Abstract

X-ray phase contrast imaging is a very promising technique that may lead to significant advancements in a variety of fields, perhaps most notably, medical imaging. The radiation physics group at University College London is currently developing an x-ray phase contrast imaging technique that works with laboratory x-ray sources. This system essentially measures the degree to which photons are refracted by regions of an imaged object. The amount of refraction that may be expected to be encountered in practice impacts strongly upon the design of the imaging system. In this paper, we derive an approximate expression between the properties of archetypal imaged objects encountered in practice and the resulting distribution of refracted photons. This is used to derive constraints governing the design of the system.

© 2010 Optical Society of America

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References

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  1. A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by x-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
    [CrossRef]
  2. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nature Phys. 2, 258–261 (2006).
    [CrossRef]
  3. A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. 91, 074106 (2007).
    [CrossRef]
  4. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
  5. P. Munro, K. Ignatyev, R. Speller, and A. Olivo, “The relationship between wave and geometrical optics models of coded aperture type x-ray phase contrast imaging systems,” Opt. Express 18, 4103–4117 (2010).
    [CrossRef] [PubMed]
  6. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
    [CrossRef]

2010 (1)

2007 (1)

A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. 91, 074106 (2007).
[CrossRef]

2006 (2)

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by x-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
[CrossRef]

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

2003 (1)

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Bunk, O.

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

David, C.

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

Hamaishi, Y.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[CrossRef]

Hattori, T.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by x-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
[CrossRef]

Ignatyev, K.

Kawamoto, S.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[CrossRef]

Koyama, I.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[CrossRef]

Momose, A.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by x-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
[CrossRef]

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[CrossRef]

Munro, P.

Olivo, A.

Pfeiffer, F.

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

Speller, R.

Suzuki, Y.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by x-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
[CrossRef]

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[CrossRef]

Takai, K.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[CrossRef]

Takeda, Y.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by x-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
[CrossRef]

Weitkamp, T.

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

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Yashiro, W.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by x-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
[CrossRef]

Appl. Phys. Lett. (1)

A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. 91, 074106 (2007).
[CrossRef]

Jpn. J. Appl. Phys. (2)

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by x-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
[CrossRef]

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[CrossRef]

Nature Phys. (1)

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

Opt. Express (1)

Other (1)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

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Figures (4)

Fig. 1
Fig. 1

System diagram of an XPCi system employing a point source. P is the detector pixel width and Δ P is the displacement of the detector aperture/detector arrangement relative to the rojection of the sample apertures. The detector apertures are always centered upon the detector pixels. The object to be imaged would normally by placed adjacent to the sample apertures on the detector side.

Fig. 2
Fig. 2

Sample profile of a cylinder with radius 0.5 mm , δ = 10 6 , P = 100 μm , Δ P = P / 4 , x-ray photon energy 17.5 ke V , z so = 1.6 m , z od = 0.4 m , and a scanning step size of 2 μm . The plot shows a spurious positive peak, indicated by a box, resulting from refracted x-rays, which are detected by a pixel not matched with the part of the sample aperture through which the x-rays entered the system.

Fig. 3
Fig. 3

Plots of | U in ( x ) + U s ( x ) | 2 | U in ( x ) | 2 (solid curve, normalized by | U 1 p | 2 ) and I N ( x ) (broken curve) for a variety of values of δ and R. The left- and right-hand columns correspond to delta values of 10 6 and 10 7 , respectively. The rows correspond to values of radii, from the top down, of 10, 100, and 1000 μm , respectively. Note that | U in ( x ) + U s ( x ) | 2 | U in ( x ) | 2 is highly oscillatory and is plotted with equal line widths in each set of axes. These results were obtained for a detector periodicity, P, of 85 μm , η = 0.5 , z so = 1.6 m , and z od = 0.4 m . These values match our experimental setup, but the correspondence shown above does not depend on these values.

Fig. 4
Fig. 4

Contours of Δ x for a range of radii and values of δ for a threshold of ζ = 0.01 .

Equations (8)

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U s ( Q ) = C O A [ exp ( i ϕ ( ξ ) ) 1 ] exp ( i k ξ 2 M 2 z od i k ξ x z od ) d ξ ,
A j ( ξ ) = { 1 j P M η P 2 M < ξ < j P M + η P 2 M 0 otherwise , D j ( x ) = { 1 j P η P 2 < x < j P + η P 2 0 otherwise ,
j P M + η P 2 M Δ A = R ,
U s ( x ) C 2 π k | g 1 ¨ ( x , ξ 0 ) | exp ( i [ k g 1 ( x , ξ 0 ) + π 4 ] ) ,
g 1 ( x , ξ ) = M ξ 2 2 z od x ξ z od 2 δ R 2 ξ 2 ,
| U s ( x ) | 2 | C | 2 λ M z od + 2 δ R [ ( x M R ) 3 ( 2 z od δ ) 3 + x M R 2 z od δ ] .
I N ( x ) = M R + 2 δ z od M R + 2 δ z od [ ( x M R 2 z od δ ) 3 + x M R 2 z od δ ] ,
y 3 + y = 1 ζ + M R ( 1 / ζ 1 ) 2 δ z od ,

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