Abstract

X-ray coherence evolution in the imaging process plays a key role for x-ray phase-sensitive imaging. In this work we present a phase-space formulation for the phase-sensitive imaging. The theory is reformulated in terms of the cross-spectral density and associated Wigner distribution. The phase-space formulation enables an explicit and quantitative account of partial coherence effects on phase-sensitive imaging. The presented formulas for x-ray spectral density at the detector can be used for performing accurate phase retrieval and optimizing the phase-contrast visibility. The concept of phase-space shearing length derived from this phase-space formulation clarifies the spatial coherence requirement for phase-sensitive imaging with incoherent sources. The theory has been applied to x-ray Talbot interferometric imaging as well. The peak coherence condition derived reveals new insights into three-grating-based Talbot-interferometric imaging and gratings-based x-ray dark-field imaging.

© 2008 Optical Society of America

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  1. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast micro-imaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486-5492 (1995).
    [CrossRef]
  2. 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]
  3. S. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase contrast imaging using polychromatic hard x-ray,” Nature 384, 335-338 (1996).
    [CrossRef]
  4. P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D Appl. Phys. 29, 133-146 (1996).
  5. F. Arfelli, and V. Bonvicini, A. Bravin,G. Cantatore, E. Castelli,, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography with synchrotron radiation: phase-detected techniques,” Radiology (Oak Brook, Ill.) 215, 286-293 (2000).
  6. C. Kotre and I. Birch, “Phase contrast enhancement of x-ray mammography: a design study,” Phys. Med. Biol. 44, 2853-2866 (1999).
    [CrossRef]
  7. E. Donnelly and R. Price, “Effect of kVp on edge-enhancement index in phase-contrast radiography,” Med. Phys. 29, 999-1002 (2002).
    [CrossRef]
  8. X.Wu and H. Liu, “A general formalism for x-ray phase contrast imaging,” J. X-Ray Sci. Technol. 11, 33-42 (2003).
  9. X. Wu and H. Liu, “Clarification of aspects in in-line phase-sensitive x-ray imaging,” Med. Phys. 34, 737-743 (2007)
    [CrossRef]
  10. R. G. Littlejohn, “The semiclassical evolution of wave packets,” Phys. Rep. 138, 193-291 (1986).
    [CrossRef]
  11. M. Bastiaans,“Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227-1238 (1986).
  12. X. Wu and H. Liu, “A new theory of phase-contrast x-ray imaging based on Wigner distributions,” Med. Phys. 31, 2378-2384 (2004).
    [CrossRef]
  13. X. Wu and H. Liu, “A phase-space formulation for phase-contrast X-ray imaging,” Appl. Opt. 44, 5847-5854 (2005).
    [CrossRef]
  14. M. Teague, “Deterministic phase retrieval: a Green's function solution,” J. Opt. Soc. Am. 73, 1434-1441 (1983).
  15. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
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  17. R. Coisson, “Spatial coherence of synchrotron radiation,” Appl. Opt. 34, 904-908 (1995).
  18. J. Guigay, “Fourier transform analysis of Fresnel diffraction and in-line holograms,” Optik (Jena) 49, 121-125 (1977).
  19. K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961-2965 (1996).
    [CrossRef]
  20. A. Pogany, D. Gao, and S. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774-2782 (1997).
    [CrossRef]
  21. 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]
  22. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “Mixed transfer function and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett. 32, 1617-1619 (2007)
    [CrossRef]
  23. 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]
  24. T. Weitkamp, A. Daiz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296-6304 (2005).
    [CrossRef]
  25. 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]
  26. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray source,” Nature Phys. 2, 258-261 (2006).
    [CrossRef]
  27. F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134-137 (2008).
    [CrossRef]
  28. J. P. Guigay, S. Zabler, P. Cloetens, C. David, R. Mokso, and M. Schlenker, “ The partial Talbot effect and its use in measuring the coherence of synchrotron x-rays,” J. Synchrotron Radiat. 11, 476-482 (2004).
    [CrossRef]

2008 (1)

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134-137 (2008).
[CrossRef]

2007 (2)

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 source,” Nature Phys. 2, 258-261 (2006).
[CrossRef]

2005 (2)

2004 (1)

J. P. Guigay, S. Zabler, P. Cloetens, C. David, R. Mokso, and M. Schlenker, “ The partial Talbot effect and its use in measuring the coherence of synchrotron x-rays,” J. Synchrotron Radiat. 11, 476-482 (2004).
[CrossRef]

2003 (2)

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]

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

2002 (1)

E. Donnelly and R. Price, “Effect of kVp on edge-enhancement index in phase-contrast radiography,” Med. Phys. 29, 999-1002 (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]

2000 (1)

F. Arfelli, and V. Bonvicini, A. Bravin,G. Cantatore, E. Castelli,, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography with synchrotron radiation: phase-detected techniques,” Radiology (Oak Brook, Ill.) 215, 286-293 (2000).

1999 (1)

C. Kotre and I. Birch, “Phase contrast enhancement of x-ray mammography: a design study,” Phys. Med. Biol. 44, 2853-2866 (1999).
[CrossRef]

1997 (1)

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

1996 (3)

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961-2965 (1996).
[CrossRef]

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

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D Appl. Phys. 29, 133-146 (1996).

1995 (4)

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast micro-imaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486-5492 (1995).
[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]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

R. Coisson, “Spatial coherence of synchrotron radiation,” Appl. Opt. 34, 904-908 (1995).

1986 (2)

M. Bastiaans,“Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227-1238 (1986).

R. G. Littlejohn, “The semiclassical evolution of wave packets,” Phys. Rep. 138, 193-291 (1986).
[CrossRef]

1985 (1)

J. Goodman, Statistical Optics (Wiley,1985).

1983 (1)

1977 (1)

J. Guigay, “Fourier transform analysis of Fresnel diffraction and in-line holograms,” Optik (Jena) 49, 121-125 (1977).

Arfelli, F.

F. Arfelli, and V. Bonvicini, A. Bravin,G. Cantatore, E. Castelli,, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography with synchrotron radiation: phase-detected techniques,” Radiology (Oak Brook, Ill.) 215, 286-293 (2000).

Barnea, Z.

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961-2965 (1996).
[CrossRef]

Barrett, R.

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D Appl. Phys. 29, 133-146 (1996).

Baruchel, J.

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D Appl. Phys. 29, 133-146 (1996).

Bastiaans, M.

Bech, M.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134-137 (2008).
[CrossRef]

Birch, I.

C. Kotre and I. Birch, “Phase contrast enhancement of x-ray mammography: a design study,” Phys. Med. Biol. 44, 2853-2866 (1999).
[CrossRef]

Boistel, R.

Bonvicini, V.

F. Arfelli, and V. Bonvicini, A. Bravin,G. Cantatore, E. Castelli,, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography with synchrotron radiation: phase-detected techniques,” Radiology (Oak Brook, Ill.) 215, 286-293 (2000).

Bravin,, A.

F. Arfelli, and V. Bonvicini, A. Bravin,G. Cantatore, E. Castelli,, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography with synchrotron radiation: phase-detected techniques,” Radiology (Oak Brook, Ill.) 215, 286-293 (2000).

Brönnimann, Ch.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134-137 (2008).
[CrossRef]

Bunk, O.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134-137 (2008).
[CrossRef]

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

Cantatore, G.

F. Arfelli, and V. Bonvicini, A. Bravin,G. Cantatore, E. Castelli,, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography with synchrotron radiation: phase-detected techniques,” Radiology (Oak Brook, Ill.) 215, 286-293 (2000).

Castelli, E.

F. Arfelli, and V. Bonvicini, A. Bravin,G. Cantatore, E. Castelli,, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography with synchrotron radiation: phase-detected techniques,” Radiology (Oak Brook, Ill.) 215, 286-293 (2000).

Cloetens, P.

P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “Mixed transfer function and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett. 32, 1617-1619 (2007)
[CrossRef]

T. Weitkamp, A. Daiz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296-6304 (2005).
[CrossRef]

J. P. Guigay, S. Zabler, P. Cloetens, C. David, R. Mokso, and M. Schlenker, “ The partial Talbot effect and its use in measuring the coherence of synchrotron x-rays,” J. Synchrotron Radiat. 11, 476-482 (2004).
[CrossRef]

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D Appl. Phys. 29, 133-146 (1996).

Coisson, R.

Cookson, D.

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961-2965 (1996).
[CrossRef]

Daiz, A.

Dalla Palma, L.

F. Arfelli, and V. Bonvicini, A. Bravin,G. Cantatore, E. Castelli,, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography with synchrotron radiation: phase-detected techniques,” Radiology (Oak Brook, Ill.) 215, 286-293 (2000).

David, C.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134-137 (2008).
[CrossRef]

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

T. Weitkamp, A. Daiz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296-6304 (2005).
[CrossRef]

J. P. Guigay, S. Zabler, P. Cloetens, C. David, R. Mokso, and M. Schlenker, “ The partial Talbot effect and its use in measuring the coherence of synchrotron x-rays,” J. Synchrotron Radiat. 11, 476-482 (2004).
[CrossRef]

Di Michiel, M.

F. Arfelli, and V. Bonvicini, A. Bravin,G. Cantatore, E. Castelli,, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography with synchrotron radiation: phase-detected techniques,” Radiology (Oak Brook, Ill.) 215, 286-293 (2000).

Donnelly, E.

E. Donnelly and R. Price, “Effect of kVp on edge-enhancement index in phase-contrast radiography,” Med. Phys. 29, 999-1002 (2002).
[CrossRef]

Eikenberry, E. F.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134-137 (2008).
[CrossRef]

Gao, D.

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

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

Goodman, J.

J. Goodman, Statistical Optics (Wiley,1985).

Grünzweig, C.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134-137 (2008).
[CrossRef]

Guigay, J.

J. Guigay, “Fourier transform analysis of Fresnel diffraction and in-line holograms,” Optik (Jena) 49, 121-125 (1977).

Guigay, J. P.

J. P. Guigay, S. Zabler, P. Cloetens, C. David, R. Mokso, and M. Schlenker, “ The partial Talbot effect and its use in measuring the coherence of synchrotron x-rays,” J. Synchrotron Radiat. 11, 476-482 (2004).
[CrossRef]

Guigay, J.-P.

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D Appl. Phys. 29, 133-146 (1996).

Guigay, P.

Gureyev, T.

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

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961-2965 (1996).
[CrossRef]

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]

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]

Itai, Y.

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]

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]

Kohn, V.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast micro-imaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486-5492 (1995).
[CrossRef]

Kotre, C.

C. Kotre and I. Birch, “Phase contrast enhancement of x-ray mammography: a design study,” Phys. Med. Biol. 44, 2853-2866 (1999).
[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]

Kraft, P.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134-137 (2008).
[CrossRef]

Kuznetsov, S.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast micro-imaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486-5492 (1995).
[CrossRef]

Langer, M.

Littlejohn, R. G.

R. G. Littlejohn, “The semiclassical evolution of wave packets,” Phys. Rep. 138, 193-291 (1986).
[CrossRef]

Liu, H.

X. Wu and H. Liu, “Clarification of aspects in in-line phase-sensitive x-ray imaging,” Med. Phys. 34, 737-743 (2007)
[CrossRef]

X. Wu and H. Liu, “A phase-space formulation for phase-contrast X-ray imaging,” Appl. Opt. 44, 5847-5854 (2005).
[CrossRef]

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

X. Wu and H. Liu, “A new theory of phase-contrast x-ray imaging based on Wigner distributions,” Med. Phys. 31, 2378-2384 (2004).
[CrossRef]

Longo, R.

F. Arfelli, and V. Bonvicini, A. Bravin,G. Cantatore, E. Castelli,, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography with synchrotron radiation: phase-detected techniques,” Radiology (Oak Brook, Ill.) 215, 286-293 (2000).

M. Schlenker,

J. P. Guigay, S. Zabler, P. Cloetens, C. David, R. Mokso, and M. Schlenker, “ The partial Talbot effect and its use in measuring the coherence of synchrotron x-rays,” J. Synchrotron Radiat. 11, 476-482 (2004).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

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]

Mokso, R.

J. P. Guigay, S. Zabler, P. Cloetens, C. David, R. Mokso, and M. Schlenker, “ The partial Talbot effect and its use in measuring the coherence of synchrotron x-rays,” J. Synchrotron Radiat. 11, 476-482 (2004).
[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]

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]

Nugent, K.

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961-2965 (1996).
[CrossRef]

Olivo, A.

F. Arfelli, and V. Bonvicini, A. Bravin,G. Cantatore, E. Castelli,, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography with synchrotron radiation: phase-detected techniques,” Radiology (Oak Brook, Ill.) 215, 286-293 (2000).

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. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961-2965 (1996).
[CrossRef]

Pani, S.

F. Arfelli, and V. Bonvicini, A. Bravin,G. Cantatore, E. Castelli,, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography with synchrotron radiation: phase-detected techniques,” Radiology (Oak Brook, Ill.) 215, 286-293 (2000).

Pfeiffer, F.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134-137 (2008).
[CrossRef]

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

T. Weitkamp, A. Daiz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296-6304 (2005).
[CrossRef]

Pogany, A.

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

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

Pontoni, D.

F. Arfelli, and V. Bonvicini, A. Bravin,G. Cantatore, E. Castelli,, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography with synchrotron radiation: phase-detected techniques,” Radiology (Oak Brook, Ill.) 215, 286-293 (2000).

Poropat, P.

F. Arfelli, and V. Bonvicini, A. Bravin,G. Cantatore, E. Castelli,, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography with synchrotron radiation: phase-detected techniques,” Radiology (Oak Brook, Ill.) 215, 286-293 (2000).

Prest, M.

F. Arfelli, and V. Bonvicini, A. Bravin,G. Cantatore, E. Castelli,, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography with synchrotron radiation: phase-detected techniques,” Radiology (Oak Brook, Ill.) 215, 286-293 (2000).

Price, R.

E. Donnelly and R. Price, “Effect of kVp on edge-enhancement index in phase-contrast radiography,” Med. Phys. 29, 999-1002 (2002).
[CrossRef]

Rashevsky, A.

F. Arfelli, and V. Bonvicini, A. Bravin,G. Cantatore, E. Castelli,, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography with synchrotron radiation: phase-detected techniques,” Radiology (Oak Brook, Ill.) 215, 286-293 (2000).

Schelokov, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast micro-imaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486-5492 (1995).
[CrossRef]

Schlenker, M.

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D Appl. Phys. 29, 133-146 (1996).

Snigirev, A.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast micro-imaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486-5492 (1995).
[CrossRef]

Snigireva, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast micro-imaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486-5492 (1995).
[CrossRef]

Stampanoni, M.

Stevenson, A.

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

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, T.

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]

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]

Teague, M.

Tromba, G.

F. Arfelli, and V. Bonvicini, A. Bravin,G. Cantatore, E. Castelli,, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography with synchrotron radiation: phase-detected techniques,” Radiology (Oak Brook, Ill.) 215, 286-293 (2000).

Vacchi, A.

F. Arfelli, and V. Bonvicini, A. Bravin,G. Cantatore, E. Castelli,, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography with synchrotron radiation: phase-detected techniques,” Radiology (Oak Brook, Ill.) 215, 286-293 (2000).

Weitkamp, T.

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

T. Weitkamp, A. Daiz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296-6304 (2005).
[CrossRef]

Wilkins, S.

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

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

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Wu,

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

Wu, X.

X. Wu and H. Liu, “Clarification of aspects in in-line phase-sensitive x-ray imaging,” Med. Phys. 34, 737-743 (2007)
[CrossRef]

X. Wu and H. Liu, “A phase-space formulation for phase-contrast X-ray imaging,” Appl. Opt. 44, 5847-5854 (2005).
[CrossRef]

X. Wu and H. Liu, “A new theory of phase-contrast x-ray imaging based on Wigner distributions,” Med. Phys. 31, 2378-2384 (2004).
[CrossRef]

X.,

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

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]

Zabler, S.

J. P. Guigay, S. Zabler, P. Cloetens, C. David, R. Mokso, and M. Schlenker, “ The partial Talbot effect and its use in measuring the coherence of synchrotron x-rays,” J. Synchrotron Radiat. 11, 476-482 (2004).
[CrossRef]

Ziegler, E.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Phys. D Appl. Phys. (1)

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D Appl. Phys. 29, 133-146 (1996).

J. Synchrotron Radiat. (1)

J. P. Guigay, S. Zabler, P. Cloetens, C. David, R. Mokso, and M. Schlenker, “ The partial Talbot effect and its use in measuring the coherence of synchrotron x-rays,” J. Synchrotron Radiat. 11, 476-482 (2004).
[CrossRef]

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

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

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]

Med. Phys. (3)

E. Donnelly and R. Price, “Effect of kVp on edge-enhancement index in phase-contrast radiography,” Med. Phys. 29, 999-1002 (2002).
[CrossRef]

X. Wu and H. Liu, “A new theory of phase-contrast x-ray imaging based on Wigner distributions,” Med. Phys. 31, 2378-2384 (2004).
[CrossRef]

X. Wu and H. Liu, “Clarification of aspects in in-line phase-sensitive x-ray imaging,” Med. Phys. 34, 737-743 (2007)
[CrossRef]

Nat. Mater. (1)

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134-137 (2008).
[CrossRef]

Nature (1)

S. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase contrast imaging using polychromatic hard x-ray,” Nature 384, 335-338 (1996).
[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 source,” Nature Phys. 2, 258-261 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Optik (Jena) (1)

J. Guigay, “Fourier transform analysis of Fresnel diffraction and in-line holograms,” Optik (Jena) 49, 121-125 (1977).

Phys. Med. Biol. (1)

C. Kotre and I. Birch, “Phase contrast enhancement of x-ray mammography: a design study,” Phys. Med. Biol. 44, 2853-2866 (1999).
[CrossRef]

Phys. Rep. (1)

R. G. Littlejohn, “The semiclassical evolution of wave packets,” Phys. Rep. 138, 193-291 (1986).
[CrossRef]

Phys. Rev. Lett. (2)

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961-2965 (1996).
[CrossRef]

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]

Radiology (Oak Brook, Ill.) (1)

F. Arfelli, and V. Bonvicini, A. Bravin,G. Cantatore, E. Castelli,, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography with synchrotron radiation: phase-detected techniques,” Radiology (Oak Brook, Ill.) 215, 286-293 (2000).

Rev. Sci. Instrum. (3)

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast micro-imaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486-5492 (1995).
[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]

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

Other (2)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

J. Goodman, Statistical Optics (Wiley,1985).

Cited By

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Figures (2)

Fig. 1
Fig. 1

(a) Phase-contrast image of a breast lumpectomy specimen. For details see text. (b) Conventional radiograph of the specimen. For details see text.

Fig. 2
Fig. 2

Schematic of the three-grating Talbot-interferometric imaging setting.

Equations (30)

Equations on this page are rendered with MathJax. Learn more.

i 1 k z E ( r , z ) = H ^ E ( r , z ) , H ^ 1 2 k 2 2 + 1 2 ( 1 n 2 ( r , z ) ) ,
Γ τ ( X 1 , X 2 ) = E ( X 1 , t + τ ) · E * ( X 2 , t ) ,
Γ ( X 1 , X 2 ; ν ) = + Γ τ ( X 1 , X 2 ) · exp ( i 2 π ν τ ) d τ ,
W ( r , u ; z ) = Γ ( r + q / 2 , r q / 2 ; z ; ν ) exp ( i 2 π q · u ) d q .
W z + { W , H } = 0 , { W , H } r W · u H u W · r H .
W z + λ u · r W = 0 ,
S ( r ; z ; ν ) = W ( r , u ; z ) d u .
S ( r ; z ; ν ) z = i λ 2 π r · ( q Γ ( r + q / 2 , r q / 2 ; z ; ν ) ) q = 0 ,
S ( r ; z ; ν ) z = λ 2 π ( S ( r ; z ; ν ) · ϕ ( r ; z ) ) ,
ϕ o ( r ) = λ r e ρ e ( r , z ) d z = λ r e ρ e , p ( r ) ,
W ( r , u ; z ) = W ( r λ ( z R 1 ) u , u ; R 1 ) .
S ( r ; R 1 + R 2 ; ν ) = W ( r , u ; R 1 + R 2 ) d u = W ( r λ R 2 u , u ; R 1 ) d u .
S ˜ ( u ; R 1 + R 2 ; ν ) = S i n ( r + λ R 2 u / 2 ; ν ) S i n ( r λ R 2 u / 2 ; ν ) · μ i n ( r + λ R 2 u / 2 , r λ R 2 u / 2 ; ν ) × T ( r + λ R 2 u / 2 ) T * ( r λ R 2 u / 2 ) exp ( i 2 π r · u ) d r ,
μ i n ( r 1 , r 2 ; ν ) = Γ i n ( r 1 , r 2 ; ν ) S i n ( r 1 ; ν ) · S i n ( r 2 ; ν ) .
μ i n ( r 1 , r 2 ; ν ) = exp ( i π r 1 2 r 2 2 λ R 1 ) μ ˜ i n ( r 1 r 2 ; ν ) ,
μ ˜ i n ( r 1 r 2 ; ν ) = 2 J 1 ( π f | r 1 r 2 | / λ R 1 ) π f | r 1 r 2 | / λ R 1 ,
μ ˜ i n ( r 1 r 2 ; ν ) = exp [ 1 2 ( ( x 1 x 2 ) 2 σ μ x 2 + ( y 1 y 2 ) 2 σ μ y ) ] ,
σ μ x 2 = ( λ R 1 2 π ) 1 ( σ e x 2 + 1 4 u 1 x 2 ) 1 4 ( 2 π σ e x / λ ) 2 + 4 u 1 x 2 ,
S ˜ ( u M ; R 1 + R 2 ; ν ) = S i n ( ν ) μ ˜ i n ( λ R 2 u M ; ν ) T ( r + λ R 2 u 2 M ) T * ( r λ R 2 u 2 M ) exp ( i 2 π r · u ) d r .
L shear L = R 2 s | u | M R 1 = ( M 1 ) s | u | M .
μ ˜ i n ( λ R 2 u M ; ν ) = 2 J 1 ( π L shear / L ) ( π L shear / L ) .
| ϕ ( r + λ R 2 u 2 M ) ϕ ( r λ R 2 u 2 M ) | < < 1.
S ˜ ( u M ; R 1 + R 2 ; ν ) = S i n ( ν ) · μ ˜ i n ( λ R 2 u M ; ν ) · O T F det ( u M ) × { cos ( π λ R 2 u 2 M ) · ( F ^ ( A o 2 ) i λ R 2 M u · F ^ ( ϕ A o 2 ) ) + 2 sin ( π λ R 2 u 2 M ) · ( F ^ ( A o 2 ϕ ) + i λ R 2 4 M u · F ^ ( A o 2 ) ) } ,
S ˜ ( u M ; R 1 + R 2 ; ν ) = S i n ( ν ) · μ ˜ i n ( λ R 2 u M ) · O T F det ( u M ) · { F ^ ( A o 2 ) λ R 2 2 π M F ^ ( · ( A o 2 ϕ ) ) } .
S ( r ; R 1 + R 2 ; ν ) = S i n ( ν ) M 2 F ^ 1 ( μ ˜ i n ( λ R 2 u M ; ν ) O T F det ( u M ) ) { A o 2 ( r M ) λ R 2 2 π M · ( A o 2 ϕ ( r M ) ) } ,
I ( r ; R 1 + R 2 ) = I i n M 2 F ^ 1 ( O T F G . U . ( u M ) · O T F det ( u M ) ) { A T 2 ( r M ) R 2 λ 2 e x i t 2 π M λ e x i t · ( A T 2 ϕ T ( r M ) ) } ,
R P F ( u ) = c 2 h 2 O T F G . U . ( u M ) O T F det ( u M ) [ π R 2 u 2 M E 2 w exit ( E ) d E ] ,
G o ( s ) = m = f · sin c ( π m f ) · exp ( i 2 π s / p o ) ,
μ ˜ i n ( λ R 2 u / M ; ν ) = 1 L o m = sin c ( π m f ) · δ ( R 2 M R 1 u m p o ) ,
p o p 2 = R 1 R 2 .

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