Abstract

In this paper a non-interferometric, non-iterative method for phase retrieval by Green’s functions is presented. The theory is based on the parabolic wave equation that describes propagation of light in the Fresnel approximation in homogeneous media. Green’s first identity will be used to derive an algorithm for phase retrieval considering different boundary conditions. Finally it will be shown that a commonly used solution of the transport-of-intensity equation can be obtained as a special case of the more general Green’s function formulation derived here.

© 2010 Optical Society of America

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  1. G. Wernicke and W. Osten, Holografische Interferometrie (Physik-Verlag, 1982).
  2. T. Kreis, Handbook of Holographic Interferometry (Wiley-VCH, 2005).
  3. U. Schnars and W. Jueptner, Digital Holography (Springer Verlag, 2005).
  4. G. Wernicke, J. Frank, H. Gruber, M. Duerr, A. Langner, S. Eisebitt, C. Guenther, L. Bouamama, S. Krueger, and A. Hermerschmidt, “Applications of the high-resolution optical reconstruction of digital holograms,” Proc. SPIE 6136, 61360Q (2006).
    [CrossRef]
  5. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, 1998).
  6. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  7. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  8. F. Roddier and C. Roddier, “Wavefront reconstruction using iterative Fourier transforms,” Appl. Opt. 30, 1325–1327 (1991).
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  9. L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
    [CrossRef]
  10. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
  11. D. Paganin and K. A. Nugent, “Noninterferometric Phase Imaging with Partially Coherent Light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
    [CrossRef]
  12. A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
    [CrossRef]
  13. E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
    [CrossRef] [PubMed]
  14. M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
    [CrossRef] [PubMed]
  15. C. Dorrer and J. D. Zuegel, “Optical testing using the transport-of-intensity equation,” Opt. Express 15, 7165–7175 (2007).
    [CrossRef] [PubMed]
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  21. N. Nakajima, “Phase retrieval using an aperture-array filter under partially coherent illumination,” Opt. Commun. 282, 2128–2135 (2009).
    [CrossRef]
  22. J. F. Nye and M. F. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
    [CrossRef]
  23. 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. A 12, 1932–1941 (1995).
    [CrossRef]
  24. T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13, 1670–1682 (1996).
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  25. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2007).
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    [CrossRef]
  28. A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists (Chapman & Hall, 2002).
  29. P. M. Morse and H. Feshbach, Methods of Theoretical Physics Part 1 (McGraw-Hill, 1953).
  30. E. G. Abramochkin and V. G. Volostnikov, “Relationship between two-dimensional intensity and phase in a Fresnel diffraction zone,” Opt. Commun. 74, 144–148 (1989).
    [CrossRef]
  31. H. Maitre and I. Lyuboshenko, “Robust algorithms for phase unwrapping in SAR interferometry,” Proc. SPIE 3217, 176–187 (1997).
    [CrossRef]
  32. I. Lyuboshenko and H. Maitre, “Phase unwrapping for interferometric synthetic aperture radar by use of Helmholtz equation eigenfunctions and the first Green’s identity,” J. Opt. Soc. Am. A 16, 378–395 (1999).
    [CrossRef]
  33. B. E. Allman and K. A. Nugent, “Shape imaging in defence operations,” in Proceedings Land Warfare Conference (2006).
  34. V. V. Volkov, Y. Zhu, and M. De Graef, “A new symmetrized solution for phase retrieval using the transport of intensity equation,” Micron 33, 411–416 (2002).
    [CrossRef] [PubMed]
  35. A. V. Martin, F.-R. Chen, W.-K. Hsieh, J.-J. Kai, S. D. Findlay, and L. J. Allen, “Spatial incoherence in phase retrieval based on focus variation,” Ultramicroscopy 106, 914–924 (2006).
    [CrossRef] [PubMed]
  36. N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
    [CrossRef]
  37. T. E. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220, 49–58 (2003).
    [CrossRef]
  38. National Institute of Standards and Technology, “Green’s functions tutorial, introduction to Green’s functions,” http://www.boulder.nist.gov/div853/greenfn/tutorial.html.

2009 (2)

N. Nakajima, “Phase retrieval using an aperture-array filter under partially coherent illumination,” Opt. Commun. 282, 2128–2135 (2009).
[CrossRef]

N. Nakajima, “Phase retrieval from a high-numerical-aperture intensity distribution by use of an aperture-array filter,” J. Opt. Soc. Am. A 26, 2172–2180 (2009).
[CrossRef]

2008 (1)

2007 (3)

N. Nakajima, “Noniterative phase retrieval from a single diffraction intensity pattern by use of an aperture array,” Phys. Rev. Lett. 98, 223901 (2007).
[CrossRef] [PubMed]

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2007).

C. Dorrer and J. D. Zuegel, “Optical testing using the transport-of-intensity equation,” Opt. Express 15, 7165–7175 (2007).
[CrossRef] [PubMed]

2006 (3)

G. Wernicke, J. Frank, H. Gruber, M. Duerr, A. Langner, S. Eisebitt, C. Guenther, L. Bouamama, S. Krueger, and A. Hermerschmidt, “Applications of the high-resolution optical reconstruction of digital holograms,” Proc. SPIE 6136, 61360Q (2006).
[CrossRef]

B. E. Allman and K. A. Nugent, “Shape imaging in defence operations,” in Proceedings Land Warfare Conference (2006).

A. V. Martin, F.-R. Chen, W.-K. Hsieh, J.-J. Kai, S. D. Findlay, and L. J. Allen, “Spatial incoherence in phase retrieval based on focus variation,” Ultramicroscopy 106, 914–924 (2006).
[CrossRef] [PubMed]

2005 (3)

T. Kreis, Handbook of Holographic Interferometry (Wiley-VCH, 2005).

U. Schnars and W. Jueptner, Digital Holography (Springer Verlag, 2005).

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

2004 (2)

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[CrossRef] [PubMed]

N. Nakajima, “Lensless imaging from diffraction intensity measurements by use of a noniterative phase-retrieval method,” Appl. Opt. 43, 1710–1718 (2004).
[CrossRef] [PubMed]

2003 (2)

2002 (3)

V. V. Volkov, Y. Zhu, and M. De Graef, “A new symmetrized solution for phase retrieval using the transport of intensity equation,” Micron 33, 411–416 (2002).
[CrossRef] [PubMed]

A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists (Chapman & Hall, 2002).

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
[CrossRef] [PubMed]

2001 (1)

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[CrossRef]

1999 (1)

1998 (3)

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
[CrossRef]

D. Paganin and K. A. Nugent, “Noninterferometric Phase Imaging with Partially Coherent Light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[CrossRef]

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, 1998).

1997 (1)

H. Maitre and I. Lyuboshenko, “Robust algorithms for phase unwrapping in SAR interferometry,” Proc. SPIE 3217, 176–187 (1997).
[CrossRef]

1996 (2)

T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13, 1670–1682 (1996).
[CrossRef]

G. Fornaro, G. Franceschetti, and R. Lanari, “Interferometric SAR phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
[CrossRef]

1995 (1)

1991 (1)

1989 (1)

E. G. Abramochkin and V. G. Volostnikov, “Relationship between two-dimensional intensity and phase in a Fresnel diffraction zone,” Opt. Commun. 74, 144–148 (1989).
[CrossRef]

1984 (1)

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

1983 (1)

1982 (2)

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

G. Wernicke and W. Osten, Holografische Interferometrie (Physik-Verlag, 1982).

1974 (1)

J. F. Nye and M. F. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

1953 (1)

P. M. Morse and H. Feshbach, Methods of Theoretical Physics Part 1 (McGraw-Hill, 1953).

Abramochkin, E. G.

E. G. Abramochkin and V. G. Volostnikov, “Relationship between two-dimensional intensity and phase in a Fresnel diffraction zone,” Opt. Commun. 74, 144–148 (1989).
[CrossRef]

Allen, L. J.

A. V. Martin, F.-R. Chen, W.-K. Hsieh, J.-J. Kai, S. D. Findlay, and L. J. Allen, “Spatial incoherence in phase retrieval based on focus variation,” Ultramicroscopy 106, 914–924 (2006).
[CrossRef] [PubMed]

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[CrossRef]

Allman, B. E.

B. E. Allman and K. A. Nugent, “Shape imaging in defence operations,” in Proceedings Land Warfare Conference (2006).

Barone-Nugent, E. D.

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
[CrossRef] [PubMed]

Barty, A.

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
[CrossRef] [PubMed]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
[CrossRef]

Beleggia, M.

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[CrossRef] [PubMed]

Berry, M. F.

J. F. Nye and M. F. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

Bouamama, L.

G. Wernicke, J. Frank, H. Gruber, M. Duerr, A. Langner, S. Eisebitt, C. Guenther, L. Bouamama, S. Krueger, and A. Hermerschmidt, “Applications of the high-resolution optical reconstruction of digital holograms,” Proc. SPIE 6136, 61360Q (2006).
[CrossRef]

Chen, F. -R.

A. V. Martin, F.-R. Chen, W.-K. Hsieh, J.-J. Kai, S. D. Findlay, and L. J. Allen, “Spatial incoherence in phase retrieval based on focus variation,” Ultramicroscopy 106, 914–924 (2006).
[CrossRef] [PubMed]

De Graef, M.

V. V. Volkov, Y. Zhu, and M. De Graef, “A new symmetrized solution for phase retrieval using the transport of intensity equation,” Micron 33, 411–416 (2002).
[CrossRef] [PubMed]

Dorrer, C.

Duerr, M.

G. Wernicke, J. Frank, H. Gruber, M. Duerr, A. Langner, S. Eisebitt, C. Guenther, L. Bouamama, S. Krueger, and A. Hermerschmidt, “Applications of the high-resolution optical reconstruction of digital holograms,” Proc. SPIE 6136, 61360Q (2006).
[CrossRef]

Eisebitt, S.

G. Wernicke, J. Frank, H. Gruber, M. Duerr, A. Langner, S. Eisebitt, C. Guenther, L. Bouamama, S. Krueger, and A. Hermerschmidt, “Applications of the high-resolution optical reconstruction of digital holograms,” Proc. SPIE 6136, 61360Q (2006).
[CrossRef]

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics Part 1 (McGraw-Hill, 1953).

Fienup, J. R.

Findlay, S. D.

A. V. Martin, F.-R. Chen, W.-K. Hsieh, J.-J. Kai, S. D. Findlay, and L. J. Allen, “Spatial incoherence in phase retrieval based on focus variation,” Ultramicroscopy 106, 914–924 (2006).
[CrossRef] [PubMed]

Fornaro, G.

G. Fornaro, G. Franceschetti, and R. Lanari, “Interferometric SAR phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
[CrossRef]

Franceschetti, G.

G. Fornaro, G. Franceschetti, and R. Lanari, “Interferometric SAR phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
[CrossRef]

Frank, J.

G. Wernicke, J. Frank, H. Gruber, M. Duerr, A. Langner, S. Eisebitt, C. Guenther, L. Bouamama, S. Krueger, and A. Hermerschmidt, “Applications of the high-resolution optical reconstruction of digital holograms,” Proc. SPIE 6136, 61360Q (2006).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Ghiglia, D. C.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, 1998).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

Gruber, H.

G. Wernicke, J. Frank, H. Gruber, M. Duerr, A. Langner, S. Eisebitt, C. Guenther, L. Bouamama, S. Krueger, and A. Hermerschmidt, “Applications of the high-resolution optical reconstruction of digital holograms,” Proc. SPIE 6136, 61360Q (2006).
[CrossRef]

Guenther, C.

G. Wernicke, J. Frank, H. Gruber, M. Duerr, A. Langner, S. Eisebitt, C. Guenther, L. Bouamama, S. Krueger, and A. Hermerschmidt, “Applications of the high-resolution optical reconstruction of digital holograms,” Proc. SPIE 6136, 61360Q (2006).
[CrossRef]

Gureyev, T. E.

Hermerschmidt, A.

G. Wernicke, J. Frank, H. Gruber, M. Duerr, A. Langner, S. Eisebitt, C. Guenther, L. Bouamama, S. Krueger, and A. Hermerschmidt, “Applications of the high-resolution optical reconstruction of digital holograms,” Proc. SPIE 6136, 61360Q (2006).
[CrossRef]

Hsieh, W. -K.

A. V. Martin, F.-R. Chen, W.-K. Hsieh, J.-J. Kai, S. D. Findlay, and L. J. Allen, “Spatial incoherence in phase retrieval based on focus variation,” Ultramicroscopy 106, 914–924 (2006).
[CrossRef] [PubMed]

Jueptner, W.

U. Schnars and W. Jueptner, Digital Holography (Springer Verlag, 2005).

Kai, J. -J.

A. V. Martin, F.-R. Chen, W.-K. Hsieh, J.-J. Kai, S. D. Findlay, and L. J. Allen, “Spatial incoherence in phase retrieval based on focus variation,” Ultramicroscopy 106, 914–924 (2006).
[CrossRef] [PubMed]

Kreis, T.

T. Kreis, Handbook of Holographic Interferometry (Wiley-VCH, 2005).

Krueger, S.

G. Wernicke, J. Frank, H. Gruber, M. Duerr, A. Langner, S. Eisebitt, C. Guenther, L. Bouamama, S. Krueger, and A. Hermerschmidt, “Applications of the high-resolution optical reconstruction of digital holograms,” Proc. SPIE 6136, 61360Q (2006).
[CrossRef]

Lanari, R.

G. Fornaro, G. Franceschetti, and R. Lanari, “Interferometric SAR phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
[CrossRef]

Langner, A.

G. Wernicke, J. Frank, H. Gruber, M. Duerr, A. Langner, S. Eisebitt, C. Guenther, L. Bouamama, S. Krueger, and A. Hermerschmidt, “Applications of the high-resolution optical reconstruction of digital holograms,” Proc. SPIE 6136, 61360Q (2006).
[CrossRef]

Lyuboshenko, I.

Maitre, H.

Martin, A. V.

A. V. Martin, F.-R. Chen, W.-K. Hsieh, J.-J. Kai, S. D. Findlay, and L. J. Allen, “Spatial incoherence in phase retrieval based on focus variation,” Ultramicroscopy 106, 914–924 (2006).
[CrossRef] [PubMed]

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics Part 1 (McGraw-Hill, 1953).

Nakajima, N.

Nugent, K. A.

B. E. Allman and K. A. Nugent, “Shape imaging in defence operations,” in Proceedings Land Warfare Conference (2006).

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
[CrossRef] [PubMed]

D. Paganin and K. A. Nugent, “Noninterferometric Phase Imaging with Partially Coherent Light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[CrossRef]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
[CrossRef]

T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13, 1670–1682 (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. A 12, 1932–1941 (1995).
[CrossRef]

Nye, J. F.

J. F. Nye and M. F. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

Osten, W.

G. Wernicke and W. Osten, Holografische Interferometrie (Physik-Verlag, 1982).

Oxley, M. P.

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[CrossRef]

Paganin, D.

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
[CrossRef]

D. Paganin and K. A. Nugent, “Noninterferometric Phase Imaging with Partially Coherent Light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[CrossRef]

Polyanin, A. D.

A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists (Chapman & Hall, 2002).

Pritt, M. D.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, 1998).

Roberts, A.

Roddier, C.

Roddier, F.

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2007).

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Schnars, U.

U. Schnars and W. Jueptner, Digital Holography (Springer Verlag, 2005).

Schofield, M. A.

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[CrossRef] [PubMed]

Streibl, N.

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

Teague, M. R.

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2007).

Volkov, V. V.

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[CrossRef] [PubMed]

V. V. Volkov, Y. Zhu, and M. De Graef, “A new symmetrized solution for phase retrieval using the transport of intensity equation,” Micron 33, 411–416 (2002).
[CrossRef] [PubMed]

Volostnikov, V. G.

E. G. Abramochkin and V. G. Volostnikov, “Relationship between two-dimensional intensity and phase in a Fresnel diffraction zone,” Opt. Commun. 74, 144–148 (1989).
[CrossRef]

Wernicke, G.

G. Wernicke, J. Frank, H. Gruber, M. Duerr, A. Langner, S. Eisebitt, C. Guenther, L. Bouamama, S. Krueger, and A. Hermerschmidt, “Applications of the high-resolution optical reconstruction of digital holograms,” Proc. SPIE 6136, 61360Q (2006).
[CrossRef]

G. Wernicke and W. Osten, Holografische Interferometrie (Physik-Verlag, 1982).

Zhu, Y.

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[CrossRef] [PubMed]

V. V. Volkov, Y. Zhu, and M. De Graef, “A new symmetrized solution for phase retrieval using the transport of intensity equation,” Micron 33, 411–416 (2002).
[CrossRef] [PubMed]

Zuegel, J. D.

Appl. Opt. (4)

IEEE Trans. Geosci. Remote Sens. (1)

G. Fornaro, G. Franceschetti, and R. Lanari, “Interferometric SAR phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
[CrossRef]

J. Microsc. (1)

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Micron (1)

V. V. Volkov, Y. Zhu, and M. De Graef, “A new symmetrized solution for phase retrieval using the transport of intensity equation,” Micron 33, 411–416 (2002).
[CrossRef] [PubMed]

Opt. Commun. (5)

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

T. E. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220, 49–58 (2003).
[CrossRef]

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, “Relationship between two-dimensional intensity and phase in a Fresnel diffraction zone,” Opt. Commun. 74, 144–148 (1989).
[CrossRef]

N. Nakajima, “Phase retrieval using an aperture-array filter under partially coherent illumination,” Opt. Commun. 282, 2128–2135 (2009).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Optik (Stuttgart) (1)

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Phys. Rev. Lett. (2)

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[CrossRef] [PubMed]

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[CrossRef]

Proc. R. Soc. London, Ser. A (1)

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[CrossRef]

Proc. SPIE (2)

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[CrossRef]

G. Wernicke, J. Frank, H. Gruber, M. Duerr, A. Langner, S. Eisebitt, C. Guenther, L. Bouamama, S. Krueger, and A. Hermerschmidt, “Applications of the high-resolution optical reconstruction of digital holograms,” Proc. SPIE 6136, 61360Q (2006).
[CrossRef]

Ultramicroscopy (2)

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

Intensity and phase distribution of the test object together with two intensity distributions after propagation of the complex field: (a) shows the intensity distribution of the object (Joseph Fourier) with a contrast K = 10 % . The phase distribution, shown in (b), with phase values of [ 0 , π ]   rad , corresponding to the gray values black and white, was chosen to infringe periodic boundary conditions. The two defocused images, shown in (c) and (d), were obtained by propagating the complex object field by applying the ASPW method [26]. Parameters of the simulation have been field size a × b = 512 × 512   pixel , a pixel size of d x = 4 μ m , wavelength λ = 632.8   nm , and defocus distance Δ z / 2 = ± 800 μ m .

Fig. 2
Fig. 2

Original and reconstructed phase distributions of the test object in inverted gray scale: (a) shows the original two-dimensional phase distribution of the test object. (b) is the reconstructed phase distribution considering unbounded free space and (c) considering Neumann boundary conditions. (b) and (c) were obtained by using Figs. 1a, 1c, 1d as input data in the presented phase retrieval algorithms. The scale bar at the right shows the phase values of [ 0 , π ]   rad , while the gray values have been inverted for a better visibility of the discrepancies. Obviously (b) shows remarkable artifacts at the boundaries on the upper left and lower right. This is in agreement with the already mentioned problems of this retrieval algorithm in case of non-periodic objects. In contradistinction to the reconstruction considering unbounded free space (b), the determined phase distribution using the presented Green’s formalism with Neumann boundary conditions (c) shows no such deviations from the original phase distribution at the boundaries.

Fig. 3
Fig. 3

Comparison of the reconstructed phase distributions together with the original phase of the test object: The figure shows cross-sections through Figs. 2a, 2b, 2c from the bottom left to the top left close to the boundary. The original phase distribution of the test object is plotted in black (square), while the reconstructed phase distribution considering unbounded free space is colored light gray (circle). The reconstruction considering Neumann boundary conditions is tinted gray (triangle). The artifacts, caused by the non-periodic object, are clearly visible for the unbounded free space algorithm (light gray, circle). The maximal phase shift recovered is ≈2.5 rad instead of the desired π   rad . Furthermore the reconstructed distribution shows discrepancies at the onset of the phase disturbance and an elevation of ≈0.6 rad, which is not present in the test object phase, occurs due to the periodic assumption made by this retrieval method. The Green’s method with Neumann boundary conditions (gray, triangle) almost reaches the desired π   rad phase shift and shows relevant deviations from the original test phase at the onset of the phase disturbance only.

Fig. 4
Fig. 4

Comparison of the reconstructed phase distributions together with the original phase of the test object: The figure shows diagonal cross-sections through Figs. 2a, 2b, 2c from the bottom left to the top right. The original phase distribution of the test object is plotted in black (square), while the reconstructed phase distribution considering unbounded free space is colored light gray (circle). The reconstruction considering Neumann boundary conditions is tinted gray (triangle). In contrast to the sections shown in Fig. 3, the discrepancies between the reconstructed functions and the object phase in this figure are negligible. Both algorithms yield comparable results away from boundaries.

Tables (1)

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Table 1 Relative and Absolute RMS Errors for Both Reconstruction Methods

Equations (41)

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U z ( x , y ) = [ I z ( x , y ) ] 1 / 2 exp [ i Φ z ( x , y ) ] .
( i z + 2 2 k + k ) U z ( x , y ) = 0 ,
{ I z ( x , y ) Φ z ( x , y ) } = k I z ( x , y ) z ,
R ( Φ G + Φ 2 G ) d r = C Φ ( G n R ) d r ,
2 G ( r , r ) = δ ( r r ) .
Φ ( r ) = R ( Φ G ) d r + C Φ ( G n R ) d r .
G ( r , r ) = 0 ,     r C ,
G ( r , r ) n R = 0 ,     r C .
unbounded   free   space
G ( r , r ) = m , n = 0 κ m , n 0 Ψ m , n ( r ) Ψ m , n ( r ) κ m , n .
Φ ( r ) = R ( Φ G ) d r .
Φ ( r ) = k I ( r ) R ( I z G ) d r .
Φ ( r ) = k R { [ I 1 R ( I z G ) d r ] G } d r .
Φ ( x , y ) = k 0 a 0 b { [ I 1 0 a 0 b ( I z G ) d y d x ] G } d y d x .
2 Ψ m , n ( r ) + κ m , n Ψ m , n ( r ) = 0.
Ψ D , m , n = a m , n   sin ( m π a x ) sin ( n π b y ) ,
Ψ N , m , n = a m , n   cos ( m π a x ) cos ( n π b y ) ,
a m , n = { 2 a b , m , n 0 2 a b , m   or   n = 0 , }
κ m , n = ( m π a ) 2 + ( n π b ) 2 .
G N = ( G N x , G N y ) ,
G N x ( x , y , x , y ) = 2 a m = 1 cos ( m π a x ) sin ( m π a x ) × { cosh ( π m a [ b y ] ) cosh ( π m a y ) sinh ( π b a m ) , for   y > y cosh ( π m a [ b y ] ) cosh ( π m a y ) sinh ( π b a m ) , for   y < y , }
G N y ( x , y , x , y ) = 2 b n = 1 cos ( n π b y ) sin ( n π b y ) × { cosh ( π n b [ a x ] ) cosh ( π n b x ) sinh ( π a b n ) , for   x > x cosh ( π n b [ a x ] ) cosh ( π n b x ) sinh ( π a b n ) , for   x < x . }
Φ ( r ) = k { [ I 1 ( I z G ) d r ] G } d r .
F { p q } = F { p } F { q } ,
Φ ( r ) = k F 1 { F { G } F { I 1 F 1 { F { G } F { z I } } } } .
F { G } = i k x k x 2 + k y 2 e x + i k y k x 2 + k y 2 e y ,
Φ = Φ x + Φ y ,
Φ x = k F 1 { k x k x 2 + k y 2 F { I 1 F 1 { k x k x 2 + k y 2 F { z I } } } } ,
Φ y = k F 1 { k y k x 2 + k y 2 F { I 1 F 1 { k y k x 2 + k y 2 F { z I } } } } .
I z I + I Δ z ,
rel .   RMS-error = 100 % × x , y [ Φ rec ( x , y ) Φ obj ( x , y ) ] 2 x , y [ Φ obj ( x , y ) ] 2 ,
abs .   RMS-error = x , y [ Φ rec ( x , y ) Φ obj ( x , y ) ] 2 a × b ,
F { p ( x , y ) } = P ( k x , k y ) = 1 2 π p ( x , y ) exp [ i ( k x x + k y y ) ] d x d y ,
F 1 { P ( k x , k y ) } = p ( x , y ) = 1 2 π P ( k x , k y ) exp [ i ( k x x + k y y ) ] d k x d k y .
n x n p ( x , y ) = F 1 { ( i k x ) n P ( k x , k y ) } = F 1 { ( i k x ) n F { p ( x , y ) } } .
p ( x , y ) = ( F 1 { i k x F { p ( x , y ) } } F 1 { i k y F { p ( x , y ) } } ) ,
2 p ( x , y ) = F 1 { ( k x 2 + k y 2 ) F { p ( x , y ) } } .
2 G ( r , r ) = δ ( r r ) ,
F { 2 G } = F { δ } = ( k x 2 + k y 2 ) F { G } = 1 F { G } = 1 k x 2 + k y 2 .
F { G } = ( i k x F { G } i k y F { G } ) .
F { G } = ( i k x k x 2 + k y 2 , i k y k x 2 + k y 2 ) = i k x k x 2 + k y 2 e x + i k y k x 2 + k y 2 e y

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