Abstract

We present a simple solution to the Fresnel–Kirchoff diffraction integral that is appropriate for x-ray radiography of strongly absorbing and phase-shifting objects in the geometrical optics regime, where phase contrast enhancements can be considered to be caused by refraction by a semi-opaque object. We demonstrate its accuracy by comparison to brute-force numerical ray trace and diffraction calculations of a representative simulated object, and show excellent agreement for spatial scales corresponding to Fresnel numbers greater than unity. The result represents a significant improvement over approximate formulas typically used in analysis of refraction-enhanced radiographs, particularly for radiography of transient phenomena in objects that strongly refract and show significant absorption.

© 2013 Optical Society of America

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  1. D. H. Kalantar, S. W. Haan, B. A. Hammel, C. J. Keane, O. L. Landen, and D. H. Munro, “X-ray backlit imaging measurement of in-flight pusher density for an indirect drive capsule implosion,” Rev. Sci. Instrum. 68, 814–816 (1997).
    [CrossRef]
  2. F. J. Marshall, P. W. McKenty, J. A. Delettrez, R. Epstein, J. P. Knauer, V. A. Smalyuk, J. A. Frenje, C. K. Li, R. D. Petrasso, F. H. Sequin, and R. C. Mancini, “Plasma-density determination from x-ray radiography of laser-driven spherical implosions,” Phys. Rev. Lett. 102, 185004 (2009).
    [CrossRef]
  3. D. G. Hicks, B. K. Spears, D. G. Braun, R. E. Olson, C. M. Source, P. M. Celliers, G. W. Collins, and O. L. Landen, “Convergent ablator performance measurements,” Phys. Plasmas 17, 102703 (2010).
    [CrossRef]
  4. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase contrast imaging using polychromatic hard x-rays,” Nature 384, 335–338 (1996).
    [CrossRef]
  5. D. Chapman, W. Thomlinson, R. E. Johnson, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
    [CrossRef]
  6. N. Yagi, Y. Suzuki, K. Umetani, Y. Kohmura, and K. Yamasaki, “Refraction-enhanced x-ray imaging of mouse lung using synchrotron radiation source,” Med. Phys. 26, 2190–2193 (1999).
    [CrossRef]
  7. J. Keyriläinen, M. Fernández, and P. Suortti, “Refraction contrast in x-ray imaging,” Nucl. Instrum. Methods Phys. Res. A 488, 419–427 (2002).
    [CrossRef]
  8. M. N. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from x-ray refraction-angle images,” Phys. Med. Biol. 51, 1769–1778 (2006).
    [CrossRef]
  9. A. Clegg, A. L. Fey, and T. J. W. Lazio, “The Gaussian plasma lens in astrophysics: refraction,” Astrophys. J. 496, 253–266 (1998).
    [CrossRef]
  10. J. A. Koch, O. L. Landen, B. J. Kozioziemski, N. Izumi, E. L. Dewald, J. D. Salmonson, and B. A. Hammel, “Refraction-enhanced x-ray radiography for inertial confinement fusion and laser-produced plasma applications,” J. Appl. Phys. 105, 113112 (2009).
    [CrossRef]
  11. 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]
  12. T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999).
    [CrossRef]
  13. K. Nugent, “X-ray noninterferometric phase imaging: a unified picture,” J. Opt. Soc. Am. A 24, 536–547 (2007).
    [CrossRef]
  14. Y. Suzuki, N. Yagi, and K. Uesugi, “X-ray refraction-enhanced imaging and a method for phase retrieval for a simple object,” J. Synchrotron Radiat. 9, 160–165 (2002).
    [CrossRef]
  15. 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]
  16. T. E. Gureyev and S. W. Wilkins, “On x-ray phase imaging with a point source,” J. Opt. Soc. Am. A 15, 579–585 (1998).
    [CrossRef]
  17. T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative in-line phase contrast imaging with multienergy x-rays,” Phys. Rev. Lett. 86, 5827–5830 (2001).
    [CrossRef]
  18. X. Wu and H. Liu, “A general theoretical formalism for x-ray phase contrast imaging,” J. X-Ray Sci. Technol. 11, 33–42 (2003).
  19. E. Hecht and A. Zajac, Optics (Addison-Wesley, 1974).
  20. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, 1964).
  21. I. Golovkin, R. Mancini, S. Louis, Y. Ochi, K. Fujita, H. Nishimura, H. Shirga, N. Miyanaga, H. Azechi, R. Butzback, I. Uschmann, E. Förster, J. Delettrez, J. Koch, R. W. Lee, and L. Klein, “Spectroscopic determination of dynamic plasma gradients in implosion cores,” Phys. Rev. Lett. 88, 045002 (2002).
    [CrossRef]
  22. I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Elsevier, 2007).
  23. J. A. Koch, O. L. Landen, L. J. Suter, L. P. Masse, D. S. Clark, J. S. Ross, A. J. MacKinnon, N. B. Meezan, C. A. Thomas, and Y. Ping, “Refraction-enhanced backlit imaging of axially symmetric inertial confinement fusion plasmas,” Appl. Opt. 52, 3538–3556 (2013).
    [CrossRef]
  24. J. A. Koch, J. D. Sater, A. J. MacKinnon, T. P. Bernat, D. N. Bittner, G. W. Collins, B. A. Hammel, E. R. Mapoles, and C. H. Still, “Numerical raytrace verification of optical diagnostics of ice surface roughness for inertial confinement fusion experiments,” Fusion Sci. Technol. 43, 55–66 (2003).

2013 (1)

2010 (1)

D. G. Hicks, B. K. Spears, D. G. Braun, R. E. Olson, C. M. Source, P. M. Celliers, G. W. Collins, and O. L. Landen, “Convergent ablator performance measurements,” Phys. Plasmas 17, 102703 (2010).
[CrossRef]

2009 (2)

F. J. Marshall, P. W. McKenty, J. A. Delettrez, R. Epstein, J. P. Knauer, V. A. Smalyuk, J. A. Frenje, C. K. Li, R. D. Petrasso, F. H. Sequin, and R. C. Mancini, “Plasma-density determination from x-ray radiography of laser-driven spherical implosions,” Phys. Rev. Lett. 102, 185004 (2009).
[CrossRef]

J. A. Koch, O. L. Landen, B. J. Kozioziemski, N. Izumi, E. L. Dewald, J. D. Salmonson, and B. A. Hammel, “Refraction-enhanced x-ray radiography for inertial confinement fusion and laser-produced plasma applications,” J. Appl. Phys. 105, 113112 (2009).
[CrossRef]

2007 (1)

2006 (1)

M. N. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from x-ray refraction-angle images,” Phys. Med. Biol. 51, 1769–1778 (2006).
[CrossRef]

2003 (2)

X. Wu and H. Liu, “A general theoretical formalism for x-ray phase contrast imaging,” J. X-Ray Sci. Technol. 11, 33–42 (2003).

J. A. Koch, J. D. Sater, A. J. MacKinnon, T. P. Bernat, D. N. Bittner, G. W. Collins, B. A. Hammel, E. R. Mapoles, and C. H. Still, “Numerical raytrace verification of optical diagnostics of ice surface roughness for inertial confinement fusion experiments,” Fusion Sci. Technol. 43, 55–66 (2003).

2002 (3)

I. Golovkin, R. Mancini, S. Louis, Y. Ochi, K. Fujita, H. Nishimura, H. Shirga, N. Miyanaga, H. Azechi, R. Butzback, I. Uschmann, E. Förster, J. Delettrez, J. Koch, R. W. Lee, and L. Klein, “Spectroscopic determination of dynamic plasma gradients in implosion cores,” Phys. Rev. Lett. 88, 045002 (2002).
[CrossRef]

Y. Suzuki, N. Yagi, and K. Uesugi, “X-ray refraction-enhanced imaging and a method for phase retrieval for a simple object,” J. Synchrotron Radiat. 9, 160–165 (2002).
[CrossRef]

J. Keyriläinen, M. Fernández, and P. Suortti, “Refraction contrast in x-ray imaging,” Nucl. Instrum. Methods Phys. Res. A 488, 419–427 (2002).
[CrossRef]

2001 (1)

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative in-line phase contrast imaging with multienergy x-rays,” Phys. Rev. Lett. 86, 5827–5830 (2001).
[CrossRef]

1999 (2)

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999).
[CrossRef]

N. Yagi, Y. Suzuki, K. Umetani, Y. Kohmura, and K. Yamasaki, “Refraction-enhanced x-ray imaging of mouse lung using synchrotron radiation source,” Med. Phys. 26, 2190–2193 (1999).
[CrossRef]

1998 (2)

A. Clegg, A. L. Fey, and T. J. W. Lazio, “The Gaussian plasma lens in astrophysics: refraction,” Astrophys. J. 496, 253–266 (1998).
[CrossRef]

T. E. Gureyev and S. W. Wilkins, “On x-ray phase imaging with a point source,” J. Opt. Soc. Am. A 15, 579–585 (1998).
[CrossRef]

1997 (3)

D. Chapman, W. Thomlinson, R. E. Johnson, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef]

D. H. Kalantar, S. W. Haan, B. A. Hammel, C. J. Keane, O. L. Landen, and D. H. Munro, “X-ray backlit imaging measurement of in-flight pusher density for an indirect drive capsule implosion,” Rev. Sci. Instrum. 68, 814–816 (1997).
[CrossRef]

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]

1996 (2)

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]

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

Arfelli, F.

D. Chapman, W. Thomlinson, R. E. Johnson, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef]

Azechi, H.

I. Golovkin, R. Mancini, S. Louis, Y. Ochi, K. Fujita, H. Nishimura, H. Shirga, N. Miyanaga, H. Azechi, R. Butzback, I. Uschmann, E. Förster, J. Delettrez, J. Koch, R. W. Lee, and L. Klein, “Spectroscopic determination of dynamic plasma gradients in implosion cores,” Phys. Rev. Lett. 88, 045002 (2002).
[CrossRef]

Barnea, Z.

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]

Bernat, T. P.

J. A. Koch, J. D. Sater, A. J. MacKinnon, T. P. Bernat, D. N. Bittner, G. W. Collins, B. A. Hammel, E. R. Mapoles, and C. H. Still, “Numerical raytrace verification of optical diagnostics of ice surface roughness for inertial confinement fusion experiments,” Fusion Sci. Technol. 43, 55–66 (2003).

Bittner, D. N.

J. A. Koch, J. D. Sater, A. J. MacKinnon, T. P. Bernat, D. N. Bittner, G. W. Collins, B. A. Hammel, E. R. Mapoles, and C. H. Still, “Numerical raytrace verification of optical diagnostics of ice surface roughness for inertial confinement fusion experiments,” Fusion Sci. Technol. 43, 55–66 (2003).

Born, M.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, 1964).

Braun, D. G.

D. G. Hicks, B. K. Spears, D. G. Braun, R. E. Olson, C. M. Source, P. M. Celliers, G. W. Collins, and O. L. Landen, “Convergent ablator performance measurements,” Phys. Plasmas 17, 102703 (2010).
[CrossRef]

Butzback, R.

I. Golovkin, R. Mancini, S. Louis, Y. Ochi, K. Fujita, H. Nishimura, H. Shirga, N. Miyanaga, H. Azechi, R. Butzback, I. Uschmann, E. Förster, J. Delettrez, J. Koch, R. W. Lee, and L. Klein, “Spectroscopic determination of dynamic plasma gradients in implosion cores,” Phys. Rev. Lett. 88, 045002 (2002).
[CrossRef]

Celliers, P. M.

D. G. Hicks, B. K. Spears, D. G. Braun, R. E. Olson, C. M. Source, P. M. Celliers, G. W. Collins, and O. L. Landen, “Convergent ablator performance measurements,” Phys. Plasmas 17, 102703 (2010).
[CrossRef]

Chapman, D.

M. N. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from x-ray refraction-angle images,” Phys. Med. Biol. 51, 1769–1778 (2006).
[CrossRef]

D. Chapman, W. Thomlinson, R. E. Johnson, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef]

Clark, D. S.

Clegg, A.

A. Clegg, A. L. Fey, and T. J. W. Lazio, “The Gaussian plasma lens in astrophysics: refraction,” Astrophys. J. 496, 253–266 (1998).
[CrossRef]

Collins, G. W.

D. G. Hicks, B. K. Spears, D. G. Braun, R. E. Olson, C. M. Source, P. M. Celliers, G. W. Collins, and O. L. Landen, “Convergent ablator performance measurements,” Phys. Plasmas 17, 102703 (2010).
[CrossRef]

J. A. Koch, J. D. Sater, A. J. MacKinnon, T. P. Bernat, D. N. Bittner, G. W. Collins, B. A. Hammel, E. R. Mapoles, and C. H. Still, “Numerical raytrace verification of optical diagnostics of ice surface roughness for inertial confinement fusion experiments,” Fusion Sci. Technol. 43, 55–66 (2003).

Cookson, D. F.

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]

Delettrez, J.

I. Golovkin, R. Mancini, S. Louis, Y. Ochi, K. Fujita, H. Nishimura, H. Shirga, N. Miyanaga, H. Azechi, R. Butzback, I. Uschmann, E. Förster, J. Delettrez, J. Koch, R. W. Lee, and L. Klein, “Spectroscopic determination of dynamic plasma gradients in implosion cores,” Phys. Rev. Lett. 88, 045002 (2002).
[CrossRef]

Delettrez, J. A.

F. J. Marshall, P. W. McKenty, J. A. Delettrez, R. Epstein, J. P. Knauer, V. A. Smalyuk, J. A. Frenje, C. K. Li, R. D. Petrasso, F. H. Sequin, and R. C. Mancini, “Plasma-density determination from x-ray radiography of laser-driven spherical implosions,” Phys. Rev. Lett. 102, 185004 (2009).
[CrossRef]

Dewald, E. L.

J. A. Koch, O. L. Landen, B. J. Kozioziemski, N. Izumi, E. L. Dewald, J. D. Salmonson, and B. A. Hammel, “Refraction-enhanced x-ray radiography for inertial confinement fusion and laser-produced plasma applications,” J. Appl. Phys. 105, 113112 (2009).
[CrossRef]

Epstein, R.

F. J. Marshall, P. W. McKenty, J. A. Delettrez, R. Epstein, J. P. Knauer, V. A. Smalyuk, J. A. Frenje, C. K. Li, R. D. Petrasso, F. H. Sequin, and R. C. Mancini, “Plasma-density determination from x-ray radiography of laser-driven spherical implosions,” Phys. Rev. Lett. 102, 185004 (2009).
[CrossRef]

Fernández, M.

J. Keyriläinen, M. Fernández, and P. Suortti, “Refraction contrast in x-ray imaging,” Nucl. Instrum. Methods Phys. Res. A 488, 419–427 (2002).
[CrossRef]

Fey, A. L.

A. Clegg, A. L. Fey, and T. J. W. Lazio, “The Gaussian plasma lens in astrophysics: refraction,” Astrophys. J. 496, 253–266 (1998).
[CrossRef]

Förster, E.

I. Golovkin, R. Mancini, S. Louis, Y. Ochi, K. Fujita, H. Nishimura, H. Shirga, N. Miyanaga, H. Azechi, R. Butzback, I. Uschmann, E. Förster, J. Delettrez, J. Koch, R. W. Lee, and L. Klein, “Spectroscopic determination of dynamic plasma gradients in implosion cores,” Phys. Rev. Lett. 88, 045002 (2002).
[CrossRef]

Frenje, J. A.

F. J. Marshall, P. W. McKenty, J. A. Delettrez, R. Epstein, J. P. Knauer, V. A. Smalyuk, J. A. Frenje, C. K. Li, R. D. Petrasso, F. H. Sequin, and R. C. Mancini, “Plasma-density determination from x-ray radiography of laser-driven spherical implosions,” Phys. Rev. Lett. 102, 185004 (2009).
[CrossRef]

Fujita, K.

I. Golovkin, R. Mancini, S. Louis, Y. Ochi, K. Fujita, H. Nishimura, H. Shirga, N. Miyanaga, H. Azechi, R. Butzback, I. Uschmann, E. Förster, J. Delettrez, J. Koch, R. W. Lee, and L. Klein, “Spectroscopic determination of dynamic plasma gradients in implosion cores,” Phys. Rev. Lett. 88, 045002 (2002).
[CrossRef]

Gao, D.

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]

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

Gmür, N.

D. Chapman, W. Thomlinson, R. E. Johnson, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef]

Golovkin, I.

I. Golovkin, R. Mancini, S. Louis, Y. Ochi, K. Fujita, H. Nishimura, H. Shirga, N. Miyanaga, H. Azechi, R. Butzback, I. Uschmann, E. Förster, J. Delettrez, J. Koch, R. W. Lee, and L. Klein, “Spectroscopic determination of dynamic plasma gradients in implosion cores,” Phys. Rev. Lett. 88, 045002 (2002).
[CrossRef]

Gradshteyn, I.

I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Elsevier, 2007).

Gureyev, T. E.

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative in-line phase contrast imaging with multienergy x-rays,” Phys. Rev. Lett. 86, 5827–5830 (2001).
[CrossRef]

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999).
[CrossRef]

T. E. Gureyev and S. W. Wilkins, “On x-ray phase imaging with a point source,” J. Opt. Soc. Am. A 15, 579–585 (1998).
[CrossRef]

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

Haan, S. W.

D. H. Kalantar, S. W. Haan, B. A. Hammel, C. J. Keane, O. L. Landen, and D. H. Munro, “X-ray backlit imaging measurement of in-flight pusher density for an indirect drive capsule implosion,” Rev. Sci. Instrum. 68, 814–816 (1997).
[CrossRef]

Hammel, B. A.

J. A. Koch, O. L. Landen, B. J. Kozioziemski, N. Izumi, E. L. Dewald, J. D. Salmonson, and B. A. Hammel, “Refraction-enhanced x-ray radiography for inertial confinement fusion and laser-produced plasma applications,” J. Appl. Phys. 105, 113112 (2009).
[CrossRef]

J. A. Koch, J. D. Sater, A. J. MacKinnon, T. P. Bernat, D. N. Bittner, G. W. Collins, B. A. Hammel, E. R. Mapoles, and C. H. Still, “Numerical raytrace verification of optical diagnostics of ice surface roughness for inertial confinement fusion experiments,” Fusion Sci. Technol. 43, 55–66 (2003).

D. H. Kalantar, S. W. Haan, B. A. Hammel, C. J. Keane, O. L. Landen, and D. H. Munro, “X-ray backlit imaging measurement of in-flight pusher density for an indirect drive capsule implosion,” Rev. Sci. Instrum. 68, 814–816 (1997).
[CrossRef]

Hasnah, M.

M. N. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from x-ray refraction-angle images,” Phys. Med. Biol. 51, 1769–1778 (2006).
[CrossRef]

Hecht, E.

E. Hecht and A. Zajac, Optics (Addison-Wesley, 1974).

Hicks, D. G.

D. G. Hicks, B. K. Spears, D. G. Braun, R. E. Olson, C. M. Source, P. M. Celliers, G. W. Collins, and O. L. Landen, “Convergent ablator performance measurements,” Phys. Plasmas 17, 102703 (2010).
[CrossRef]

Izumi, N.

J. A. Koch, O. L. Landen, B. J. Kozioziemski, N. Izumi, E. L. Dewald, J. D. Salmonson, and B. A. Hammel, “Refraction-enhanced x-ray radiography for inertial confinement fusion and laser-produced plasma applications,” J. Appl. Phys. 105, 113112 (2009).
[CrossRef]

Johnson, R. E.

D. Chapman, W. Thomlinson, R. E. Johnson, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef]

Kalantar, D. H.

D. H. Kalantar, S. W. Haan, B. A. Hammel, C. J. Keane, O. L. Landen, and D. H. Munro, “X-ray backlit imaging measurement of in-flight pusher density for an indirect drive capsule implosion,” Rev. Sci. Instrum. 68, 814–816 (1997).
[CrossRef]

Keane, C. J.

D. H. Kalantar, S. W. Haan, B. A. Hammel, C. J. Keane, O. L. Landen, and D. H. Munro, “X-ray backlit imaging measurement of in-flight pusher density for an indirect drive capsule implosion,” Rev. Sci. Instrum. 68, 814–816 (1997).
[CrossRef]

Keyriläinen, J.

J. Keyriläinen, M. Fernández, and P. Suortti, “Refraction contrast in x-ray imaging,” Nucl. Instrum. Methods Phys. Res. A 488, 419–427 (2002).
[CrossRef]

Klein, L.

I. Golovkin, R. Mancini, S. Louis, Y. Ochi, K. Fujita, H. Nishimura, H. Shirga, N. Miyanaga, H. Azechi, R. Butzback, I. Uschmann, E. Förster, J. Delettrez, J. Koch, R. W. Lee, and L. Klein, “Spectroscopic determination of dynamic plasma gradients in implosion cores,” Phys. Rev. Lett. 88, 045002 (2002).
[CrossRef]

Knauer, J. P.

F. J. Marshall, P. W. McKenty, J. A. Delettrez, R. Epstein, J. P. Knauer, V. A. Smalyuk, J. A. Frenje, C. K. Li, R. D. Petrasso, F. H. Sequin, and R. C. Mancini, “Plasma-density determination from x-ray radiography of laser-driven spherical implosions,” Phys. Rev. Lett. 102, 185004 (2009).
[CrossRef]

Koch, J.

I. Golovkin, R. Mancini, S. Louis, Y. Ochi, K. Fujita, H. Nishimura, H. Shirga, N. Miyanaga, H. Azechi, R. Butzback, I. Uschmann, E. Förster, J. Delettrez, J. Koch, R. W. Lee, and L. Klein, “Spectroscopic determination of dynamic plasma gradients in implosion cores,” Phys. Rev. Lett. 88, 045002 (2002).
[CrossRef]

Koch, J. A.

J. A. Koch, O. L. Landen, L. J. Suter, L. P. Masse, D. S. Clark, J. S. Ross, A. J. MacKinnon, N. B. Meezan, C. A. Thomas, and Y. Ping, “Refraction-enhanced backlit imaging of axially symmetric inertial confinement fusion plasmas,” Appl. Opt. 52, 3538–3556 (2013).
[CrossRef]

J. A. Koch, O. L. Landen, B. J. Kozioziemski, N. Izumi, E. L. Dewald, J. D. Salmonson, and B. A. Hammel, “Refraction-enhanced x-ray radiography for inertial confinement fusion and laser-produced plasma applications,” J. Appl. Phys. 105, 113112 (2009).
[CrossRef]

J. A. Koch, J. D. Sater, A. J. MacKinnon, T. P. Bernat, D. N. Bittner, G. W. Collins, B. A. Hammel, E. R. Mapoles, and C. H. Still, “Numerical raytrace verification of optical diagnostics of ice surface roughness for inertial confinement fusion experiments,” Fusion Sci. Technol. 43, 55–66 (2003).

Kohmura, Y.

N. Yagi, Y. Suzuki, K. Umetani, Y. Kohmura, and K. Yamasaki, “Refraction-enhanced x-ray imaging of mouse lung using synchrotron radiation source,” Med. Phys. 26, 2190–2193 (1999).
[CrossRef]

Kozioziemski, B. J.

J. A. Koch, O. L. Landen, B. J. Kozioziemski, N. Izumi, E. L. Dewald, J. D. Salmonson, and B. A. Hammel, “Refraction-enhanced x-ray radiography for inertial confinement fusion and laser-produced plasma applications,” J. Appl. Phys. 105, 113112 (2009).
[CrossRef]

Landen, O. L.

J. A. Koch, O. L. Landen, L. J. Suter, L. P. Masse, D. S. Clark, J. S. Ross, A. J. MacKinnon, N. B. Meezan, C. A. Thomas, and Y. Ping, “Refraction-enhanced backlit imaging of axially symmetric inertial confinement fusion plasmas,” Appl. Opt. 52, 3538–3556 (2013).
[CrossRef]

D. G. Hicks, B. K. Spears, D. G. Braun, R. E. Olson, C. M. Source, P. M. Celliers, G. W. Collins, and O. L. Landen, “Convergent ablator performance measurements,” Phys. Plasmas 17, 102703 (2010).
[CrossRef]

J. A. Koch, O. L. Landen, B. J. Kozioziemski, N. Izumi, E. L. Dewald, J. D. Salmonson, and B. A. Hammel, “Refraction-enhanced x-ray radiography for inertial confinement fusion and laser-produced plasma applications,” J. Appl. Phys. 105, 113112 (2009).
[CrossRef]

D. H. Kalantar, S. W. Haan, B. A. Hammel, C. J. Keane, O. L. Landen, and D. H. Munro, “X-ray backlit imaging measurement of in-flight pusher density for an indirect drive capsule implosion,” Rev. Sci. Instrum. 68, 814–816 (1997).
[CrossRef]

Lazio, T. J. W.

A. Clegg, A. L. Fey, and T. J. W. Lazio, “The Gaussian plasma lens in astrophysics: refraction,” Astrophys. J. 496, 253–266 (1998).
[CrossRef]

Lee, R. W.

I. Golovkin, R. Mancini, S. Louis, Y. Ochi, K. Fujita, H. Nishimura, H. Shirga, N. Miyanaga, H. Azechi, R. Butzback, I. Uschmann, E. Förster, J. Delettrez, J. Koch, R. W. Lee, and L. Klein, “Spectroscopic determination of dynamic plasma gradients in implosion cores,” Phys. Rev. Lett. 88, 045002 (2002).
[CrossRef]

Li, C. K.

F. J. Marshall, P. W. McKenty, J. A. Delettrez, R. Epstein, J. P. Knauer, V. A. Smalyuk, J. A. Frenje, C. K. Li, R. D. Petrasso, F. H. Sequin, and R. C. Mancini, “Plasma-density determination from x-ray radiography of laser-driven spherical implosions,” Phys. Rev. Lett. 102, 185004 (2009).
[CrossRef]

Liu, H.

X. Wu and H. Liu, “A general theoretical formalism for x-ray phase contrast imaging,” J. X-Ray Sci. Technol. 11, 33–42 (2003).

Louis, S.

I. Golovkin, R. Mancini, S. Louis, Y. Ochi, K. Fujita, H. Nishimura, H. Shirga, N. Miyanaga, H. Azechi, R. Butzback, I. Uschmann, E. Förster, J. Delettrez, J. Koch, R. W. Lee, and L. Klein, “Spectroscopic determination of dynamic plasma gradients in implosion cores,” Phys. Rev. Lett. 88, 045002 (2002).
[CrossRef]

MacKinnon, A. J.

J. A. Koch, O. L. Landen, L. J. Suter, L. P. Masse, D. S. Clark, J. S. Ross, A. J. MacKinnon, N. B. Meezan, C. A. Thomas, and Y. Ping, “Refraction-enhanced backlit imaging of axially symmetric inertial confinement fusion plasmas,” Appl. Opt. 52, 3538–3556 (2013).
[CrossRef]

J. A. Koch, J. D. Sater, A. J. MacKinnon, T. P. Bernat, D. N. Bittner, G. W. Collins, B. A. Hammel, E. R. Mapoles, and C. H. Still, “Numerical raytrace verification of optical diagnostics of ice surface roughness for inertial confinement fusion experiments,” Fusion Sci. Technol. 43, 55–66 (2003).

Mancini, R.

I. Golovkin, R. Mancini, S. Louis, Y. Ochi, K. Fujita, H. Nishimura, H. Shirga, N. Miyanaga, H. Azechi, R. Butzback, I. Uschmann, E. Förster, J. Delettrez, J. Koch, R. W. Lee, and L. Klein, “Spectroscopic determination of dynamic plasma gradients in implosion cores,” Phys. Rev. Lett. 88, 045002 (2002).
[CrossRef]

Mancini, R. C.

F. J. Marshall, P. W. McKenty, J. A. Delettrez, R. Epstein, J. P. Knauer, V. A. Smalyuk, J. A. Frenje, C. K. Li, R. D. Petrasso, F. H. Sequin, and R. C. Mancini, “Plasma-density determination from x-ray radiography of laser-driven spherical implosions,” Phys. Rev. Lett. 102, 185004 (2009).
[CrossRef]

Mapoles, E. R.

J. A. Koch, J. D. Sater, A. J. MacKinnon, T. P. Bernat, D. N. Bittner, G. W. Collins, B. A. Hammel, E. R. Mapoles, and C. H. Still, “Numerical raytrace verification of optical diagnostics of ice surface roughness for inertial confinement fusion experiments,” Fusion Sci. Technol. 43, 55–66 (2003).

Marshall, F. J.

F. J. Marshall, P. W. McKenty, J. A. Delettrez, R. Epstein, J. P. Knauer, V. A. Smalyuk, J. A. Frenje, C. K. Li, R. D. Petrasso, F. H. Sequin, and R. C. Mancini, “Plasma-density determination from x-ray radiography of laser-driven spherical implosions,” Phys. Rev. Lett. 102, 185004 (2009).
[CrossRef]

Masse, L. P.

Mayo, S.

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative in-line phase contrast imaging with multienergy x-rays,” Phys. Rev. Lett. 86, 5827–5830 (2001).
[CrossRef]

McKenty, P. W.

F. J. Marshall, P. W. McKenty, J. A. Delettrez, R. Epstein, J. P. Knauer, V. A. Smalyuk, J. A. Frenje, C. K. Li, R. D. Petrasso, F. H. Sequin, and R. C. Mancini, “Plasma-density determination from x-ray radiography of laser-driven spherical implosions,” Phys. Rev. Lett. 102, 185004 (2009).
[CrossRef]

Meezan, N. B.

Menk, R.

D. Chapman, W. Thomlinson, R. E. Johnson, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef]

Miyanaga, N.

I. Golovkin, R. Mancini, S. Louis, Y. Ochi, K. Fujita, H. Nishimura, H. Shirga, N. Miyanaga, H. Azechi, R. Butzback, I. Uschmann, E. Förster, J. Delettrez, J. Koch, R. W. Lee, and L. Klein, “Spectroscopic determination of dynamic plasma gradients in implosion cores,” Phys. Rev. Lett. 88, 045002 (2002).
[CrossRef]

Mondal, I.

M. N. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from x-ray refraction-angle images,” Phys. Med. Biol. 51, 1769–1778 (2006).
[CrossRef]

Munro, D. H.

D. H. Kalantar, S. W. Haan, B. A. Hammel, C. J. Keane, O. L. Landen, and D. H. Munro, “X-ray backlit imaging measurement of in-flight pusher density for an indirect drive capsule implosion,” Rev. Sci. Instrum. 68, 814–816 (1997).
[CrossRef]

Nishimura, H.

I. Golovkin, R. Mancini, S. Louis, Y. Ochi, K. Fujita, H. Nishimura, H. Shirga, N. Miyanaga, H. Azechi, R. Butzback, I. Uschmann, E. Förster, J. Delettrez, J. Koch, R. W. Lee, and L. Klein, “Spectroscopic determination of dynamic plasma gradients in implosion cores,” Phys. Rev. Lett. 88, 045002 (2002).
[CrossRef]

Nugent, K.

Nugent, K. A.

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]

Ochi, Y.

I. Golovkin, R. Mancini, S. Louis, Y. Ochi, K. Fujita, H. Nishimura, H. Shirga, N. Miyanaga, H. Azechi, R. Butzback, I. Uschmann, E. Förster, J. Delettrez, J. Koch, R. W. Lee, and L. Klein, “Spectroscopic determination of dynamic plasma gradients in implosion cores,” Phys. Rev. Lett. 88, 045002 (2002).
[CrossRef]

Olson, R. E.

D. G. Hicks, B. K. Spears, D. G. Braun, R. E. Olson, C. M. Source, P. M. Celliers, G. W. Collins, and O. L. Landen, “Convergent ablator performance measurements,” Phys. Plasmas 17, 102703 (2010).
[CrossRef]

Paganin, D.

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative in-line phase contrast imaging with multienergy x-rays,” Phys. Rev. Lett. 86, 5827–5830 (2001).
[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]

Parham, C.

M. N. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from x-ray refraction-angle images,” Phys. Med. Biol. 51, 1769–1778 (2006).
[CrossRef]

Petrasso, R. D.

F. J. Marshall, P. W. McKenty, J. A. Delettrez, R. Epstein, J. P. Knauer, V. A. Smalyuk, J. A. Frenje, C. K. Li, R. D. Petrasso, F. H. Sequin, and R. C. Mancini, “Plasma-density determination from x-ray radiography of laser-driven spherical implosions,” Phys. Rev. Lett. 102, 185004 (2009).
[CrossRef]

Ping, Y.

Pisano, E.

M. N. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from x-ray refraction-angle images,” Phys. Med. Biol. 51, 1769–1778 (2006).
[CrossRef]

D. Chapman, W. Thomlinson, R. E. Johnson, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef]

Pogany, A.

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]

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

Raven, C.

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999).
[CrossRef]

Ross, J. S.

Ryzhik, I.

I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Elsevier, 2007).

Salmonson, J. D.

J. A. Koch, O. L. Landen, B. J. Kozioziemski, N. Izumi, E. L. Dewald, J. D. Salmonson, and B. A. Hammel, “Refraction-enhanced x-ray radiography for inertial confinement fusion and laser-produced plasma applications,” J. Appl. Phys. 105, 113112 (2009).
[CrossRef]

Sater, J. D.

J. A. Koch, J. D. Sater, A. J. MacKinnon, T. P. Bernat, D. N. Bittner, G. W. Collins, B. A. Hammel, E. R. Mapoles, and C. H. Still, “Numerical raytrace verification of optical diagnostics of ice surface roughness for inertial confinement fusion experiments,” Fusion Sci. Technol. 43, 55–66 (2003).

Sayers, D.

D. Chapman, W. Thomlinson, R. E. Johnson, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef]

Sequin, F. H.

F. J. Marshall, P. W. McKenty, J. A. Delettrez, R. Epstein, J. P. Knauer, V. A. Smalyuk, J. A. Frenje, C. K. Li, R. D. Petrasso, F. H. Sequin, and R. C. Mancini, “Plasma-density determination from x-ray radiography of laser-driven spherical implosions,” Phys. Rev. Lett. 102, 185004 (2009).
[CrossRef]

Shirga, H.

I. Golovkin, R. Mancini, S. Louis, Y. Ochi, K. Fujita, H. Nishimura, H. Shirga, N. Miyanaga, H. Azechi, R. Butzback, I. Uschmann, E. Förster, J. Delettrez, J. Koch, R. W. Lee, and L. Klein, “Spectroscopic determination of dynamic plasma gradients in implosion cores,” Phys. Rev. Lett. 88, 045002 (2002).
[CrossRef]

Smalyuk, V. A.

F. J. Marshall, P. W. McKenty, J. A. Delettrez, R. Epstein, J. P. Knauer, V. A. Smalyuk, J. A. Frenje, C. K. Li, R. D. Petrasso, F. H. Sequin, and R. C. Mancini, “Plasma-density determination from x-ray radiography of laser-driven spherical implosions,” Phys. Rev. Lett. 102, 185004 (2009).
[CrossRef]

Snigirev, A.

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999).
[CrossRef]

Snigireva, I.

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999).
[CrossRef]

Source, C. M.

D. G. Hicks, B. K. Spears, D. G. Braun, R. E. Olson, C. M. Source, P. M. Celliers, G. W. Collins, and O. L. Landen, “Convergent ablator performance measurements,” Phys. Plasmas 17, 102703 (2010).
[CrossRef]

Spears, B. K.

D. G. Hicks, B. K. Spears, D. G. Braun, R. E. Olson, C. M. Source, P. M. Celliers, G. W. Collins, and O. L. Landen, “Convergent ablator performance measurements,” Phys. Plasmas 17, 102703 (2010).
[CrossRef]

Stevenson, A. W.

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative in-line phase contrast imaging with multienergy x-rays,” Phys. Rev. Lett. 86, 5827–5830 (2001).
[CrossRef]

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

Still, C. H.

J. A. Koch, J. D. Sater, A. J. MacKinnon, T. P. Bernat, D. N. Bittner, G. W. Collins, B. A. Hammel, E. R. Mapoles, and C. H. Still, “Numerical raytrace verification of optical diagnostics of ice surface roughness for inertial confinement fusion experiments,” Fusion Sci. Technol. 43, 55–66 (2003).

Suortti, P.

J. Keyriläinen, M. Fernández, and P. Suortti, “Refraction contrast in x-ray imaging,” Nucl. Instrum. Methods Phys. Res. A 488, 419–427 (2002).
[CrossRef]

Suter, L. J.

Suzuki, Y.

Y. Suzuki, N. Yagi, and K. Uesugi, “X-ray refraction-enhanced imaging and a method for phase retrieval for a simple object,” J. Synchrotron Radiat. 9, 160–165 (2002).
[CrossRef]

N. Yagi, Y. Suzuki, K. Umetani, Y. Kohmura, and K. Yamasaki, “Refraction-enhanced x-ray imaging of mouse lung using synchrotron radiation source,” Med. Phys. 26, 2190–2193 (1999).
[CrossRef]

Thomas, C. A.

Thomlinson, W.

D. Chapman, W. Thomlinson, R. E. Johnson, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef]

Uesugi, K.

Y. Suzuki, N. Yagi, and K. Uesugi, “X-ray refraction-enhanced imaging and a method for phase retrieval for a simple object,” J. Synchrotron Radiat. 9, 160–165 (2002).
[CrossRef]

Umetani, K.

N. Yagi, Y. Suzuki, K. Umetani, Y. Kohmura, and K. Yamasaki, “Refraction-enhanced x-ray imaging of mouse lung using synchrotron radiation source,” Med. Phys. 26, 2190–2193 (1999).
[CrossRef]

Uschmann, I.

I. Golovkin, R. Mancini, S. Louis, Y. Ochi, K. Fujita, H. Nishimura, H. Shirga, N. Miyanaga, H. Azechi, R. Butzback, I. Uschmann, E. Förster, J. Delettrez, J. Koch, R. W. Lee, and L. Klein, “Spectroscopic determination of dynamic plasma gradients in implosion cores,” Phys. Rev. Lett. 88, 045002 (2002).
[CrossRef]

Washburn, D.

D. Chapman, W. Thomlinson, R. E. Johnson, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef]

Wernick, M. N.

M. N. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from x-ray refraction-angle images,” Phys. Med. Biol. 51, 1769–1778 (2006).
[CrossRef]

Wilkins, S. W.

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative in-line phase contrast imaging with multienergy x-rays,” Phys. Rev. Lett. 86, 5827–5830 (2001).
[CrossRef]

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999).
[CrossRef]

T. E. Gureyev and S. W. Wilkins, “On x-ray phase imaging with a point source,” J. Opt. Soc. Am. A 15, 579–585 (1998).
[CrossRef]

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]

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

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, 1964).

Wu, X.

X. Wu and H. Liu, “A general theoretical formalism for x-ray phase contrast imaging,” J. X-Ray Sci. Technol. 11, 33–42 (2003).

Yagi, N.

Y. Suzuki, N. Yagi, and K. Uesugi, “X-ray refraction-enhanced imaging and a method for phase retrieval for a simple object,” J. Synchrotron Radiat. 9, 160–165 (2002).
[CrossRef]

N. Yagi, Y. Suzuki, K. Umetani, Y. Kohmura, and K. Yamasaki, “Refraction-enhanced x-ray imaging of mouse lung using synchrotron radiation source,” Med. Phys. 26, 2190–2193 (1999).
[CrossRef]

Yamasaki, K.

N. Yagi, Y. Suzuki, K. Umetani, Y. Kohmura, and K. Yamasaki, “Refraction-enhanced x-ray imaging of mouse lung using synchrotron radiation source,” Med. Phys. 26, 2190–2193 (1999).
[CrossRef]

Yang, Y.

M. N. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from x-ray refraction-angle images,” Phys. Med. Biol. 51, 1769–1778 (2006).
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E. Hecht and A. Zajac, Optics (Addison-Wesley, 1974).

Zhong, Z.

M. N. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from x-ray refraction-angle images,” Phys. Med. Biol. 51, 1769–1778 (2006).
[CrossRef]

D. Chapman, W. Thomlinson, R. E. Johnson, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef]

Appl. Opt. (1)

Astrophys. J. (1)

A. Clegg, A. L. Fey, and T. J. W. Lazio, “The Gaussian plasma lens in astrophysics: refraction,” Astrophys. J. 496, 253–266 (1998).
[CrossRef]

Fusion Sci. Technol. (1)

J. A. Koch, J. D. Sater, A. J. MacKinnon, T. P. Bernat, D. N. Bittner, G. W. Collins, B. A. Hammel, E. R. Mapoles, and C. H. Still, “Numerical raytrace verification of optical diagnostics of ice surface roughness for inertial confinement fusion experiments,” Fusion Sci. Technol. 43, 55–66 (2003).

J. Appl. Phys. (1)

J. A. Koch, O. L. Landen, B. J. Kozioziemski, N. Izumi, E. L. Dewald, J. D. Salmonson, and B. A. Hammel, “Refraction-enhanced x-ray radiography for inertial confinement fusion and laser-produced plasma applications,” J. Appl. Phys. 105, 113112 (2009).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Phys. D (1)

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999).
[CrossRef]

J. Synchrotron Radiat. (1)

Y. Suzuki, N. Yagi, and K. Uesugi, “X-ray refraction-enhanced imaging and a method for phase retrieval for a simple object,” J. Synchrotron Radiat. 9, 160–165 (2002).
[CrossRef]

J. X-Ray Sci. Technol. (1)

X. Wu and H. Liu, “A general theoretical formalism for x-ray phase contrast imaging,” J. X-Ray Sci. Technol. 11, 33–42 (2003).

Med. Phys. (1)

N. Yagi, Y. Suzuki, K. Umetani, Y. Kohmura, and K. Yamasaki, “Refraction-enhanced x-ray imaging of mouse lung using synchrotron radiation source,” Med. Phys. 26, 2190–2193 (1999).
[CrossRef]

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, 335–338 (1996).
[CrossRef]

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

J. Keyriläinen, M. Fernández, and P. Suortti, “Refraction contrast in x-ray imaging,” Nucl. Instrum. Methods Phys. Res. A 488, 419–427 (2002).
[CrossRef]

Phys. Med. Biol. (2)

M. N. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from x-ray refraction-angle images,” Phys. Med. Biol. 51, 1769–1778 (2006).
[CrossRef]

D. Chapman, W. Thomlinson, R. E. Johnson, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef]

Phys. Plasmas (1)

D. G. Hicks, B. K. Spears, D. G. Braun, R. E. Olson, C. M. Source, P. M. Celliers, G. W. Collins, and O. L. Landen, “Convergent ablator performance measurements,” Phys. Plasmas 17, 102703 (2010).
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Phys. Rev. Lett. (4)

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Figures (2)

Fig. 1.
Fig. 1.

(a) Radial electron density N e ( r ) and radial attenuation coefficient α ( r ) from a hydrodynamics simulation of a particular ICF implosion at a particular time and (b) corresponding transmitted phase shift ϕ ( x ) and optical depth τ ( x ) through a spherical shell plasma having these radial profiles.

Fig. 2.
Fig. 2.

(a) Radiograph profiles of the strongly absorbing and refracting object of Fig. 1(b), calculated by numerical ray tracing, exact FK diffraction integration, Eq. (10) of this work, and Eq. (33) of [18], and (b) a magnified view showing just the x = 466 478 μm region where the differences are most apparent. For comparison, we also show the absorption-alone radiograph, without refractive enhancements, in (a).

Equations (11)

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I z = λ 2 π ( I 2 ϕ + I ϕ ) .
E ( x ) = C S ( x 0 ) exp ( i ( 2 π q λ 1 + ( x x o ) 2 q 2 + ϕ ( x o ) π 2 ) τ ( x o ) 2 ) d x o ,
E ( x ) = 1 λ q exp ( i ( ϕ ( x o ) + π ( x x o ) 2 λ q ) τ ( x o ) 2 ) d x o .
E ( x ) = 1 π exp ( i ( ϕ ( x a u ) + u 2 ) τ ( x a u ) 2 ) d u
E ( x ) = 1 π exp ( i ( ϕ ( x ) a u ϕ 1 ( x ) + ( 1 + a 2 ϕ 2 ( x ) 2 ) u 2 ) τ ( x a u ) 2 ) d u .
E ( x ) = 1 π exp ( τ ( x a u ) 2 ) exp ( i ( ( 1 + a 2 ϕ 2 ( x ) 2 ) u 2 a u ϕ 1 ( x ) ) ) d u = 1 λ q exp ( τ ( x o ) 2 ) exp ( i ( 1 + a 2 ϕ 2 ( x ) 2 ( x x o ) a a ϕ 1 ( x ) 2 1 + a 2 ϕ 2 ( x ) 2 ) 2 ) d x o .
E ( x ) = 1 λ q exp ( τ ( x o ) 2 ) exp ( i ( 1 + a 2 ϕ 2 ( x ) 2 ) ( x x o ) 2 a 2 ) d x o = 1 1 + a 2 2 ϕ 2 ( x ) 1 π exp ( τ ( x b u ) 2 ) exp ( i u 2 ) d u ,
E ( x ) = b exp ( τ ( x ) 2 ) λ q exp ( ( b 2 τ 2 4 u 2 b τ 1 2 u ) ) exp ( i u 2 ) d u .
I ( x ) = I ( x + λ q ϕ 1 2 π ) = exp ( τ ( x ) ) | 1 + λ q ϕ 2 2 π | K ( x ) , K ( x ) = exp ( λ 2 q 2 32 π 2 τ 1 2 τ 2 ( ( 1 + λ q ϕ 2 2 π ) 2 + λ 2 q 2 τ 2 2 16 π 2 ) ) 1 + λ 2 q 2 τ 2 2 16 π 2 ( 1 + λ q ϕ 2 2 π ) 2 1 + 1 8 ( λ q 2 π ) 2 τ 2 ( τ 1 2 τ 2 ) ( 1 + λ q ϕ 2 2 π ) 2 ,
I ( x + λ q 2 π d ϕ ( x ) d x ) = exp ( τ ( x ) ) | 1 + λ q 2 π d 2 ϕ ( x ) d x 2 | .
I ( x , y ) = I ( x + λ q 2 π d ϕ ( x , y ) d x , y + λ q 2 π d ϕ ( x , y ) d y ) = exp ( τ ( x , y ) ) | 1 + λ q 2 π 2 ϕ ( x , y ) | .

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