Abstract

X-ray phase imaging with illumination by a partially coherent source with a setup similar to in-line holography is considered. Using the optical transform function, we consider the effects of partial coherence on this x-ray phase imaging for a weak phase object. The optimal contrast and the resolution of phase imaging are analyzed. As the coherence decreases, the imaging contrast and the optimal contrast frequency decrease, and the resolution degrades. It is shown that this contrast-enhanced phase-imaging method can be regarded as a linear bandpass filter and that the bandwidth and the image contrast are changeable. The frequency property of the imaging system can be improved if an incoherent x-ray source with the proper shape is used.

© 1999 Optical Society of America

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  1. V. N. Ingal and E. A. Beliaevskaya, J. Phys. D 28, 2314 (1995).
    [CrossRef]
  2. A. Momose, Nucl. Instrum. Methods Res. Sect. A 352, 622 (1995).
    [CrossRef]
  3. P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, J. Phys. D 29, 133 (1996).
    [CrossRef]
  4. T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, Nature (London)373, 595 (1995).
    [CrossRef]
  5. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
    [CrossRef] [PubMed]
  6. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, Rev. Sci. Instrum. 66, 5486 (1995).
    [CrossRef]
  7. C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, and V. Kohn, Appl. Phys. Lett. 69, 1826 (1996).
    [CrossRef]
  8. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, Nature (London)384, 335 (1996).
    [CrossRef]
  9. A. Pogany, D. Gao, and S. W. Wilkins, Rev. Sci. Instrum. 68, 2774 (1997).
    [CrossRef]
  10. D. Paganin and K. A. Nugent, Phys. Rev. Lett. 80, 2586 (1998).
    [CrossRef]
  11. L. Jochum and W. Meyer-Llse, Appl. Opt. 34, 4944 (1995).
    [CrossRef] [PubMed]
  12. A. Snigirev, I. Snigireva, P. Bosecke, S. Lequien, and I. Schelokov, Opt. Commun. 135, 378 (1997).
    [CrossRef]
  13. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

1998 (1)

D. Paganin and K. A. Nugent, Phys. Rev. Lett. 80, 2586 (1998).
[CrossRef]

1997 (2)

A. Snigirev, I. Snigireva, P. Bosecke, S. Lequien, and I. Schelokov, Opt. Commun. 135, 378 (1997).
[CrossRef]

A. Pogany, D. Gao, and S. W. Wilkins, Rev. Sci. Instrum. 68, 2774 (1997).
[CrossRef]

1996 (3)

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, and V. Kohn, Appl. Phys. Lett. 69, 1826 (1996).
[CrossRef]

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, J. Phys. D 29, 133 (1996).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
[CrossRef] [PubMed]

1995 (4)

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, Rev. Sci. Instrum. 66, 5486 (1995).
[CrossRef]

V. N. Ingal and E. A. Beliaevskaya, J. Phys. D 28, 2314 (1995).
[CrossRef]

A. Momose, Nucl. Instrum. Methods Res. Sect. A 352, 622 (1995).
[CrossRef]

L. Jochum and W. Meyer-Llse, Appl. Opt. 34, 4944 (1995).
[CrossRef] [PubMed]

Barnea, Z.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
[CrossRef] [PubMed]

Barrett, R.

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, J. Phys. D 29, 133 (1996).
[CrossRef]

Baruchel, J.

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, J. Phys. D 29, 133 (1996).
[CrossRef]

Beliaevskaya, E. A.

V. N. Ingal and E. A. Beliaevskaya, J. Phys. D 28, 2314 (1995).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

Bosecke, P.

A. Snigirev, I. Snigireva, P. Bosecke, S. Lequien, and I. Schelokov, Opt. Commun. 135, 378 (1997).
[CrossRef]

Cloetens, P.

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, J. Phys. D 29, 133 (1996).
[CrossRef]

Cookson, D. F.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
[CrossRef] [PubMed]

Davis, T. J.

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, Nature (London)373, 595 (1995).
[CrossRef]

Gao, D.

A. Pogany, D. Gao, and S. W. Wilkins, Rev. Sci. Instrum. 68, 2774 (1997).
[CrossRef]

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, Nature (London)373, 595 (1995).
[CrossRef]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, Nature (London)384, 335 (1996).
[CrossRef]

Guigay, J.-P.

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, J. Phys. D 29, 133 (1996).
[CrossRef]

Gureyev, T. E.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
[CrossRef] [PubMed]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, Nature (London)384, 335 (1996).
[CrossRef]

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, Nature (London)373, 595 (1995).
[CrossRef]

Ingal, V. N.

V. N. Ingal and E. A. Beliaevskaya, J. Phys. D 28, 2314 (1995).
[CrossRef]

Jochum, L.

Kohn, V.

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, and V. Kohn, Appl. Phys. Lett. 69, 1826 (1996).
[CrossRef]

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, Rev. Sci. Instrum. 66, 5486 (1995).
[CrossRef]

Kuznetsov, S.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, Rev. Sci. Instrum. 66, 5486 (1995).
[CrossRef]

Lequien, S.

A. Snigirev, I. Snigireva, P. Bosecke, S. Lequien, and I. Schelokov, Opt. Commun. 135, 378 (1997).
[CrossRef]

Meyer-Llse, W.

Momose, A.

A. Momose, Nucl. Instrum. Methods Res. Sect. A 352, 622 (1995).
[CrossRef]

Nugent, K. A.

D. Paganin and K. A. Nugent, Phys. Rev. Lett. 80, 2586 (1998).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
[CrossRef] [PubMed]

Paganin, D.

D. Paganin and K. A. Nugent, Phys. Rev. Lett. 80, 2586 (1998).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
[CrossRef] [PubMed]

Pogany, A.

A. Pogany, D. Gao, and S. W. Wilkins, Rev. Sci. Instrum. 68, 2774 (1997).
[CrossRef]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, Nature (London)384, 335 (1996).
[CrossRef]

Raven, C.

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, and V. Kohn, Appl. Phys. Lett. 69, 1826 (1996).
[CrossRef]

Schelokov, I.

A. Snigirev, I. Snigireva, P. Bosecke, S. Lequien, and I. Schelokov, Opt. Commun. 135, 378 (1997).
[CrossRef]

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, Rev. Sci. Instrum. 66, 5486 (1995).
[CrossRef]

Schlenker, M.

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, J. Phys. D 29, 133 (1996).
[CrossRef]

Snigirev, A.

A. Snigirev, I. Snigireva, P. Bosecke, S. Lequien, and I. Schelokov, Opt. Commun. 135, 378 (1997).
[CrossRef]

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, and V. Kohn, Appl. Phys. Lett. 69, 1826 (1996).
[CrossRef]

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, Rev. Sci. Instrum. 66, 5486 (1995).
[CrossRef]

Snigireva, I.

A. Snigirev, I. Snigireva, P. Bosecke, S. Lequien, and I. Schelokov, Opt. Commun. 135, 378 (1997).
[CrossRef]

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, and V. Kohn, Appl. Phys. Lett. 69, 1826 (1996).
[CrossRef]

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, Rev. Sci. Instrum. 66, 5486 (1995).
[CrossRef]

Souvorov, A.

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, and V. Kohn, Appl. Phys. Lett. 69, 1826 (1996).
[CrossRef]

Spanne, P.

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, and V. Kohn, Appl. Phys. Lett. 69, 1826 (1996).
[CrossRef]

Stevenson, A. W.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, Nature (London)384, 335 (1996).
[CrossRef]

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, Nature (London)373, 595 (1995).
[CrossRef]

Wilkins, S. W.

A. Pogany, D. Gao, and S. W. Wilkins, Rev. Sci. Instrum. 68, 2774 (1997).
[CrossRef]

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, Nature (London)373, 595 (1995).
[CrossRef]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, Nature (London)384, 335 (1996).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

Appl. Opt. (1)

Appl. Phys. Lett. (1)

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, and V. Kohn, Appl. Phys. Lett. 69, 1826 (1996).
[CrossRef]

J. Phys. D (2)

V. N. Ingal and E. A. Beliaevskaya, J. Phys. D 28, 2314 (1995).
[CrossRef]

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, J. Phys. D 29, 133 (1996).
[CrossRef]

Nucl. Instrum. Methods Res. Sect. A (1)

A. Momose, Nucl. Instrum. Methods Res. Sect. A 352, 622 (1995).
[CrossRef]

Opt. Commun. (1)

A. Snigirev, I. Snigireva, P. Bosecke, S. Lequien, and I. Schelokov, Opt. Commun. 135, 378 (1997).
[CrossRef]

Phys. Rev. Lett. (2)

D. Paganin and K. A. Nugent, Phys. Rev. Lett. 80, 2586 (1998).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (2)

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, Rev. Sci. Instrum. 66, 5486 (1995).
[CrossRef]

A. Pogany, D. Gao, and S. W. Wilkins, Rev. Sci. Instrum. 68, 2774 (1997).
[CrossRef]

Other (3)

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, Nature (London)384, 335 (1996).
[CrossRef]

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, Nature (London)373, 595 (1995).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Schematic setup of x-ray phase imaging.

Fig. 2
Fig. 2

OTF’s for (a) large and (b) small l when the source is of uniform intensity. The units of the vertical axes are 10 / 2 λ z.

Fig. 3
Fig. 3

OTF improved to yield the ideal bandpass filter. (a) Relative intensity distribution of the source: Because of the size limit, only a 1-mm-wide source is displayed. To obtain the result shown in (b) we use a 4-mm-wide source. (b) OTF. In our calculation, l = z = 1 m. The units of the vertical axis in (a) are in micrometers, and in (b) the units are 250/m.

Fig. 4
Fig. 4

Same as Fig. 3 but for an improved OTF with an easily produced source.

Equations (7)

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I ( x i , y i ) = A d x 1 d y 1 A d x 2 d y 2 1 λ 2 d 1 i d 2 i J 12 ( x 1 , y 1 , x 2 , y 2 ) × exp ( j 2 π λ d 1 i ) exp ( - j 2 π λ d 2 i ) t ( x 1 , y 1 ) t * ( x 2 , y 2 ) ,
J 12 ( x 1 , y 1 , x 2 , y 2 ) = exp [ j π ( x 1 2 + y 1 2 - x 2 2 - y 2 2 ) λ l ] × I ˜ S ( x 1 - x 2 λ l , y 1 - y 2 λ l ) 1 l 2 ,
I ( r ) = A d r 1 A d r 2 1 λ 2 z 2 I ˜ S [ 1 λ l ( r 1 - r 2 ) ] t ( r 1 ) t * ( r 2 ) × exp [ j π λ ( 1 z + 1 l ) ( r 1 2 - r 2 2 ) ] × exp [ - j 2 π λ z ( r 1 - r 2 ) · r ] ,
I ˜ ( k ) = 1 l 2 I ˜ S ( z l k ) d q t ( q + λ z k 2 ) t * ( q - λ z k 2 ) × exp ( j 2 π z f k · q ) ,
I ˜ ( k ) = 1 l 2 I ˜ S ( z l k ) δ ( z f k ) + 2 1 l 2 I ˜ S ( z l k ) × sin ( π λ z z 2 k 2 f 2 ) Φ ( z f k ) ,
H ˜ ( k ) = 2 1 l 2 I ˜ S ( z l k ) sin ( π λ z z 2 k 2 f 2 ) .
I ˜ ( k ) = c 1 δ ( k ) + c 2 Φ ( k z / f ) ,             k [ k d , k u ] ,

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