Abstract

We discuss contrast formation in a propagating x-ray beam. We consider the validity conditions for linear relations based on the transport-of-intensity equation (TIE) and on contrast transfer functions (CTFs). From a single diffracted image, we recover the thickness of a homogeneous object which has substantial absorption and a phase-shift of -0.37radian.

© 2004 Optical Society of America

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References

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  1. G. Schmahl, D. Rudolph and P. Guttmann, ???Phase contrast x-ray microscopy ??? experiments at the BESSY storage ring,??? in X-ray Microscopy II, D. Sayre, M. Howells, J. Kirz and H. Rarback, eds., Vol 56 in Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1988), pp. 228???232
  2. D. Sayre and H. N. Chapman, ???X-ray microscopy,??? Acta. Cryst. A51, 237???252 (1995)
  3. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin and Z. Barnea, ???Quantitative phase imaging using hard x rays,??? Phys. Rev. Lett. 77, 2961???2964 (1996)
    [CrossRef] [PubMed]
  4. A. Pogany, D. Gao and S.W.Wilkins, ???Contrast and resolution in imaging with a microfocus x-ray source,??? Rev. Sci. Instrum. 68, 2774???2782 (1997)
    [CrossRef]
  5. U. Bonse and M. Hart, ???An x-ray interferometer,??? Appl. Phys. Lett. 6, 155???157 (1965)
    [CrossRef]
  6. A. Momose, T. Takeda and Y. Itai, ???Phase-contrast x-ray computed tomography for observing biological specimens and organic materials,??? Rev. Sci. Instrum. 66, 1434???1436 (1995)
    [CrossRef]
  7. Y. Kohmura, H. Takano, Y. Suzuki and T. Ishikawa, ???Shearing x-ray interferometer with an x-ray prism and it improvement,??? in Proc. 7th Intern. Conf. on X-ray Microscopy, D. Joyeux, F. Polack, eds., J. de Physique IV, Vol 104, J. Susini (EDP Sciences, Les Ulis, 2003), pp. 571???574
  8. T.Wilhein, B. Kaulich, E. Di Fabrizio, F. Romanato, M. Altissimo, J. Susini, B. Fayard, U. Neuh¨ausler, S. Cabrini and F. Polack, ???Differential interference contrast x-ray microscopy with twin zone plates at ESRF beamline ID21,??? in Proc. 7th Intern. Conf. on X-ray microscopy, ibid
  9. E. M. Di Fabrizio, D. Cojoc, S. Cabrini, B. Kaulich, J. Susini, P. Facci and T. Wilhein,???Diffractive optical elements for differential interference contrast x-ray microscopy,??? Opt. Express 11, 2278???2288 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-19-2278">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-19-2278</a>
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  10. E. Forster, K. Goetz and P. Zaumseil, ???Double crystal diffractometry for the characterization of targets for laser fusion experiments,??? Krist. Tech. 15, 937???945 (1980)
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  11. J. R. Palmer and G. R. Morrison, ???Differential phase contrast imaging in the scanning transmission x-ray microscope,??? in Short Wavelength Coherent Radiation, P. H. Bucksbaum and N. M. Ceglio, eds., Vol. 11 of OSA Proceedings Series (Optical Society of America, Washington, D. C., 1991), pp. 141???145
  12. C. Jacobsen, M. Howells, J. Kirz and S. Rothman, ???X-ray holographic microscopy using photoresists,??? J. Opt. Soc. Am. A 7, 1847???1861 (1990)
    [CrossRef]
  13. L. J. Allen, W. McBride and M. P. Oxley, ???Exit wave reconstruction using soft x-rays,??? Opt. Commun. 233, 77???82 (2004)
    [CrossRef]
  14. M. R. Teague, ???Deterministic phase retrieval: Greens function solution,??? J. Opt. Soc. Am. A 73, 1434???41 (1983)
    [CrossRef]
  15. D. Paganin and K. A. Nugent, ???Noninterferometric phase imaging with partially coherent light,??? Phys. Rev. Lett. 80, 2586???2589 (1998)
    [CrossRef]
  16. T. E. Gureyev, A. Pogany, D. M. Paganin and S.W. Wilkins, ???Linear algorithms for phase retrieval in the Fresnel region,??? Opt. Commun. 231, 53???70 (2004)
    [CrossRef]
  17. X. Wu and H. Li, ???A general theoretical formalism for x-ray phase contrast imaging,??? J. X-ray Sci. Tech. 11, 33???42 (2003)
  18. J.-P. Guigay, R. H. Wade and C. Delpha, ???Optical diffraction of Lorentz microscope images,??? in Proceedings of the 25th meeting of the Electron Microscopy and Analysis Group, W. C. Nixon, ed. (The Institute of Physics, London, 1971), pp. 238???239
  19. J.-P. Guigay, ???Fourier transform analysis of Fresnel diffraction patterns,??? Optik 49, 121???125 (1977)
  20. D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller and S.W. Wilkins, ???Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,??? J. Microsc 206, 33???40 (2001)
    [CrossRef]
  21. P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J.-P. Guigay and M. Schlenker, ???Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,??? Appl. Phys. Lett. 75, 2912???2914 (1999)
    [CrossRef]
  22. M. H. Maleki and A. J. Devaney, ???Noniterative reconstruction of complex-valued objects from two intensity measurements,??? Opt. Eng. 33, 3243???3253 (1994)
    [CrossRef]
  23. A. N. Tikhonov and V.Y. Arsenin, ???Solutions of Ill-posed Problems??? (V. H. Winston, Washington D.C., 1977)
  24. E. C. Harvey and 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. Chang and S. W. Pang, eds., Proc. SPIE 3223, 26???33 (1997)
  25. L. D. Turner, K. P.Weber, D. Paganin and R. E. Scholten, ???Off-resonant defocus-contrast imaging of cold atoms,??? Opt. Lett. 29, 232???234 (2004)
    [CrossRef] [PubMed]
  26. L. D. Turner, K. F. E. M. Domen,W. Rooijakkers and R. E. Scholten, School of Physics, University of Melbourne 3010, Australia are preparing a manuscript to be called ???Holographic imaging of cold atoms???
  27. M. Centurion, Y. Pu, Z. Liu, D. Psaltis and T.W. H¨ansch, ???Holographic recording of laser-induced plasma,??? Opt. Lett. 29, 772???774 (2004)
    [CrossRef] [PubMed]

Acta. Cryst.

D. Sayre and H. N. Chapman, ???X-ray microscopy,??? Acta. Cryst. A51, 237???252 (1995)

Appl. Phys. Lett.

U. Bonse and M. Hart, ???An x-ray interferometer,??? Appl. Phys. Lett. 6, 155???157 (1965)
[CrossRef]

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J.-P. Guigay and M. Schlenker, ???Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,??? Appl. Phys. Lett. 75, 2912???2914 (1999)
[CrossRef]

EDP Sciences

Y. Kohmura, H. Takano, Y. Suzuki and T. Ishikawa, ???Shearing x-ray interferometer with an x-ray prism and it improvement,??? in Proc. 7th Intern. Conf. on X-ray Microscopy, D. Joyeux, F. Polack, eds., J. de Physique IV, Vol 104, J. Susini (EDP Sciences, Les Ulis, 2003), pp. 571???574

J. Microsc

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller and S.W. Wilkins, ???Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,??? J. Microsc 206, 33???40 (2001)
[CrossRef]

J. Opt. Soc. Am. A

C. Jacobsen, M. Howells, J. Kirz and S. Rothman, ???X-ray holographic microscopy using photoresists,??? J. Opt. Soc. Am. A 7, 1847???1861 (1990)
[CrossRef]

M. R. Teague, ???Deterministic phase retrieval: Greens function solution,??? J. Opt. Soc. Am. A 73, 1434???41 (1983)
[CrossRef]

J. X-ray Sci. Tech.

X. Wu and H. Li, ???A general theoretical formalism for x-ray phase contrast imaging,??? J. X-ray Sci. Tech. 11, 33???42 (2003)

Krist. Tech.

E. Forster, K. Goetz and P. Zaumseil, ???Double crystal diffractometry for the characterization of targets for laser fusion experiments,??? Krist. Tech. 15, 937???945 (1980)
[CrossRef]

Micromachining and Microfabrication Proc

E. C. Harvey and 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. Chang and S. W. Pang, eds., Proc. SPIE 3223, 26???33 (1997)

Opt. Commun.

T. E. Gureyev, A. Pogany, D. M. Paganin and S.W. Wilkins, ???Linear algorithms for phase retrieval in the Fresnel region,??? Opt. Commun. 231, 53???70 (2004)
[CrossRef]

L. J. Allen, W. McBride and M. P. Oxley, ???Exit wave reconstruction using soft x-rays,??? Opt. Commun. 233, 77???82 (2004)
[CrossRef]

Opt. Eng.

M. H. Maleki and A. J. Devaney, ???Noniterative reconstruction of complex-valued objects from two intensity measurements,??? Opt. Eng. 33, 3243???3253 (1994)
[CrossRef]

Opt. Express

Opt. Lett.

Optik

J.-P. Guigay, ???Fourier transform analysis of Fresnel diffraction patterns,??? Optik 49, 121???125 (1977)

OSA

J. R. Palmer and G. R. Morrison, ???Differential phase contrast imaging in the scanning transmission x-ray microscope,??? in Short Wavelength Coherent Radiation, P. H. Bucksbaum and N. M. Ceglio, eds., Vol. 11 of OSA Proceedings Series (Optical Society of America, Washington, D. C., 1991), pp. 141???145

Phys. Rev. Lett.

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

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin and Z. Barnea, ???Quantitative phase imaging using hard x rays,??? Phys. Rev. Lett. 77, 2961???2964 (1996)
[CrossRef] [PubMed]

Rev. Sci. Instrum.

A. Pogany, D. Gao and S.W.Wilkins, ???Contrast and resolution in imaging with a microfocus x-ray source,??? Rev. Sci. Instrum. 68, 2774???2782 (1997)
[CrossRef]

A. Momose, T. Takeda and Y. Itai, ???Phase-contrast x-ray computed tomography for observing biological specimens and organic materials,??? Rev. Sci. Instrum. 66, 1434???1436 (1995)
[CrossRef]

Springer Series in Optical Sciences

G. Schmahl, D. Rudolph and P. Guttmann, ???Phase contrast x-ray microscopy ??? experiments at the BESSY storage ring,??? in X-ray Microscopy II, D. Sayre, M. Howells, J. Kirz and H. Rarback, eds., Vol 56 in Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1988), pp. 228???232

Other

T.Wilhein, B. Kaulich, E. Di Fabrizio, F. Romanato, M. Altissimo, J. Susini, B. Fayard, U. Neuh¨ausler, S. Cabrini and F. Polack, ???Differential interference contrast x-ray microscopy with twin zone plates at ESRF beamline ID21,??? in Proc. 7th Intern. Conf. on X-ray microscopy, ibid

J.-P. Guigay, R. H. Wade and C. Delpha, ???Optical diffraction of Lorentz microscope images,??? in Proceedings of the 25th meeting of the Electron Microscopy and Analysis Group, W. C. Nixon, ed. (The Institute of Physics, London, 1971), pp. 238???239

A. N. Tikhonov and V.Y. Arsenin, ???Solutions of Ill-posed Problems??? (V. H. Winston, Washington D.C., 1977)

L. D. Turner, K. F. E. M. Domen,W. Rooijakkers and R. E. Scholten, School of Physics, University of Melbourne 3010, Australia are preparing a manuscript to be called ???Holographic imaging of cold atoms???

Supplementary Material (1)

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Figures (2)

Fig. 1.
Fig. 1.

Inverse of the contrast transfer functions for the TIE (blue dotted line) and CTF (red line) forms calculated for an infinite grid with spatial feature size of 1.9µm and the experimental conditions. The green dashed line is at the experimental distance. The embedded movie shows the inverse CTFs as shown here with a slider indicating the z position corresponding to diffraction distance. Also shown in the movie is a plot of an input amplitude (green) and the amplitude retrieved using either the CTF (red) or TIE (blue) methods for the indicated propagation distance. The CTF method correctly accounts for the contrast reversals that arise on propagation. The TIE method should only be applied for z closer than the first contrast reversal; it may retrieve inverted amplitudes if applied at greater z. [Media 1]

Fig. 2.
Fig. 2.

(a) CTF-retrieved thickness map for the square with grid lines.(b) Column-average of retrieved thickness for the grid pattern in the region shown in (a) for the TIE solution (blue) and the CTF solution (red). The AFM result (green) shows excellent agreement. AFM measurements also confirm the presence of grid lines outside the square. These are not a retrieval artefact, unlike the circular fringes around the contaminant at centre right. The contaminating material presumably violates the assumption of an homogeneous object.

Equations (20)

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𝓕 [ I ( r , z ) ] = + f * ( r + λ z u 2 ) f ( r λ z u 2 ) exp ( 2 π i r · u ) d r .
f ( r + λ z u 2 ) = f ( r ) + 1 2 λ z u · f ( r ) ,
· ( I ( r , z ) ϕ ( r , z ) ) = 2 π λ z I ( r , z ) ,
j = 2 1 j ! ( 1 2 λ z u · ) j f ( r ) 1 .
f ( r ) = f 0 exp ( μ ( r ) + i ϕ ( r ) ) ,
f ( r ) = f 0 ( 1 μ ( r ) + i ϕ ( r ) ) .
𝓕 [ I ( r , z ) ] = I 0 ( δ ( u ) 2 cos ( π λ z u 2 ) 𝓕 [ μ ( r ) ] + 2 sin ( π λ z u 2 ) 𝓕 [ ϕ ( r ) ] )
ϕ ( r + λ z u 2 ) ϕ ( r λ z u 2 ) 1 .
μ ( r ) = k β T ( r ) and ϕ ( r ) = k δ T ( r ) ,
𝓕 [ I ( r , z ) ] = I 0 + exp ( 2 k β T ( r ) ) exp ( i k δ λ z u · T ( r ) ) exp ( 2 π i r · u ) d r .
𝓕 [ I ( r , z ) ] = I 0 + exp ( 2 k β T ( r ) ) ( 1 i k δ λ z u · T ( r ) ) exp ( 2 π i r · u ) d r .
[ I ( r , z ) ] = I 0 [ exp ( 2 k β T ( r ) ) ] ( 1 + δ β λ z u 2 ) .
T ( r ) = 1 2 k β ln 𝓕 1 [ β β + δ π λ z u 2 𝓕 [ I ( r , z ) I 0 ] ] .
𝓕 [ I ( r , z ) ] = I 0 + exp ( μ ( r + λ z u 2 ) μ ( r λ z u 2 ) + i ( ϕ ( r λ z u 2 ) ϕ ( r + λ z u 2 ) ) )
× exp ( 2 π i r · u ) d r .
𝓕 [ I ( r , z ) ] I 0 = δ ( u ) 𝓕 [ μ ( r + λ z u 2 ) + μ ( r λ z u 2 ) ]
+ i 𝓕 [ ϕ ( r λ z u 2 ) ϕ ( r + λ z u 2 ) ] .
T ( r ) = 𝓕 1 [ 1 2 k ( δ sin ( π λ z u 2 ) + β cos ( π λ z u 2 ) ) 𝓕 [ I ( r , z ) I 0 1 ] ] .
2 μ ( r ) 1 and ϕ ( r + λ z u 2 ) ϕ ( r λ z u 2 ) 1 .
I R 1 ( r , R 2 ) = 1 M 2 I ( r M , R 2 M ) ,

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