Abstract

Transport-of-intensity and transport-of-spectrum equations are derived using the coherent mode decomposition for paraxial fields having an arbitrary state of coherence. We give a simple example that demonstrates the difference between a partially coherent and a fully coherent transport of intensity or spectrum. The results presented here may be used to estimate the intensity response in a variety of phase-contrast imaging modalities and may form the basis for improved phase-retrieval techniques.

© 2010 Optical Society of America

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  1. A. H. Bennett, H. Osterberg, H. Jupnik, and O. W. Richards, Phase Microscopy: Principles and Applications(Wiley, 1951).
  2. S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, Opt. Express 11, 2289 (2003).
    [CrossRef] [PubMed]
  3. T. C. Petersen, V. J. Keast, and D. M. Paganin, Ultramicroscopy 108, 805 (2008).
    [CrossRef] [PubMed]
  4. M. R. Teague, J. Opt. Soc. Am. 73, 1434 (1983).
    [CrossRef]
  5. E. Wolf, J. Opt. Soc. Am. 72, 343 (1982).
    [CrossRef]
  6. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).
  7. A. M. Zysk, P. S. Carney, and J. C. Schotland, Phys. Rev. Lett. 95, 043904 (2005).
    [CrossRef] [PubMed]
  8. R. W. Schoonover, A. M. Zysk, P. S. Carney, J. C. Schotland, and E. Wolf, Phys. Rev. A 77, 043831 (2008).
    [CrossRef]
  9. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
    [CrossRef] [PubMed]
  10. T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, Phys. Rev. Lett. 93, 068103 (2004).
    [CrossRef] [PubMed]
  11. S. Flewett, H. M. Quiney, C. Q. Tran, and K. A. Nugent, Opt. Lett. 34, 2198 (2009).
    [CrossRef] [PubMed]
  12. B. J. Davis and R. W. Schoonover, Opt. Lett. 34, 923 (2009).
    [CrossRef] [PubMed]
  13. T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, Opt. Commun. 231, 53 (2004).
    [CrossRef]
  14. D. Paganin and K. A. Nugent, Phys. Rev. Lett. 80, 2586 (1998).
    [CrossRef]
  15. A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, Opt. Lett. 23, 817 (1998).
    [CrossRef]
  16. L. W. Whitehead, G. J. Williams, H. M. Quiney, D. J. Vine, R. A. Dilanian, S. Flewett, K. A. Nugent, A. G. Peele, E. Balaur, and I. McNulty, Phys. Rev. Lett. 103, 243902 (2009).
    [CrossRef]

2009

S. Flewett, H. M. Quiney, C. Q. Tran, and K. A. Nugent, Opt. Lett. 34, 2198 (2009).
[CrossRef] [PubMed]

B. J. Davis and R. W. Schoonover, Opt. Lett. 34, 923 (2009).
[CrossRef] [PubMed]

L. W. Whitehead, G. J. Williams, H. M. Quiney, D. J. Vine, R. A. Dilanian, S. Flewett, K. A. Nugent, A. G. Peele, E. Balaur, and I. McNulty, Phys. Rev. Lett. 103, 243902 (2009).
[CrossRef]

2008

T. C. Petersen, V. J. Keast, and D. M. Paganin, Ultramicroscopy 108, 805 (2008).
[CrossRef] [PubMed]

R. W. Schoonover, A. M. Zysk, P. S. Carney, J. C. Schotland, and E. Wolf, Phys. Rev. A 77, 043831 (2008).
[CrossRef]

2007

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

2005

A. M. Zysk, P. S. Carney, and J. C. Schotland, Phys. Rev. Lett. 95, 043904 (2005).
[CrossRef] [PubMed]

2004

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, Phys. Rev. Lett. 93, 068103 (2004).
[CrossRef] [PubMed]

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, Opt. Commun. 231, 53 (2004).
[CrossRef]

2003

1998

1996

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

1983

1982

1951

A. H. Bennett, H. Osterberg, H. Jupnik, and O. W. Richards, Phase Microscopy: Principles and Applications(Wiley, 1951).

Balaur, E.

L. W. Whitehead, G. J. Williams, H. M. Quiney, D. J. Vine, R. A. Dilanian, S. Flewett, K. A. Nugent, A. G. Peele, E. Balaur, and I. McNulty, Phys. Rev. Lett. 103, 243902 (2009).
[CrossRef]

Barnea, Z.

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

Barty, A.

Bennett, A. H.

A. H. Bennett, H. Osterberg, H. Jupnik, and O. W. Richards, Phase Microscopy: Principles and Applications(Wiley, 1951).

Carney, P. S.

R. W. Schoonover, A. M. Zysk, P. S. Carney, J. C. Schotland, and E. Wolf, Phys. Rev. A 77, 043831 (2008).
[CrossRef]

A. M. Zysk, P. S. Carney, and J. C. Schotland, Phys. Rev. Lett. 95, 043904 (2005).
[CrossRef] [PubMed]

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, B. J.

Davis, T. J.

Dilanian, R. A.

L. W. Whitehead, G. J. Williams, H. M. Quiney, D. J. Vine, R. A. Dilanian, S. Flewett, K. A. Nugent, A. G. Peele, E. Balaur, and I. McNulty, Phys. Rev. Lett. 103, 243902 (2009).
[CrossRef]

Flewett, S.

L. W. Whitehead, G. J. Williams, H. M. Quiney, D. J. Vine, R. A. Dilanian, S. Flewett, K. A. Nugent, A. G. Peele, E. Balaur, and I. McNulty, Phys. Rev. Lett. 103, 243902 (2009).
[CrossRef]

S. Flewett, H. M. Quiney, C. Q. Tran, and K. A. Nugent, Opt. Lett. 34, 2198 (2009).
[CrossRef] [PubMed]

Gureyev, T. E.

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, Opt. Commun. 231, 53 (2004).
[CrossRef]

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, Phys. Rev. Lett. 93, 068103 (2004).
[CrossRef] [PubMed]

S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, Opt. Express 11, 2289 (2003).
[CrossRef] [PubMed]

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

Jupnik, H.

A. H. Bennett, H. Osterberg, H. Jupnik, and O. W. Richards, Phase Microscopy: Principles and Applications(Wiley, 1951).

Keast, V. J.

T. C. Petersen, V. J. Keast, and D. M. Paganin, Ultramicroscopy 108, 805 (2008).
[CrossRef] [PubMed]

Mayo, S. C.

McNulty, I.

L. W. Whitehead, G. J. Williams, H. M. Quiney, D. J. Vine, R. A. Dilanian, S. Flewett, K. A. Nugent, A. G. Peele, E. Balaur, and I. McNulty, Phys. Rev. Lett. 103, 243902 (2009).
[CrossRef]

Miller, P. R.

Nugent, K. A.

L. W. Whitehead, G. J. Williams, H. M. Quiney, D. J. Vine, R. A. Dilanian, S. Flewett, K. A. Nugent, A. G. Peele, E. Balaur, and I. McNulty, Phys. Rev. Lett. 103, 243902 (2009).
[CrossRef]

S. Flewett, H. M. Quiney, C. Q. Tran, and K. A. Nugent, Opt. Lett. 34, 2198 (2009).
[CrossRef] [PubMed]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, Opt. Lett. 23, 817 (1998).
[CrossRef]

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]

Osterberg, H.

A. H. Bennett, H. Osterberg, H. Jupnik, and O. W. Richards, Phase Microscopy: Principles and Applications(Wiley, 1951).

Paganin, D.

Paganin, D. M.

T. C. Petersen, V. J. Keast, and D. M. Paganin, Ultramicroscopy 108, 805 (2008).
[CrossRef] [PubMed]

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, Phys. Rev. Lett. 93, 068103 (2004).
[CrossRef] [PubMed]

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, Opt. Commun. 231, 53 (2004).
[CrossRef]

Peele, A. G.

L. W. Whitehead, G. J. Williams, H. M. Quiney, D. J. Vine, R. A. Dilanian, S. Flewett, K. A. Nugent, A. G. Peele, E. Balaur, and I. McNulty, Phys. Rev. Lett. 103, 243902 (2009).
[CrossRef]

Petersen, T. C.

T. C. Petersen, V. J. Keast, and D. M. Paganin, Ultramicroscopy 108, 805 (2008).
[CrossRef] [PubMed]

Pogany, A.

Quiney, H. M.

S. Flewett, H. M. Quiney, C. Q. Tran, and K. A. Nugent, Opt. Lett. 34, 2198 (2009).
[CrossRef] [PubMed]

L. W. Whitehead, G. J. Williams, H. M. Quiney, D. J. Vine, R. A. Dilanian, S. Flewett, K. A. Nugent, A. G. Peele, E. Balaur, and I. McNulty, Phys. Rev. Lett. 103, 243902 (2009).
[CrossRef]

Richards, O. W.

A. H. Bennett, H. Osterberg, H. Jupnik, and O. W. Richards, Phase Microscopy: Principles and Applications(Wiley, 1951).

Roberts, A.

Schoonover, R. W.

B. J. Davis and R. W. Schoonover, Opt. Lett. 34, 923 (2009).
[CrossRef] [PubMed]

R. W. Schoonover, A. M. Zysk, P. S. Carney, J. C. Schotland, and E. Wolf, Phys. Rev. A 77, 043831 (2008).
[CrossRef]

Schotland, J. C.

R. W. Schoonover, A. M. Zysk, P. S. Carney, J. C. Schotland, and E. Wolf, Phys. Rev. A 77, 043831 (2008).
[CrossRef]

A. M. Zysk, P. S. Carney, and J. C. Schotland, Phys. Rev. Lett. 95, 043904 (2005).
[CrossRef] [PubMed]

Stevenson, A. W.

Teague, M. R.

Tran, C. Q.

Vine, D. J.

L. W. Whitehead, G. J. Williams, H. M. Quiney, D. J. Vine, R. A. Dilanian, S. Flewett, K. A. Nugent, A. G. Peele, E. Balaur, and I. McNulty, Phys. Rev. Lett. 103, 243902 (2009).
[CrossRef]

Whitehead, L. W.

L. W. Whitehead, G. J. Williams, H. M. Quiney, D. J. Vine, R. A. Dilanian, S. Flewett, K. A. Nugent, A. G. Peele, E. Balaur, and I. McNulty, Phys. Rev. Lett. 103, 243902 (2009).
[CrossRef]

Wilkins, S. W.

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, Opt. Commun. 231, 53 (2004).
[CrossRef]

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, Phys. Rev. Lett. 93, 068103 (2004).
[CrossRef] [PubMed]

S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, Opt. Express 11, 2289 (2003).
[CrossRef] [PubMed]

Williams, G. J.

L. W. Whitehead, G. J. Williams, H. M. Quiney, D. J. Vine, R. A. Dilanian, S. Flewett, K. A. Nugent, A. G. Peele, E. Balaur, and I. McNulty, Phys. Rev. Lett. 103, 243902 (2009).
[CrossRef]

Wolf, E.

R. W. Schoonover, A. M. Zysk, P. S. Carney, J. C. Schotland, and E. Wolf, Phys. Rev. A 77, 043831 (2008).
[CrossRef]

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

E. Wolf, J. Opt. Soc. Am. 72, 343 (1982).
[CrossRef]

Zysk, A. M.

R. W. Schoonover, A. M. Zysk, P. S. Carney, J. C. Schotland, and E. Wolf, Phys. Rev. A 77, 043831 (2008).
[CrossRef]

A. M. Zysk, P. S. Carney, and J. C. Schotland, Phys. Rev. Lett. 95, 043904 (2005).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

Opt. Commun.

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, Opt. Commun. 231, 53 (2004).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

R. W. Schoonover, A. M. Zysk, P. S. Carney, J. C. Schotland, and E. Wolf, Phys. Rev. A 77, 043831 (2008).
[CrossRef]

Phys. Rev. Lett.

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

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, Phys. Rev. Lett. 93, 068103 (2004).
[CrossRef] [PubMed]

A. M. Zysk, P. S. Carney, and J. C. Schotland, Phys. Rev. Lett. 95, 043904 (2005).
[CrossRef] [PubMed]

L. W. Whitehead, G. J. Williams, H. M. Quiney, D. J. Vine, R. A. Dilanian, S. Flewett, K. A. Nugent, A. G. Peele, E. Balaur, and I. McNulty, Phys. Rev. Lett. 103, 243902 (2009).
[CrossRef]

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

Ultramicroscopy

T. C. Petersen, V. J. Keast, and D. M. Paganin, Ultramicroscopy 108, 805 (2008).
[CrossRef] [PubMed]

Other

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

A. H. Bennett, H. Osterberg, H. Jupnik, and O. W. Richards, Phase Microscopy: Principles and Applications(Wiley, 1951).

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Figures (1)

Fig. 1
Fig. 1

Plot of the retrieved phases when the wave field is assumed to be fully coherent, found by inverting Eq. (13), on the line z = 100 mm and y = 0 for three cases: γ 1 ( ω ) = 1 and γ 2 ( ω ) = 0 ( ψ ¯ 1 ), γ 1 ( ω ) = 0 and γ 2 ( ω ) = 1 ( ψ ¯ 2 ), and γ 1 ( ω ) = 1 and γ 2 ( ω ) = 0 ( ψ ¯ 12 ).

Equations (15)

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

[ 2 i k z + 2 ] V ( r , ω ) = 0
S ( r , ω ) z = 1 k · S ( r , ω ) ψ ( r , ω ) ,
W ( r 1 , r 2 , ω ) = U * ( r 1 , ω ) U ( r 2 , ω ) ω ,
W ( r 1 , r 2 , ω ) = m = 1 M γ m ( ω ) ϕ m * ( r 1 , ω ) ϕ m ( r 2 , ω ) ,
ϕ m ( r , ω ) = a m ( r , ω ) e i ψ m ( r , ω ) ,
S ( r , ω ) = m = 1 M γ m ( ω ) D m ( r , ω ) ,
S ( r , ω ) z = 1 k m = 1 M γ m ( ω ) · [ D m ( r , ω ) ψ m ( r , ω ) ] .
I ( r ) z = m = 1 M d ω 1 k γ m ( ω ) · [ D m ( r , ω ) ψ m ( r , ω ) ] .
S ( r , ω ) z = · m = 1 M γ m ( ω ) D ¯ m ( r ) ψ ¯ m ( r ) ,
I ( r ) z = · m = 1 M P m D ¯ m ( r ) ψ ¯ m ( r ) ,
P m = d ω γ m ( ω )
I ( r ) z = P 1 · D 1 ¯ ( r ) ψ ¯ 1 ( r ) .
2 ψ ¯ 1 ( r , ω ) = S 1 ( r , ω ) S ( r , ω ) z .
W ( r , r ′′ , ω ) = m = 1 2 γ m ( ω ) e i k | r r m | | r r m | e i k | r ′′ r m | | r ′′ r m | ,
S ( r , ω ) 1 S ( r , ω ) z = 2 γ 1 ( ω ) z | r r 1 | 4 + 2 γ 2 ( ω ) z | r r 2 | 4 γ 1 ( ω ) | r r 1 | 2 + γ 2 ( ω ) | r r 2 | 2 .

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