging,” Opt. Lett 36(8), 1398–1400 (2011).
[Crossref] [PubMed]

M. Piponnier, G. Druart, N. Guérineau, J. L. de Bougrenet, and J. Primot, “Optimal conditions for using the binary approximation of continuously self-imaging gratings,” Opt. Express 19(23), 23054–23066 (2011).
[Crossref] [PubMed]

Durnin, J.

J. Durnin, “Continuously self-imaging fields of finite aperture,” J. Opt. Soc. Am. A 2, 110 (1985).

Fitzgerald, R.

R. Fitzgerald, “Phase sensitive X-ray imaging,” Phys. Today 53, 23–26 (2000).
[Crossref]

Freischlad, K. R.

Fried, D. L.

D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. A 67(3), 370–375 (1977).
[Crossref]

Gao, D.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996).
[Crossref]

Gao, K.

X. Ge, Z. Wang, K. Gao, K. Zhang, Y. Hong, D. Wang, Z. Zhu, P. Zhu, and Z. Wu, “Inverstigation of the partially coherent effects in a 2D Talbot interferometer,” Anal. Bioanal. Chem 401, 865–870 (2011).
[Crossref] [PubMed]

Ge, X.

X. Ge, Z. Wang, K. Gao, K. Zhang, Y. Hong, D. Wang, Z. Zhu, P. Zhu, and Z. Wu, “Inverstigation of the partially coherent effects in a 2D Talbot interferometer,” Anal. Bioanal. Chem 401, 865–870 (2011).
[Crossref] [PubMed]

Ghilia, D.

D. Ghilia and M. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998) pp. 34.

Guérineau, N.

J. Rizzi, T. Weitkamp, N. Guérineau, M. Idir, P. Mercère, G. Druart, G. Vincent, P. Da Silva, and J. Primot, “Quadriwave lateral shearing interferometry in an achromatic and continuously self-imaging regime for future x-ray phase imaging,” Opt. Lett 36(8), 1398–1400 (2011).
[Crossref] [PubMed]

M. Piponnier, G. Druart, N. Guérineau, J. L. de Bougrenet, and J. Primot, “Optimal conditions for using the binary approximation of continuously self-imaging gratings,” Opt. Express 19(23), 23054–23066 (2011).
[Crossref] [PubMed]

N. Guérineau, B. Harchaoui, K. Heggarty, and J. Primot, “Generation of achromatic and propagation-invariant spot arrays by use of continuously self-imaging gratings,” Opt. lett 26(7), 411–413 (2001).
[Crossref]

J. Primot and N. Guérineau, “Extended Hartmann test based on the pseudo-giuding property of a Hartmann mask completed by a phase chessboard,” Appl. Opt 39(31), 5715–5720 (2000).
[Crossref]

N. Guérineau, B. Harchaoui, and J. Primot, “Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime,” Opt. Com 180, 199–203 (2000).
[Crossref]

N. Guérineau and J. Primot, “Non diffracting array generation using an N-wave interferometer,” J. Opt. Soc. Am. A 16(2), 293–298 (1999).
[Crossref]

Guigay, J. P.

P. Cloetens, J. P. Guigay, C. De Martino, and J. Baruchel, “fractionnal Talbot imaging of phase gratings with hard x rays,” Opt. Lett 22(14), 1059–1061 (1997).
[Crossref] [PubMed]

Gureyev, T. E.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996).
[Crossref]

Gurineau, N.

J. Rizzi, P. Mercre, M. Idir, N. Gurineau, E. Sakat, R. Hadar, G. Vincent, P. Da Silva, and J. Primot, “X-ray phase contrast imaging using a broadband X-ray beam and a single phase grating used in its achromatic and propagation-invariant regime,” J. Phys.: Conf. Ser. 425,192002 (2013).
[Crossref]

Hadar, R.

J. Rizzi, P. Mercre, M. Idir, N. Gurineau, E. Sakat, R. Hadar, G. Vincent, P. Da Silva, and J. Primot, “X-ray phase contrast imaging using a broadband X-ray beam and a single phase grating used in its achromatic and propagation-invariant regime,” J. Phys.: Conf. Ser. 425,192002 (2013).
[Crossref]

Hamaishi, Y.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jap. Jour. Appl. Phys 42(7B), 866–868 (2003).
[Crossref]

Harchaoui, B.

N. Guérineau, B. Harchaoui, K. Heggarty, and J. Primot, “Generation of achromatic and propagation-invariant spot arrays by use of continuously self-imaging gratings,” Opt. lett 26(7), 411–413 (2001).
[Crossref]

N. Guérineau, B. Harchaoui, and J. Primot, “Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime,” Opt. Com 180, 199–203 (2000).
[Crossref]

Hart, M.

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

Hattori, T.

Y. Takeda, W. Yashiro, T. Hattori, A. Takeuchi, Y. Suzuki, and A. Momose, “Differential phase X-ray imaging microscopy with Talbot interferometer,” Appl. Phys. Express 1, 117002 (2008).
[Crossref]

Heggarty, K.

N. Guérineau, B. Harchaoui, K. Heggarty, and J. Primot, “Generation of achromatic and propagation-invariant spot arrays by use of continuously self-imaging gratings,” Opt. lett 26(7), 411–413 (2001).
[Crossref]

Hong, Y.

X. Ge, Z. Wang, K. Gao, K. Zhang, Y. Hong, D. Wang, Z. Zhu, P. Zhu, and Z. Wu, “Inverstigation of the partially coherent effects in a 2D Talbot interferometer,” Anal. Bioanal. Chem 401, 865–870 (2011).
[Crossref] [PubMed]

Idir, M.

J. Rizzi, P. Mercre, M. Idir, N. Gurineau, E. Sakat, R. Hadar, G. Vincent, P. Da Silva, and J. Primot, “X-ray phase contrast imaging using a broadband X-ray beam and a single phase grating used in its achromatic and propagation-invariant regime,” J. Phys.: Conf. Ser. 425,192002 (2013).
[Crossref]

J. Rizzi, T. Weitkamp, N. Guérineau, M. Idir, P. Mercère, G. Druart, G. Vincent, P. Da Silva, and J. Primot, “Quadriwave lateral shearing interferometry in an achromatic and continuously self-imaging regime for future x-ray phase imaging,” Opt. Lett 36(8), 1398–1400 (2011).
[Crossref] [PubMed]

Ina, H.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72(1), 156–160 (1982).
[Crossref]

Itoh, H.

Jennison, R. C.

R. C. Jennison, “A phase sensitive interferometer technique for the measurement of the Fourier Transfoms of spatial brightness distributions of small angular extent,” Mon. Not. Roy. Astron. Soc. 118(3), 276–284(1958).

Jerjen, I.

V. Revol, C. Kottler, R. Kaufmann, I. Jerjen, T. lüthi, F. Cardot, P. Niedermann, U. Straumann, U. Sennhauser, and C. Urban, “X-ray interferometer with bent grating: toward larger fields of view,” Nucl. Instr. Meth. Phys. Res. A 648, 302–305 (2011).
[Crossref]

Kang, H. C.

Kaufmann, R.

V. Revol, C. Kottler, R. Kaufmann, I. Jerjen, T. lüthi, F. Cardot, P. Niedermann, U. Straumann, U. Sennhauser, and C. Urban, “X-ray interferometer with bent grating: toward larger fields of view,” Nucl. Instr. Meth. Phys. Res. A 648, 302–305 (2011).
[Crossref]

Kawamoto, S.

A. Momose and S. Kawamoto, “X-ray Talbot interferometry with capillary plates,” Jap. Jour. Appl. Phys 45(1A), 314–316 (2006).
[Crossref]

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jap. Jour. Appl. Phys 42(7B), 866–868 (2003).
[Crossref]

Kim, J. M.

Kobayashi, S.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72(1), 156–160 (1982).
[Crossref]

Kohn, V.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum 66(12), 5486–5492 (1995).
[Crossref]

Koliopoulos, C. L.

Kondoh, T.

Kopace, R.

H. Wen, E. E. Bennett, R. Kopace, A. F. Stein, and V. Pai, “Single-shot x-ray differential phase-contrast and diffraction imaging using two-dimensional transmission grating,” Opt. Lett 35(12), 1932–1934 (2010).
[Crossref] [PubMed]

Kottler, C.

V. Revol, C. Kottler, R. Kaufmann, I. Jerjen, T. lüthi, F. Cardot, P. Niedermann, U. Straumann, U. Sennhauser, and C. Urban, “X-ray interferometer with bent grating: toward larger fields of view,” Nucl. Instr. Meth. Phys. Res. A 648, 302–305 (2011).
[Crossref]

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard X-ray phase tomography with low-brillance sources,” Phys. Rev. Lett 98, 108105 (2007).
[Crossref] [PubMed]

C. Kottler, C. David, F. Pfeiffer, and O. Bunk, “A two-directional approach for grating based differential phase contrast imaging using hard x-rays,” Opt. Express 15(3), 1175–1181 (2007).
[Crossref] [PubMed]

T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brillance sources,” Proc. of SPIE 631863180S (2006).
[Crossref]

Koyama, I.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jap. Jour. Appl. Phys 42(7B), 866–868 (2003).
[Crossref]

Kuznetsov, S.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum 66(12), 5486–5492 (1995).
[Crossref]

Lee, S. Y.

Leger, J. R.

J. R. Leger and G. J. Swanson, “Efficient array illuminator using binary-optics phase plates at fractional-Talbot planes,” Opt. Lett 15(5), 288–290 (1990).
[Crossref] [PubMed]

Liu, C.

lüthi, T.

V. Revol, C. Kottler, R. Kaufmann, I. Jerjen, T. lüthi, F. Cardot, P. Niedermann, U. Straumann, U. Sennhauser, and C. Urban, “X-ray interferometer with bent grating: toward larger fields of view,” Nucl. Instr. Meth. Phys. Res. A 648, 302–305 (2011).
[Crossref]

Macrander, A. T.

Maikusa, H.

Mercère, P.

J. Rizzi, T. Weitkamp, N. Guérineau, M. Idir, P. Mercère, G. Druart, G. Vincent, P. Da Silva, and J. Primot, “Quadriwave lateral shearing interferometry in an achromatic and continuously self-imaging regime for future x-ray phase imaging,” Opt. Lett 36(8), 1398–1400 (2011).
[Crossref] [PubMed]

Mercre, P.

J. Rizzi, P. Mercre, M. Idir, N. Gurineau, E. Sakat, R. Hadar, G. Vincent, P. Da Silva, and J. Primot, “X-ray phase contrast imaging using a broadband X-ray beam and a single phase grating used in its achromatic and propagation-invariant regime,” J. Phys.: Conf. Ser. 425,192002 (2013).
[Crossref]

Michelson, A. A.

A. A. Michelson, Studies in Optics (University of Chicago Press, Chicago, 1927).

Momose, A.

A. Momose, W. Yashiro, H. Maikusa, and Y. Takeda, “High-speed X-ray phase imaging and X-ray phase tomography with Talbot interferometer and white synchrotron radiation,” Opt. Express 17(15), 12540–12545 (2009).
[Crossref] [PubMed]

Y. Takeda, W. Yashiro, T. Hattori, A. Takeuchi, Y. Suzuki, and A. Momose, “Differential phase X-ray imaging microscopy with Talbot interferometer,” Appl. Phys. Express 1, 117002 (2008).
[Crossref]

A. Momose and S. Kawamoto, “X-ray Talbot interferometry with capillary plates,” Jap. Jour. Appl. Phys 45(1A), 314–316 (2006).
[Crossref]

A. Momose, “Recent advances in X-ray phase imaging,” Jap. Jour. Appl. Phys 44(9A), 6355–6367 (2005).
[Crossref]

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jap. Jour. Appl. Phys 42(7B), 866–868 (2003).
[Crossref]

A. Momose, “Phase-sensitive imaging and phase tomography using X-ray interferometers,” Opt. Express 11(19), 2303–2314 (2003).
[Crossref] [PubMed]

Monneret, S.

P. Bon, S. Monneret, and B. Wattelier, “Noninterative boundary-artifact-free wavefront reconstruction from its derivatives,” Appl. Opt 51(23), 5698–5704 (2012).
[Crossref] [PubMed]

Morgan, K. S.

Nagai, K.

Nakamura, T.

Niedermann, P.

V. Revol, C. Kottler, R. Kaufmann, I. Jerjen, T. lüthi, F. Cardot, P. Niedermann, U. Straumann, U. Sennhauser, and C. Urban, “X-ray interferometer with bent grating: toward larger fields of view,” Nucl. Instr. Meth. Phys. Res. A 648, 302–305 (2011).
[Crossref]

Noh, D. Y.

Nomarski, G.

G. Nomarski, “Nouveau dispositif pour l’observation en contraste de phase differentiel,” J.Phys.Radium 16, S88–S88 (1955).

Ouchi, C.

Paganin, D. M.

Pai, V.

H. Wen, E. E. Bennett, R. Kopace, A. F. Stein, and V. Pai, “Single-shot x-ray differential phase-contrast and diffraction imaging using two-dimensional transmission grating,” Opt. Lett 35(12), 1932–1934 (2010).
[Crossref] [PubMed]

Pfeiffer, F.

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard X-ray phase tomography with low-brillance sources,” Phys. Rev. Lett 98, 108105 (2007).
[Crossref] [PubMed]

C. Kottler, C. David, F. Pfeiffer, and O. Bunk, “A two-directional approach for grating based differential phase contrast imaging using hard x-rays,” Opt. Express 15(3), 1175–1181 (2007).
[Crossref] [PubMed]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brillance X-ray sources,” Nature Phys. 2, 258–261 (2006).
[Crossref]

T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brillance sources,” Proc. of SPIE 631863180S (2006).
[Crossref]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005).
[Crossref] [PubMed]

Piponnier, M.

Pogany, A.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996).
[Crossref]

Primot, J.

J. Rizzi, P. Mercre, M. Idir, N. Gurineau, E. Sakat, R. Hadar, G. Vincent, P. Da Silva, and J. Primot, “X-ray phase contrast imaging using a broadband X-ray beam and a single phase grating used in its achromatic and propagation-invariant regime,” J. Phys.: Conf. Ser. 425,192002 (2013).
[Crossref]

J. Rizzi, T. Weitkamp, N. Guérineau, M. Idir, P. Mercère, G. Druart, G. Vincent, P. Da Silva, and J. Primot, “Quadriwave lateral shearing interferometry in an achromatic and continuously self-imaging regime for future x-ray phase imaging,” Opt. Lett 36(8), 1398–1400 (2011).
[Crossref] [PubMed]

M. Piponnier, G. Druart, N. Guérineau, J. L. de Bougrenet, and J. Primot, “Optimal conditions for using the binary approximation of continuously self-imaging gratings,” Opt. Express 19(23), 23054–23066 (2011).
[Crossref] [PubMed]

N. Guérineau, B. Harchaoui, K. Heggarty, and J. Primot, “Generation of achromatic and propagation-invariant spot arrays by use of continuously self-imaging gratings,” Opt. lett 26(7), 411–413 (2001).
[Crossref]

J. Primot and N. Guérineau, “Extended Hartmann test based on the pseudo-giuding property of a Hartmann mask completed by a phase chessboard,” Appl. Opt 39(31), 5715–5720 (2000).
[Crossref]

N. Guérineau, B. Harchaoui, and J. Primot, “Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime,” Opt. Com 180, 199–203 (2000).
[Crossref]

N. Guérineau and J. Primot, “Non diffracting array generation using an N-wave interferometer,” J. Opt. Soc. Am. A 16(2), 293–298 (1999).
[Crossref]

J. Primot and L. Sogno, “Achromatic three-wave (or more) lateral shearing interferometer,” J. Opt. Soc. Am. A 12(12), 2679–2685 (1995).
[Crossref]

J. Primot, “Three-wave lateral shearing interferometry,” Appl. Opt 32(31), 6242–6249 (1993).
[Crossref] [PubMed]

Pritt, M.

D. Ghilia and M. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998) pp. 34.

Revol, V.

V. Revol, C. Kottler, R. Kaufmann, I. Jerjen, T. lüthi, F. Cardot, P. Niedermann, U. Straumann, U. Sennhauser, and C. Urban, “X-ray interferometer with bent grating: toward larger fields of view,” Nucl. Instr. Meth. Phys. Res. A 648, 302–305 (2011).
[Crossref]

Rizzi, J.

J. Rizzi, P. Mercre, M. Idir, N. Gurineau, E. Sakat, R. Hadar, G. Vincent, P. Da Silva, and J. Primot, “X-ray phase contrast imaging using a broadband X-ray beam and a single phase grating used in its achromatic and propagation-invariant regime,” J. Phys.: Conf. Ser. 425,192002 (2013).
[Crossref]

J. Rizzi, T. Weitkamp, N. Guérineau, M. Idir, P. Mercère, G. Druart, G. Vincent, P. Da Silva, and J. Primot, “Quadriwave lateral shearing interferometry in an achromatic and continuously self-imaging regime for future x-ray phase imaging,” Opt. Lett 36(8), 1398–1400 (2011).
[Crossref] [PubMed]

Ruthishauer, S.

S. Ruthishauer, I. Zanette, T. Weitkamp, T. Donath, and C. David, “At-wavelength charcterization of refractive x-ray lenses using a two-dimensional grating interferometer,” Appl. Phys. Lett 99, 221104 (2011).
[Crossref]

Rutishauser, S.

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional X-ray grating interferometer,” Phys. Rev. Lett 105, 248102 (2010).
[Crossref]

Sakat, E.

J. Rizzi, P. Mercre, M. Idir, N. Gurineau, E. Sakat, R. Hadar, G. Vincent, P. Da Silva, and J. Primot, “X-ray phase contrast imaging using a broadband X-ray beam and a single phase grating used in its achromatic and propagation-invariant regime,” J. Phys.: Conf. Ser. 425,192002 (2013).
[Crossref]

sato, G.

Schelokov, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum 66(12), 5486–5492 (1995).
[Crossref]

Sennhauser, U.

V. Revol, C. Kottler, R. Kaufmann, I. Jerjen, T. lüthi, F. Cardot, P. Niedermann, U. Straumann, U. Sennhauser, and C. Urban, “X-ray interferometer with bent grating: toward larger fields of view,” Nucl. Instr. Meth. Phys. Res. A 648, 302–305 (2011).
[Crossref]

Setomoto, Y.

Siu, K. K. W.

Snigirev, A.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum 66(12), 5486–5492 (1995).
[Crossref]

Snigireva, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum 66(12), 5486–5492 (1995).
[Crossref]

Sogno, L.

Southwell, W. H.

W. H. Southwell, “Wavefront estimation from wavefront slope measurements,” J. Opt. Soc. Am. A 70(8), 998–1006 (1980).
[Crossref]

Stampanoni, M.

Stein, A. F.

H. Wen, E. E. Bennett, R. Kopace, A. F. Stein, and V. Pai, “Single-shot x-ray differential phase-contrast and diffraction imaging using two-dimensional transmission grating,” Opt. Lett 35(12), 1932–1934 (2010).
[Crossref] [PubMed]

Stevenson, A. W.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996).
[Crossref]

Straumann, U.

V. Revol, C. Kottler, R. Kaufmann, I. Jerjen, T. lüthi, F. Cardot, P. Niedermann, U. Straumann, U. Sennhauser, and C. Urban, “X-ray interferometer with bent grating: toward larger fields of view,” Nucl. Instr. Meth. Phys. Res. A 648, 302–305 (2011).
[Crossref]

Suzuki, Y.

Y. Takeda, W. Yashiro, T. Hattori, A. Takeuchi, Y. Suzuki, and A. Momose, “Differential phase X-ray imaging microscopy with Talbot interferometer,” Appl. Phys. Express 1, 117002 (2008).
[Crossref]

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jap. Jour. Appl. Phys 42(7B), 866–868 (2003).
[Crossref]

Swanson, G. J.

J. R. Leger and G. J. Swanson, “Efficient array illuminator using binary-optics phase plates at fractional-Talbot planes,” Opt. Lett 15(5), 288–290 (1990).
[Crossref] [PubMed]

Takai, K.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jap. Jour. Appl. Phys 42(7B), 866–868 (2003).
[Crossref]

Takeda, M.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72(1), 156–160 (1982).
[Crossref]

Takeda, Y.

A. Momose, W. Yashiro, H. Maikusa, and Y. Takeda, “High-speed X-ray phase imaging and X-ray phase tomography with Talbot interferometer and white synchrotron radiation,” Opt. Express 17(15), 12540–12545 (2009).
[Crossref] [PubMed]

Y. Takeda, W. Yashiro, T. Hattori, A. Takeuchi, Y. Suzuki, and A. Momose, “Differential phase X-ray imaging microscopy with Talbot interferometer,” Appl. Phys. Express 1, 117002 (2008).
[Crossref]

Takeuchi, A.

Y. Takeda, W. Yashiro, T. Hattori, A. Takeuchi, Y. Suzuki, and A. Momose, “Differential phase X-ray imaging microscopy with Talbot interferometer,” Appl. Phys. Express 1, 117002 (2008).
[Crossref]

Talbot, H. F.

H. F. Talbot, “Facts relating to optical science,” Phil. Mag. Series 3 9, 401–407 (1836).

Teshima, T.

Urban, C.

V. Revol, C. Kottler, R. Kaufmann, I. Jerjen, T. lüthi, F. Cardot, P. Niedermann, U. Straumann, U. Sennhauser, and C. Urban, “X-ray interferometer with bent grating: toward larger fields of view,” Nucl. Instr. Meth. Phys. Res. A 648, 302–305 (2011).
[Crossref]

Velghe, S.

S. Velghe, “Wave-front reconstruction from multidirectional phase derivatives generated by multilateral shearing interferometers,” Opt. Lett 30(3), 245–247 (2005).
[Crossref] [PubMed]

Vincent, G.

J. Rizzi, P. Mercre, M. Idir, N. Gurineau, E. Sakat, R. Hadar, G. Vincent, P. Da Silva, and J. Primot, “X-ray phase contrast imaging using a broadband X-ray beam and a single phase grating used in its achromatic and propagation-invariant regime,” J. Phys.: Conf. Ser. 425,192002 (2013).
[Crossref]

J. Rizzi, T. Weitkamp, N. Guérineau, M. Idir, P. Mercère, G. Druart, G. Vincent, P. Da Silva, and J. Primot, “Quadriwave lateral shearing interferometry in an achromatic and continuously self-imaging regime for future x-ray phase imaging,” Opt. Lett 36(8), 1398–1400 (2011).
[Crossref] [PubMed]

Wang, D.

X. Ge, Z. Wang, K. Gao, K. Zhang, Y. Hong, D. Wang, Z. Zhu, P. Zhu, and Z. Wu, “Inverstigation of the partially coherent effects in a 2D Talbot interferometer,” Anal. Bioanal. Chem 401, 865–870 (2011).
[Crossref] [PubMed]

Wang, Z.

X. Ge, Z. Wang, K. Gao, K. Zhang, Y. Hong, D. Wang, Z. Zhu, P. Zhu, and Z. Wu, “Inverstigation of the partially coherent effects in a 2D Talbot interferometer,” Anal. Bioanal. Chem 401, 865–870 (2011).
[Crossref] [PubMed]

Wattelier, B.

P. Bon, S. Monneret, and B. Wattelier, “Noninterative boundary-artifact-free wavefront reconstruction from its derivatives,” Appl. Opt 51(23), 5698–5704 (2012).
[Crossref] [PubMed]

Weitkamp, T.

J. Rizzi, T. Weitkamp, N. Guérineau, M. Idir, P. Mercère, G. Druart, G. Vincent, P. Da Silva, and J. Primot, “Quadriwave lateral shearing interferometry in an achromatic and continuously self-imaging regime for future x-ray phase imaging,” Opt. Lett 36(8), 1398–1400 (2011).
[Crossref] [PubMed]

S. Ruthishauer, I. Zanette, T. Weitkamp, T. Donath, and C. David, “At-wavelength charcterization of refractive x-ray lenses using a two-dimensional grating interferometer,” Appl. Phys. Lett 99, 221104 (2011).
[Crossref]

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional X-ray grating interferometer,” Phys. Rev. Lett 105, 248102 (2010).
[Crossref]

T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brillance sources,” Proc. of SPIE 631863180S (2006).
[Crossref]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brillance X-ray sources,” Nature Phys. 2, 258–261 (2006).
[Crossref]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005).
[Crossref] [PubMed]

Wen, H.

H. Wen, E. E. Bennett, R. Kopace, A. F. Stein, and V. Pai, “Single-shot x-ray differential phase-contrast and diffraction imaging using two-dimensional transmission grating,” Opt. Lett 35(12), 1932–1934 (2010).
[Crossref] [PubMed]

Wilkins, S. W.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996).
[Crossref]

Wu, Z.

X. Ge, Z. Wang, K. Gao, K. Zhang, Y. Hong, D. Wang, Z. Zhu, P. Zhu, and Z. Wu, “Inverstigation of the partially coherent effects in a 2D Talbot interferometer,” Anal. Bioanal. Chem 401, 865–870 (2011).
[Crossref] [PubMed]

Yamaguchi, K.

Yashiro, W.

A. Momose, W. Yashiro, H. Maikusa, and Y. Takeda, “High-speed X-ray phase imaging and X-ray phase tomography with Talbot interferometer and white synchrotron radiation,” Opt. Express 17(15), 12540–12545 (2009).
[Crossref] [PubMed]

Y. Takeda, W. Yashiro, T. Hattori, A. Takeuchi, Y. Suzuki, and A. Momose, “Differential phase X-ray imaging microscopy with Talbot interferometer,” Appl. Phys. Express 1, 117002 (2008).
[Crossref]

Zanette, I.

S. Ruthishauer, I. Zanette, T. Weitkamp, T. Donath, and C. David, “At-wavelength charcterization of refractive x-ray lenses using a two-dimensional grating interferometer,” Appl. Phys. Lett 99, 221104 (2011).
[Crossref]

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional X-ray grating interferometer,” Phys. Rev. Lett 105, 248102 (2010).
[Crossref]

Zhang, K.

X. Ge, Z. Wang, K. Gao, K. Zhang, Y. Hong, D. Wang, Z. Zhu, P. Zhu, and Z. Wu, “Inverstigation of the partially coherent effects in a 2D Talbot interferometer,” Anal. Bioanal. Chem 401, 865–870 (2011).
[Crossref] [PubMed]

Zhu, P.

X. Ge, Z. Wang, K. Gao, K. Zhang, Y. Hong, D. Wang, Z. Zhu, P. Zhu, and Z. Wu, “Inverstigation of the partially coherent effects in a 2D Talbot interferometer,” Anal. Bioanal. Chem 401, 865–870 (2011).
[Crossref] [PubMed]

Zhu, Z.

X. Ge, Z. Wang, K. Gao, K. Zhang, Y. Hong, D. Wang, Z. Zhu, P. Zhu, and Z. Wu, “Inverstigation of the partially coherent effects in a 2D Talbot interferometer,” Anal. Bioanal. Chem 401, 865–870 (2011).
[Crossref] [PubMed]

Ziegler, E.

Anal. Bioanal. Chem (1)

X. Ge, Z. Wang, K. Gao, K. Zhang, Y. Hong, D. Wang, Z. Zhu, P. Zhu, and Z. Wu, “Inverstigation of the partially coherent effects in a 2D Talbot interferometer,” Anal. Bioanal. Chem 401, 865–870 (2011).
[Crossref] [PubMed]

Appl. Opt (3)

J. Primot and N. Guérineau, “Extended Hartmann test based on the pseudo-giuding property of a Hartmann mask completed by a phase chessboard,” Appl. Opt 39(31), 5715–5720 (2000).
[Crossref]

J. Primot, “Three-wave lateral shearing interferometry,” Appl. Opt 32(31), 6242–6249 (1993).
[Crossref] [PubMed]

P. Bon, S. Monneret, and B. Wattelier, “Noninterative boundary-artifact-free wavefront reconstruction from its derivatives,” Appl. Opt 51(23), 5698–5704 (2012).
[Crossref] [PubMed]

Appl. Phys. Express (1)

Y. Takeda, W. Yashiro, T. Hattori, A. Takeuchi, Y. Suzuki, and A. Momose, “Differential phase X-ray imaging microscopy with Talbot interferometer,” Appl. Phys. Express 1, 117002 (2008).
[Crossref]

Appl. Phys. Lett (2)

S. Ruthishauer, I. Zanette, T. Weitkamp, T. Donath, and C. David, “At-wavelength charcterization of refractive x-ray lenses using a two-dimensional grating interferometer,” Appl. Phys. Lett 99, 221104 (2011).
[Crossref]

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

J. Opt. Soc. Am. A (7)

J. Primot and L. Sogno, “Achromatic three-wave (or more) lateral shearing interferometer,” J. Opt. Soc. Am. A 12(12), 2679–2685 (1995).
[Crossref]

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72(1), 156–160 (1982).
[Crossref]

J. Durnin, “Continuously self-imaging fields of finite aperture,” J. Opt. Soc. Am. A 2, 110 (1985).

N. Guérineau and J. Primot, “Non diffracting array generation using an N-wave interferometer,” J. Opt. Soc. Am. A 16(2), 293–298 (1999).
[Crossref]

D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. A 67(3), 370–375 (1977).
[Crossref]

W. H. Southwell, “Wavefront estimation from wavefront slope measurements,” J. Opt. Soc. Am. A 70(8), 998–1006 (1980).
[Crossref]

K. R. Freischlad and C. L. Koliopoulos, “Wavefront estimation from wavefront slope measurements,” J. Opt. Soc. Am. A 3(11), 1852–1861 (1986).
[Crossref]

J. Phys.: Conf. Ser. (1)

J. Rizzi, P. Mercre, M. Idir, N. Gurineau, E. Sakat, R. Hadar, G. Vincent, P. Da Silva, and J. Primot, “X-ray phase contrast imaging using a broadband X-ray beam and a single phase grating used in its achromatic and propagation-invariant regime,” J. Phys.: Conf. Ser. 425,192002 (2013).
[Crossref]

J.Phys.Radium (1)

G. Nomarski, “Nouveau dispositif pour l’observation en contraste de phase differentiel,” J.Phys.Radium 16, S88–S88 (1955).

Jap. Jour. Appl. Phys (3)

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jap. Jour. Appl. Phys 42(7B), 866–868 (2003).
[Crossref]

A. Momose, “Recent advances in X-ray phase imaging,” Jap. Jour. Appl. Phys 44(9A), 6355–6367 (2005).
[Crossref]

A. Momose and S. Kawamoto, “X-ray Talbot interferometry with capillary plates,” Jap. Jour. Appl. Phys 45(1A), 314–316 (2006).
[Crossref]

Mon. Not. Roy. Astron. Soc. (1)

R. C. Jennison, “A phase sensitive interferometer technique for the measurement of the Fourier Transfoms of spatial brightness distributions of small angular extent,” Mon. Not. Roy. Astron. Soc. 118(3), 276–284(1958).

Nature (1)

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996).
[Crossref]

Nature Phys. (1)

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brillance X-ray sources,” Nature Phys. 2, 258–261 (2006).
[Crossref]

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

V. Revol, C. Kottler, R. Kaufmann, I. Jerjen, T. lüthi, F. Cardot, P. Niedermann, U. Straumann, U. Sennhauser, and C. Urban, “X-ray interferometer with bent grating: toward larger fields of view,” Nucl. Instr. Meth. Phys. Res. A 648, 302–305 (2011).
[Crossref]

Opt. Com (1)

N. Guérineau, B. Harchaoui, and J. Primot, “Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime,” Opt. Com 180, 199–203 (2000).
[Crossref]

Opt. Express (8)

M. Piponnier, G. Druart, N. Guérineau, J. L. de Bougrenet, and J. Primot, “Optimal conditions for using the binary approximation of continuously self-imaging gratings,” Opt. Express 19(23), 23054–23066 (2011).
[Crossref] [PubMed]

K. S. Morgan, D. M. Paganin, and K. K. W. Siu, “Quantitative single-exposure x-ray phase contrast imaging using a single attenuation grid,” Opt. Express 19(20), 19781–19789 (2011).
[Crossref] [PubMed]

C. Kottler, C. David, F. Pfeiffer, and O. Bunk, “A two-directional approach for grating based differential phase contrast imaging using hard x-rays,” Opt. Express 15(3), 1175–1181 (2007).
[Crossref] [PubMed]

H. Itoh, K. Nagai, G. sato, K. Yamaguchi, T. Nakamura, T. Kondoh, C. Ouchi, T. Teshima, Y. Setomoto, and T. Den, “Two-dimensional grating-based X-ray phase contrast imaging using Fourier transform phase retrieval,” Opt. Express 19(4), 3339–3346 (2011).
[Crossref] [PubMed]

A. Momose, W. Yashiro, H. Maikusa, and Y. Takeda, “High-speed X-ray phase imaging and X-ray phase tomography with Talbot interferometer and white synchrotron radiation,” Opt. Express 17(15), 12540–12545 (2009).
[Crossref] [PubMed]

J. M. Kim, I. H. Cho, S. Y. Lee, H. C. Kang, R. Conley, C. Liu, A. T. Macrander, and D. Y. Noh, “Observation of the Talbot effect using broadband hard x-ray beam,” Opt. Express 18(24), 24975–24982 (2010).
[Crossref] [PubMed]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005).
[Crossref] [PubMed]

A. Momose, “Phase-sensitive imaging and phase tomography using X-ray interferometers,” Opt. Express 11(19), 2303–2314 (2003).
[Crossref] [PubMed]

Opt. Lett (5)

H. Wen, E. E. Bennett, R. Kopace, A. F. Stein, and V. Pai, “Single-shot x-ray differential phase-contrast and diffraction imaging using two-dimensional transmission grating,” Opt. Lett 35(12), 1932–1934 (2010).
[Crossref] [PubMed]

J. R. Leger and G. J. Swanson, “Efficient array illuminator using binary-optics phase plates at fractional-Talbot planes,” Opt. Lett 15(5), 288–290 (1990).
[Crossref] [PubMed]

P. Cloetens, J. P. Guigay, C. De Martino, and J. Baruchel, “fractionnal Talbot imaging of phase gratings with hard x rays,” Opt. Lett 22(14), 1059–1061 (1997).
[Crossref] [PubMed]

J. Rizzi, T. Weitkamp, N. Guérineau, M. Idir, P. Mercère, G. Druart, G. Vincent, P. Da Silva, and J. Primot, “Quadriwave lateral shearing interferometry in an achromatic and continuously self-imaging regime for future x-ray phase imaging,” Opt. Lett 36(8), 1398–1400 (2011).
[Crossref] [PubMed]

N. Guérineau, B. Harchaoui, K. Heggarty, and J. Primot, “Generation of achromatic and propagation-invariant spot arrays by use of continuously self-imaging gratings,” Opt. lett 26(7), 411–413 (2001).
[Crossref]

S. Velghe, “Wave-front reconstruction from multidirectional phase derivatives generated by multilateral shearing interferometers,” Opt. Lett 30(3), 245–247 (2005).
[Crossref] [PubMed]

Phil. Mag. Series (1)

H. F. Talbot, “Facts relating to optical science,” Phil. Mag. Series 3 9, 401–407 (1836).

Phys. Rev. Lett (2)

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard X-ray phase tomography with low-brillance sources,” Phys. Rev. Lett 98, 108105 (2007).
[Crossref] [PubMed]

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional X-ray grating interferometer,” Phys. Rev. Lett 105, 248102 (2010).
[Crossref]

Phys. Today (1)

R. Fitzgerald, “Phase sensitive X-ray imaging,” Phys. Today 53, 23–26 (2000).
[Crossref]

Proc. of SPIE (1)

T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brillance sources,” Proc. of SPIE 631863180S (2006).
[Crossref]

Rev. Sci. Instrum (1)

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum 66(12), 5486–5492 (1995).
[Crossref]

Other (3)

M. Creath, “Phase-Measurement Interferometry Techniques” (Elsevier Science, 1988) pp. 349–393.

A. A. Michelson, Studies in Optics (University of Chicago Press, Chicago, 1927).

D. Ghilia and M. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998) pp. 34.

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Figures (7)

Fig. 1
Fig. 1

Experimental set-up as implemented on the Metrology and Tests Beamline at SOLEIL (S = 32 m, D = 40 cm and L = 20 cm).

Fig. 2
Fig. 2

Bevels along the (111) reticular planes of a chemically etched silicon wafer. Fig. 2(a): Sketch of the sample. Fig. 2(b): SEM measurement of the sample.

Fig. 3
Fig. 3

Interferograms. Fig. 3(a): raw interferogram with sample. Fig. 3(b): reference interferogram without sample. Fig. 3(c): enlarged part of the fringe pattern (without sample)

Fig. 4
Fig. 4

Sample Derivatives. Fig. 4(a): Dx, derivative along the X direction. Fig. 4(b): Dy, derivative along the Y direction.

Fig. 5
Fig. 5

Final reconstruction. Fig. 5(a): optical path difference (OPD) of the reference sample. Size of the reconstructed images size is (128 × 128) pixels, instead of (512 × 512) pixels for the interferograms. The resulting effective pixel size in the reconstructed images is 5.36μm. Fig. 5(b): OPD horizontal cross-section [row 80 out of the (128, 128) OPD map given in Fig. 5(a)]. The bevel height is 0.6 A.U., which corresponds to an OPD of 0.21nm as estimated from the SEM measurements. The thin green lines represent a Cartesian mapping useful to read the axes more easily. The two other thick lines are here to mark out the slope. Fig. 5(c): 20 × 20 pixels area out of the bevel, after tilt subtraction. The standard deviation in this area is equal to 2 × 10−3A.U., which corresponds to 0.7pm. It leads to a SNR of 300. We can also notice in Fig. 5(b) that the areas corresponding to the bevels are flat whereas those corresponding to the rest of the wafer are slightly non-uniform.

Fig. 6
Fig. 6

Application of the SPGI on a biological fossil. Fig. 6(a): reconstructed OPD of a mosquito trapped in amber. Fig. 6(b): mosquito viewed under a microscope. The red square marks out in Fig. 6(b) the area of the mosquito observed in Fig. 6(a). We can clearly observe some details of the anatomy of the mosquito in the OPD image.

Fig. 7
Fig. 7

Phase Derivatives Closure Map (PDCM) of the calibration phantom [Fig. 7(a)] and the mosquito [Fig. 7(b)].

Equations (13)

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Z panchro = 2 a 0 2 η 2 Δ λ
Φ ( x , y ) = F T 1 [ D ˜ x ( u , v ) + i * D ˜ y ( u , v ) u + i * v ] ( x , y )
( Φ ( x , y ) / y ) x = ( Φ ( x , y ) / x ) y
C ( x , y ) = D y x D x y
D x = Φ ( x , y ) x and D y = Φ ( x , y ) y
C ˜ ( u , v ) = 2 i π [ u * D ˜ y ( u , v ) v * D ˜ x ( u , v ) ]
D ˜ x ( u , v ) = i 2 π u Φ ˜ ( u , v ) and D ˜ y ( u , v ) = i 2 π v Φ ˜ ( u , v )
E ( Φ ˜ ) = D x ( u , v ) i 2 π u Φ ˜ 2 D y ( u , v ) i 2 π v Φ ˜
Φ ˜ L S ( u , v ) = i 2 π ( u * D ˜ x ( u , v ) + v * D ˜ y ( u , v ) ) u 2 + v 2
Φ ˜ G ( u , v ) = D ˜ x ( u , v ) + i * D ˜ y ( u , v ) u + i * v
Φ ˜ G ( u , v ) = 2 π ( H ˜ ( u , v ) + i Φ ˜ L S ( u , v ) )
H ˜ ( u , v ) = [ C ˜ ( u , v ) 4 π 2 ( u 2 + v 2 ) ]
Im [ Φ ˜ G ( u , v ) ] = 2 π Φ ˜ L S ( u , v )

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