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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    [CrossRef]
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    [CrossRef]
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  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. Express13(16), 6296–6304 (2005).
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    [CrossRef] [PubMed]
  31. N. Guérineau, B. Harchaoui, and J. Primot, “Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime,” Opt. Com180, 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. Lett15(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. Lett22(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. Opt39(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. Lett36(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 SPIE631863180S (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. Chem401, 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. lett26(7), 411–413 (2001).
    [CrossRef]
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    [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. A67(3), 370–375 (1977).
    [CrossRef]
  45. W. H. Southwell, “Wavefront estimation from wavefront slope measurements,” J. Opt. Soc. Am. A70(8), 998–1006 (1980).
    [CrossRef]
  46. K. R. Freischlad and C. L. Koliopoulos, “Wavefront estimation from wavefront slope measurements,” J. Opt. Soc. Am. A3(11), 1852–1861 (1986).
    [CrossRef]
  47. S. Velghe, “Wave-front reconstruction from multidirectional phase derivatives generated by multilateral shearing interferometers,” Opt. Lett30(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. Opt51(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. Lett99, 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. Express19(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. A648, 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. Express19(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. Express19(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. Lett36(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. Chem401, 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. Lett105, 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. Lett35(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. Express18(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. Express1, 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. Phys45(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 SPIE631863180S (2006).
[CrossRef]

2005 (3)

A. Momose, “Recent advances in X-ray phase imaging,” Jap. Jour. Appl. Phys44(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. Express13(16), 6296–6304 (2005).
[CrossRef] [PubMed]

S. Velghe, “Wave-front reconstruction from multidirectional phase derivatives generated by multilateral shearing interferometers,” Opt. Lett30(3), 245–247 (2005).
[CrossRef] [PubMed]

2003 (2)

A. Momose, “Phase-sensitive imaging and phase tomography using X-ray interferometers,” Opt. Express11(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. Phys42(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. lett26(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. Com180, 199–203 (2000).
[CrossRef]

R. Fitzgerald, “Phase sensitive X-ray imaging,” Phys. Today53, 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. Opt39(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. Lett22(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,” Nature384(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. Instrum66(12), 5486–5492 (1995).
[CrossRef]

J. Primot and L. Sogno, “Achromatic three-wave (or more) lateral shearing interferometer,” J. Opt. Soc. Am. A12(12), 2679–2685 (1995).
[CrossRef]

1993 (1)

J. Primot, “Three-wave lateral shearing interferometry,” Appl. Opt32(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. Lett15(5), 288–290 (1990).
[CrossRef] [PubMed]

1986 (1)

1985 (1)

J. Durnin, “Continuously self-imaging fields of finite aperture,” J. Opt. Soc. Am. A2, 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. A72(1), 156–160 (1982).
[CrossRef]

1980 (1)

W. H. Southwell, “Wavefront estimation from wavefront slope measurements,” J. Opt. Soc. Am. A70(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. A67(3), 370–375 (1977).
[CrossRef]

1965 (1)

U. Bonse and M. Hart, “An X-ray interferometer,” Appl. Phys. Lett6(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.Radium16, S88–S88 (1955).

1836 (1)

H. F. Talbot, “Facts relating to optical science,” Phil. Mag. Series39, 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. Lett22(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. Lett35(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. Opt51(23), 5698–5704 (2012).
[CrossRef] [PubMed]

Bonse, U.

U. Bonse and M. Hart, “An X-ray interferometer,” Appl. Phys. Lett6(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. Lett98, 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. Express15(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 SPIE631863180S (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. A648, 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. Express13(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. Lett22(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. Lett36(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. Lett99, 221104 (2011).
[CrossRef]

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional X-ray grating interferometer,” Phys. Rev. Lett105, 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. Express15(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. Lett98, 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 SPIE631863180S (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. Express13(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. Lett22(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. Lett99, 221104 (2011).
[CrossRef]

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional X-ray grating interferometer,” Phys. Rev. Lett105, 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. Lett36(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. Express19(23), 23054–23066 (2011).
[CrossRef] [PubMed]

Durnin, J.

J. Durnin, “Continuously self-imaging fields of finite aperture,” J. Opt. Soc. Am. A2, 110 (1985).

Fitzgerald, R.

R. Fitzgerald, “Phase sensitive X-ray imaging,” Phys. Today53, 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. A67(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,” Nature384(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. Chem401, 865–870 (2011).
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T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express13(16), 6296–6304 (2005).
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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. Lett99, 221104 (2011).
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[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. Chem401, 865–870 (2011).
[CrossRef] [PubMed]

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Anal. Bioanal. Chem (1)

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[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)

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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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