Abstract

We investigate the measurement of a thin sample’s optical thickness using the transport of intensity equation (TIE) and demonstrate a version of the TIE, valid for partially coherent illumination, that allows the measurement of a sample’s optical path length by the removal of illumination effects.

© 2013 OSA

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    [CrossRef] [PubMed]
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  3. E. Madelung, “Quantentheorie in hydrodynamische Form,”Z. für Phys.40, 322–326 (1926).
    [CrossRef]
  4. D. Bohm, “A suggested interpretation of the quantum theory in terms of “hidden” variables. I,” Phys. Rev.85, 166–179 (1952).
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  5. D. Paganin and K. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett.80, 2586–2589 (1998).
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  6. T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. A12, 1932–1941 (1995).
    [CrossRef]
  7. K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc., 54191–197 (2005).
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  8. T. C. Petersen, V. Keast, K. Johnson, and S. Duvall, “TEM based phase retrieval of p-n junction wafers using the transport of intensity equation,” Phil. Mag.87, 3565–3578 (2007).
    [CrossRef]
  9. L. Waller, L. Tian, and G. Barbastathis, “Transport of intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express18, 12552–12561 (2010).
    [CrossRef] [PubMed]
  10. C. Zuo, Q. Chen, Y. Yu, and A. Asundi, “Transport-of-intensity phase imaging using Savitzky-Golay differentiation filter–theory and applications,” Opt. Express21, 5346–5362 (2013).
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  12. K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett.77, 2961–2964 (1996).
    [CrossRef] [PubMed]
  13. T. C. Petersen, V. J. Keast, and D. M. Paganin, “Quantitative TEM-based phase retrieval of MgO nano-cubes using the transport of intensity equation,” Ultramicroscopy108, 805–815 (2008).
    [CrossRef] [PubMed]
  14. B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature408, 158–159 (2000).
    [CrossRef] [PubMed]
  15. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995). Ch. 4.
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  16. T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett.93, 068103 (2004).
    [CrossRef] [PubMed]
  17. A. M. Zysk, P. S. Carney, and J. C. Schotland, “Eikonal method for calculation of coherence functions,” Phys. Rev. Lett.95, 043904 (2005).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  19. T. E. Gureyev, Y. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun.259, 569–580 (2006).
    [CrossRef]
  20. K. A. Nugent, “Wave field determination using three–dimensional intensity information,” Phys. Rev. Lett.68, 2261–2264 (1992).
    [CrossRef] [PubMed]
  21. M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett.72, 1137–1140 (1994).
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  23. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), Sec. 5.7.1.
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  24. N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun.49, 6–10 (1984).
    [CrossRef]
  25. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), pp. 188–193.
  26. T. Gureyev and K. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun.133, 339–346 (1997).
    [CrossRef]
  27. D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. of Microscopy214, 51–61 (2004).
    [CrossRef]
  28. L. Tian, J. C. Petruccelli, and G. Barbastathis, “Nonlinear diffusion regularization for transport of intensity phase imaging,” Opt. Lett.37, 4131–4133 (2012).
    [CrossRef] [PubMed]
  29. J. Petruccelli, L. Tian, and G. Barbastathis, “Source diversity for transport of intensity phase imaging,” to appear in to Computational Optical Sensing and Imaging (Optical Society of America, 2013), June2013, paper CTu2C.3.

2013

2012

2010

2008

T. C. Petersen, V. J. Keast, and D. M. Paganin, “Quantitative TEM-based phase retrieval of MgO nano-cubes using the transport of intensity equation,” Ultramicroscopy108, 805–815 (2008).
[CrossRef] [PubMed]

2007

T. C. Petersen, V. Keast, K. Johnson, and S. Duvall, “TEM based phase retrieval of p-n junction wafers using the transport of intensity equation,” Phil. Mag.87, 3565–3578 (2007).
[CrossRef]

2006

T. E. Gureyev, Y. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun.259, 569–580 (2006).
[CrossRef]

2005

K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc., 54191–197 (2005).
[CrossRef]

A. M. Zysk, P. S. Carney, and J. C. Schotland, “Eikonal method for calculation of coherence functions,” 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, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett.93, 068103 (2004).
[CrossRef] [PubMed]

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. of Microscopy214, 51–61 (2004).
[CrossRef]

2000

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature408, 158–159 (2000).
[CrossRef] [PubMed]

1998

D. Paganin and K. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett.80, 2586–2589 (1998).
[CrossRef]

1997

T. Gureyev and K. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun.133, 339–346 (1997).
[CrossRef]

1996

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384, 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, 2961–2964 (1996).
[CrossRef] [PubMed]

1995

1994

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett.72, 1137–1140 (1994).
[CrossRef] [PubMed]

1992

K. A. Nugent, “Wave field determination using three–dimensional intensity information,” Phys. Rev. Lett.68, 2261–2264 (1992).
[CrossRef] [PubMed]

1984

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun.49, 6–10 (1984).
[CrossRef]

1983

M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A73, 1434–1441 (1983).
[CrossRef]

1982

1952

D. Bohm, “A suggested interpretation of the quantum theory in terms of “hidden” variables. I,” Phys. Rev.85, 166–179 (1952).
[CrossRef]

1926

E. Madelung, “Quantentheorie in hydrodynamische Form,”Z. für Phys.40, 322–326 (1926).
[CrossRef]

Allman, B.

K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc., 54191–197 (2005).
[CrossRef]

Allman, B. E.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature408, 158–159 (2000).
[CrossRef] [PubMed]

Anastasio, M. A.

Arif, M.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature408, 158–159 (2000).
[CrossRef] [PubMed]

Asundi, A.

Barbastathis, G.

Barnea, Z.

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett.77, 2961–2964 (1996).
[CrossRef] [PubMed]

Barty, A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. of Microscopy214, 51–61 (2004).
[CrossRef]

Beck, M.

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett.72, 1137–1140 (1994).
[CrossRef] [PubMed]

Bohm, D.

D. Bohm, “A suggested interpretation of the quantum theory in terms of “hidden” variables. I,” Phys. Rev.85, 166–179 (1952).
[CrossRef]

Carney, P. S.

A. M. Zysk, R. W. Schoonover, P. S. Carney, and M. A. Anastasio, “Transport of intensity and spectrum for partially coherent fields,” Opt. Lett.35, 2239–2241 (2010).
[CrossRef] [PubMed]

A. M. Zysk, P. S. Carney, and J. C. Schotland, “Eikonal method for calculation of coherence functions,” Phys. Rev. Lett.95, 043904 (2005).
[CrossRef] [PubMed]

Chen, Q.

Cookson, D.

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett.77, 2961–2964 (1996).
[CrossRef] [PubMed]

Duvall, S.

T. C. Petersen, V. Keast, K. Johnson, and S. Duvall, “TEM based phase retrieval of p-n junction wafers using the transport of intensity equation,” Phil. Mag.87, 3565–3578 (2007).
[CrossRef]

Fienup, J. R.

Gao, D.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384, 335–338 (1996).
[CrossRef]

Gureyev, T.

T. Gureyev and K. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun.133, 339–346 (1997).
[CrossRef]

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett.77, 2961–2964 (1996).
[CrossRef] [PubMed]

Gureyev, T. E.

T. E. Gureyev, Y. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun.259, 569–580 (2006).
[CrossRef]

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett.93, 068103 (2004).
[CrossRef] [PubMed]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384, 335–338 (1996).
[CrossRef]

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. A12, 1932–1941 (1995).
[CrossRef]

Ishizuka, K.

K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc., 54191–197 (2005).
[CrossRef]

Jacobson, D. L.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature408, 158–159 (2000).
[CrossRef] [PubMed]

Johnson, K.

T. C. Petersen, V. Keast, K. Johnson, and S. Duvall, “TEM based phase retrieval of p-n junction wafers using the transport of intensity equation,” Phil. Mag.87, 3565–3578 (2007).
[CrossRef]

Keast, V.

T. C. Petersen, V. Keast, K. Johnson, and S. Duvall, “TEM based phase retrieval of p-n junction wafers using the transport of intensity equation,” Phil. Mag.87, 3565–3578 (2007).
[CrossRef]

Keast, V. J.

T. C. Petersen, V. J. Keast, and D. M. Paganin, “Quantitative TEM-based phase retrieval of MgO nano-cubes using the transport of intensity equation,” Ultramicroscopy108, 805–815 (2008).
[CrossRef] [PubMed]

Lee, J.

Madelung, E.

E. Madelung, “Quantentheorie in hydrodynamische Form,”Z. für Phys.40, 322–326 (1926).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995). Ch. 4.
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), pp. 188–193.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), Sec. 5.7.1.
[CrossRef]

Mayo, S. C.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett.93, 068103 (2004).
[CrossRef] [PubMed]

McAlister, D. F.

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett.72, 1137–1140 (1994).
[CrossRef] [PubMed]

McMahon, P. J.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. of Microscopy214, 51–61 (2004).
[CrossRef]

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature408, 158–159 (2000).
[CrossRef] [PubMed]

Nesterets, Y. I.

T. E. Gureyev, Y. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun.259, 569–580 (2006).
[CrossRef]

Nugent, K.

D. Paganin and K. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett.80, 2586–2589 (1998).
[CrossRef]

T. Gureyev and K. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun.133, 339–346 (1997).
[CrossRef]

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett.77, 2961–2964 (1996).
[CrossRef] [PubMed]

Nugent, K. A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. of Microscopy214, 51–61 (2004).
[CrossRef]

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature408, 158–159 (2000).
[CrossRef] [PubMed]

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. A12, 1932–1941 (1995).
[CrossRef]

K. A. Nugent, “Wave field determination using three–dimensional intensity information,” Phys. Rev. Lett.68, 2261–2264 (1992).
[CrossRef] [PubMed]

Oh, S. B.

Paganin, D.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. of Microscopy214, 51–61 (2004).
[CrossRef]

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature408, 158–159 (2000).
[CrossRef] [PubMed]

D. Paganin and K. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett.80, 2586–2589 (1998).
[CrossRef]

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett.77, 2961–2964 (1996).
[CrossRef] [PubMed]

Paganin, D. M.

T. C. Petersen, V. J. Keast, and D. M. Paganin, “Quantitative TEM-based phase retrieval of MgO nano-cubes using the transport of intensity equation,” Ultramicroscopy108, 805–815 (2008).
[CrossRef] [PubMed]

T. E. Gureyev, Y. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun.259, 569–580 (2006).
[CrossRef]

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett.93, 068103 (2004).
[CrossRef] [PubMed]

Petersen, T. C.

T. C. Petersen, V. J. Keast, and D. M. Paganin, “Quantitative TEM-based phase retrieval of MgO nano-cubes using the transport of intensity equation,” Ultramicroscopy108, 805–815 (2008).
[CrossRef] [PubMed]

T. C. Petersen, V. Keast, K. Johnson, and S. Duvall, “TEM based phase retrieval of p-n junction wafers using the transport of intensity equation,” Phil. Mag.87, 3565–3578 (2007).
[CrossRef]

Petruccelli, J.

J. Petruccelli, L. Tian, and G. Barbastathis, “Source diversity for transport of intensity phase imaging,” to appear in to Computational Optical Sensing and Imaging (Optical Society of America, 2013), June2013, paper CTu2C.3.

Petruccelli, J. C.

Pogany, A.

T. E. Gureyev, Y. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun.259, 569–580 (2006).
[CrossRef]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384, 335–338 (1996).
[CrossRef]

Raymer, M. G.

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett.72, 1137–1140 (1994).
[CrossRef] [PubMed]

Roberts, A.

Schoonover, R. W.

Schotland, J. C.

A. M. Zysk, P. S. Carney, and J. C. Schotland, “Eikonal method for calculation of coherence functions,” Phys. Rev. Lett.95, 043904 (2005).
[CrossRef] [PubMed]

Stevenson, A. W.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett.93, 068103 (2004).
[CrossRef] [PubMed]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384, 335–338 (1996).
[CrossRef]

Streibl, N.

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun.49, 6–10 (1984).
[CrossRef]

Teague, M. R.

M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A73, 1434–1441 (1983).
[CrossRef]

Tian, L.

Waller, L.

Werner, S. A.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature408, 158–159 (2000).
[CrossRef] [PubMed]

Wilkins, S. W.

T. E. Gureyev, Y. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun.259, 569–580 (2006).
[CrossRef]

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett.93, 068103 (2004).
[CrossRef] [PubMed]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384, 335–338 (1996).
[CrossRef]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995). Ch. 4.
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), pp. 188–193.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), Sec. 5.7.1.
[CrossRef]

Yu, Y.

Zuo, C.

Zysk, A. M.

A. M. Zysk, R. W. Schoonover, P. S. Carney, and M. A. Anastasio, “Transport of intensity and spectrum for partially coherent fields,” Opt. Lett.35, 2239–2241 (2010).
[CrossRef] [PubMed]

A. M. Zysk, P. S. Carney, and J. C. Schotland, “Eikonal method for calculation of coherence functions,” Phys. Rev. Lett.95, 043904 (2005).
[CrossRef] [PubMed]

Appl. Opt.

J. Electron Microsc.

K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc., 54191–197 (2005).
[CrossRef]

J. of Microscopy

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. of Microscopy214, 51–61 (2004).
[CrossRef]

J. Opt. Soc. Am. A

Nature

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384, 335–338 (1996).
[CrossRef]

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature408, 158–159 (2000).
[CrossRef] [PubMed]

Opt. Commun.

T. E. Gureyev, Y. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun.259, 569–580 (2006).
[CrossRef]

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun.49, 6–10 (1984).
[CrossRef]

T. Gureyev and K. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun.133, 339–346 (1997).
[CrossRef]

Opt. Express

Opt. Lett.

Phil. Mag.

T. C. Petersen, V. Keast, K. Johnson, and S. Duvall, “TEM based phase retrieval of p-n junction wafers using the transport of intensity equation,” Phil. Mag.87, 3565–3578 (2007).
[CrossRef]

Phys. Rev.

D. Bohm, “A suggested interpretation of the quantum theory in terms of “hidden” variables. I,” Phys. Rev.85, 166–179 (1952).
[CrossRef]

Phys. Rev. Lett.

D. Paganin and K. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett.80, 2586–2589 (1998).
[CrossRef]

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett.77, 2961–2964 (1996).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

A. M. Zysk, P. S. Carney, and J. C. Schotland, “Eikonal method for calculation of coherence functions,” Phys. Rev. Lett.95, 043904 (2005).
[CrossRef] [PubMed]

Ultramicroscopy

T. C. Petersen, V. J. Keast, and D. M. Paganin, “Quantitative TEM-based phase retrieval of MgO nano-cubes using the transport of intensity equation,” Ultramicroscopy108, 805–815 (2008).
[CrossRef] [PubMed]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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J. Petruccelli, L. Tian, and G. Barbastathis, “Source diversity for transport of intensity phase imaging,” to appear in to Computational Optical Sensing and Imaging (Optical Society of America, 2013), June2013, paper CTu2C.3.

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Figures (3)

Fig. 1
Fig. 1

Imaging system used for the partially coherent TIE.

Fig. 2
Fig. 2

Simulation results. (a) Sample’s OPL and (b) transmittance. (c) The OPL of the modulation mask. (d) The difference in defocused intensity measurements with both masks in place. (e) Results of OPL reconstruction using Eq. (5) with both sample and modulation mask in place. (f) Reconstruction of the scalar phase of the background illumination alone. (g) The result of Eq. (14), assuming unity object transmittance. (h) The result of applying the PC–TIE, Eq. (12).

Fig. 3
Fig. 3

Experimental measurements and results. In–focus measurements of intensity for (a) the sample without illumination modulation, (b) the illumination modulation mask alone (c) both sample and modulation mask in place. (d) Difference between defocused intensity measurements with both sample and modulation mask in place. (e) Sample reconstruction using Eq. (13) with no field modulation mask in place. (f) The scalar phase reconstruction for the illumination with only the field modulation mask in place. (g) Reconstruction using Eq. (13) with both field modulation mask and sample in place. (h) Thickness reconstruction using Eq. (14), assuming a pure–phase sample. (i) Thickness reconstruction using Eq. (12).

Equations (26)

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U ( x , z ) = U parax ( x , z ) exp ( i k z ) ,
U parax = I ( x , z ) exp [ i ϕ ( x , z ) ] .
z U parax ( x , z ) = i 2 k x 2 U parax ( x , z ) ,
z I ( x , z ) + x [ I ( x , z ) v x ( x , z ) ] = 0 ,
z [ I ( x , z ) v x ( x , z ) ] + x [ I ( x , z ) v x ( x , z ) v x ( x , z ) ] = I ( x , z ) x ( x 2 I ( x , z ) 2 k 2 I ( x , z ) ) ,
z I ( x , z ) + x [ I ( x , z ) x ϕ ( x , z ) k ] = 0 .
τ ( x ) = T ( x ) exp [ i ψ ( x ) ] ,
z W ( x 1 , x 2 , z ) = ± i 2 k x j 2 W ( x 1 , x 2 , z ) ,
z W ( x x 2 , x + x 2 , z ) = i k x x W ( x x 2 , x + x 2 , z ) .
z S ( x , z ) + x F x ( x , z ) = 0 ,
F ( x , z ) = 1 i k x W ( x x 2 , x + x 2 , z ) | x = 0 .
1 k x ϕ ( x , z ) = F ( x , z ) S ( x , z ) = v ( x , z ) ,
z S tot ( x , z ) + x [ T ( x ) F inc ( x ) ] = 1 k x [ T ( x ) S inc ( x ) x ψ ( x ) ] .
S tot ( x , z ) z = 1 k x [ S inc ( x , z ) T ( x , z ) x ψ ( x , z ) ] ,
z S tot ( x , z ) T 0 S inc ( x , z ) z = T 0 k x [ S inc ( x , z ) x ψ ( x ) ] ,
W point ( x 1 , x 2 , d ; x 0 ) = I 0 ( x 0 ) f exp [ i k f ( x 2 x 1 ) x 0 ] .
W inc ( x 1 , x 2 , d ) = 1 f I 0 ( x 0 ) exp [ i k f ( x 2 x 1 ) x 0 ] d 2 x 0 .
W inc ( x 1 , x 2 , d ) = 2 π I 0 R k | x 2 x 1 | J 1 ( k R | x 2 x 1 | f ) .
1 2 ( π R z f ) 2 | u | 2 1 .
S ( x , z ) = 1 ( λ z ) 2 W ( x ¯ x 2 , x ¯ + x 2 , 0 ) exp [ i 2 π λ z ( x ¯ x ) x ] d 2 x ¯ d 2 x .
W ( x ¯ x 2 , x ¯ + x 2 , 0 ) = W inc ( x ¯ x 2 , x ¯ + x 2 , 0 ) T ( x ¯ x 2 ) T ( x ¯ + x 2 ) exp { i [ ψ ( x ¯ x 2 ) ψ ( x ¯ + x 2 ) ] } ,
S ( x , z ) = W inc ( x ¯ λ z u 2 , x ¯ + λ z u 2 , 0 ) T ( x ¯ + λ z u 2 ) T ( x ¯ λ z u 2 ) × exp { i [ ψ ( x ¯ + λ z u 2 ) ψ ( x ¯ λ z u 2 ) ] } × exp { i 2 π ( x ¯ x ) u } d 2 x ¯ d 2 u .
S ( x , z ) [ T ( x ¯ ) S inc ( x ¯ , 0 ) + i λ z T ( x ¯ ) S inc ( x ¯ ) u x ¯ ψ ( x ¯ ) + i 2 π z T ( x ¯ ) u F inc ( x ¯ , 0 ) ] × exp [ i 2 π ( x ¯ x ) u ] d 2 x ¯ d 2 u ,
u exp ( i 2 π x u ) = i 2 π x exp ( i 2 π x u ) .
S ( x , z ) T ( x ) S inc ( x , 0 ) z k x [ T ( x ) S inc ( x , 0 ) x ψ ( x ) ] z x [ T ( x ) F inc ( x , 0 ) ] .
S ^ ( u , z ) = W inc ( x ¯ λ z u 2 , x ¯ + λ z u 2 , 0 ) T ( x ¯ + λ z u 2 ) T ( x ¯ λ z u 2 ) × exp { i [ ϕ ( x ¯ + λ z u 2 ) ϕ ( x ¯ λ z u 2 ) ] } × exp ( i 2 π x ¯ u ) d 2 x ¯ ,

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