Abstract

The transport of intensity equation (TIE) is a phase retrieval method that relies on measurements of the intensity of a paraxial field under propagation between two or more closely spaced planes. A limitation of TIE is its susceptibility to low frequency noise artifacts in the reconstructed phase. Under Köhler illumination, when both illumination power and exposure time are limited, the use of larger sources can improve low–frequency performance although it introduces blurring. Appropriately combining intensity measurements taken with a diversity of source sizes can improve both low– and high–frequency performance in phase reconstruction.

© 2017 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]
  33. Y. I. Nesterets and T. E. Gureyev, “Partially coherent contrast-transfer-function approximation,” J. Opt. Soc. Am. A 33, 464–474 (2016).
    [Crossref]
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    [Crossref]
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2016 (1)

2014 (3)

2013 (3)

2012 (3)

2011 (2)

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter-wave fields: Validity of Teague’s method for solution of the transport-of-intensity equation,” Phys. Rev. A 84, 023808 (2011).
[Crossref]

J. M. Bardsley, S. Knepper, and J. Nagy, “Structured linear algebra problems in adaptive optics imaging,” Advances in Computational Mathematics 35, 103–117 (2011).
[Crossref]

2010 (2)

2007 (1)

2006 (2)

G. R. Brady and J. R. Fienup, “Nonlinear optimization algorithm for retrieving the full complex pupil function,” Opt. Express 14, 474–486 (2006).
[Crossref] [PubMed]

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

2005 (1)

K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. 54, 191–197 (2005).
[Crossref]

2004 (2)

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

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214, 51–61 (2004).
[Crossref] [PubMed]

2000 (1)

B. E. Allan, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408, 158–159 (2000).
[Crossref]

1998 (4)

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

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

K. Scheerschmidt, “Retrieval of object information by inverse problems in electron diffraction,” J. Microsc. 190, 238–248 (1998).
[Crossref]

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

1997 (1)

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

1996 (1)

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

1984 (1)

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

1983 (1)

1982 (1)

1975 (1)

R. Henderson and P. N. T. Unwin, “Three-dimensional model of purple membrane obtained by electron microscopy,” Nature 257, 28–32 (1975).
[Crossref] [PubMed]

Allan, B. E.

B. E. Allan, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408, 158–159 (2000).
[Crossref]

Allman, B.

K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. 54, 191–197 (2005).
[Crossref]

Arif, M.

B. E. Allan, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408, 158–159 (2000).
[Crossref]

Asundi, A.

Ayubi, G. A.

J. A. Ferrari, G. A. Ayubi, J. L. Flores, and C. D. Perciante, “Transport of intensity equation: Validity limits of the usually accepted solution,” Opt. Commun. 318, 133–136 (2014).
[Crossref]

Bai, X.

Barbastathis, G.

Bardsley, J. M.

J. M. Bardsley, S. Knepper, and J. Nagy, “Structured linear algebra problems in adaptive optics imaging,” Advances in Computational Mathematics 35, 103–117 (2011).
[Crossref]

Barnea, Z.

K. A. Nugent, T. E. Gureyev, D. J. 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. Microsc. 214, 51–61 (2004).
[Crossref] [PubMed]

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

Batenburg, K. J.

Bau, D.

L. N. Trefethen and D. Bau, Numerical Linear Algebra, vol. 50 (Siam, 1997), pp. 77–82.

Beleggia, M.

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

Bie, R.

Boistel, R.

Bostan, E.

E. Froustey, E. Bostan, S. Lefkimmiatis, and M. Unser, “Digital phase reconstruction via iterative solutions of transport-of-intensity equation,” in the 2014 13th Workshop on Information Optics, WIO (2014) pp. 1–3.

Brady, G. R.

Chen, Q.

Cheng, J.

J. Cheng and S. Han, “X-ray phase imaging with a finite size source,” in “International Symposium on Biomedical Optics,” (International Society for Optics and Photonics, 1999), pp. 119–123.

Claus, R. A.

Connolly, B.

Cookson, D. J.

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

Dauwels, J.

Ferrari, J. A.

J. A. Ferrari, G. A. Ayubi, J. L. Flores, and C. D. Perciante, “Transport of intensity equation: Validity limits of the usually accepted solution,” Opt. Commun. 318, 133–136 (2014).
[Crossref]

Fienup, J. R.

Flores, J. L.

J. A. Ferrari, G. A. Ayubi, J. L. Flores, and C. D. Perciante, “Transport of intensity equation: Validity limits of the usually accepted solution,” Opt. Commun. 318, 133–136 (2014).
[Crossref]

Froustey, E.

E. Froustey, E. Bostan, S. Lefkimmiatis, and M. Unser, “Digital phase reconstruction via iterative solutions of transport-of-intensity equation,” in the 2014 13th Workshop on Information Optics, WIO (2014) pp. 1–3.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (Roberts and Company Publishers, 1996), Sec. 6.6.

Guigay, J.

Gureyev, T.

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

Gureyev, T. E.

Y. I. Nesterets and T. E. Gureyev, “Partially coherent contrast-transfer-function approximation,” J. Opt. Soc. Am. A 33, 464–474 (2016).
[Crossref]

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter-wave fields: Validity of Teague’s method for solution of the transport-of-intensity equation,” Phys. Rev. A 84, 023808 (2011).
[Crossref]

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

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

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

Han, S.

J. Cheng and S. Han, “X-ray phase imaging with a finite size source,” in “International Symposium on Biomedical Optics,” (International Society for Optics and Photonics, 1999), pp. 119–123.

Henderson, R.

R. Henderson and P. N. T. Unwin, “Three-dimensional model of purple membrane obtained by electron microscopy,” Nature 257, 28–32 (1975).
[Crossref] [PubMed]

Ishizuka, K.

K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. 54, 191–197 (2005).
[Crossref]

Jacobson, D. L.

B. E. Allan, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408, 158–159 (2000).
[Crossref]

Jingshan, Z.

King, A.

Knepper, S.

J. M. Bardsley, S. Knepper, and J. Nagy, “Structured linear algebra problems in adaptive optics imaging,” Advances in Computational Mathematics 35, 103–117 (2011).
[Crossref]

Kostenko, A.

Kou, S. S.

Langer, M.

Lefkimmiatis, S.

E. Froustey, E. Bostan, S. Lefkimmiatis, and M. Unser, “Digital phase reconstruction via iterative solutions of transport-of-intensity equation,” in the 2014 13th Workshop on Information Optics, WIO (2014) pp. 1–3.

McMahon, P. J.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214, 51–61 (2004).
[Crossref] [PubMed]

B. E. Allan, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408, 158–159 (2000).
[Crossref]

Nagy, J.

J. M. Bardsley, S. Knepper, and J. Nagy, “Structured linear algebra problems in adaptive optics imaging,” Advances in Computational Mathematics 35, 103–117 (2011).
[Crossref]

Nesterets, Y. I.

Y. I. Nesterets and T. E. Gureyev, “Partially coherent contrast-transfer-function approximation,” J. Opt. Soc. Am. A 33, 464–474 (2016).
[Crossref]

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

Neureuther, A.

Nugent, K.

Nugent, K. A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214, 51–61 (2004).
[Crossref] [PubMed]

B. E. Allan, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408, 158–159 (2000).
[Crossref]

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

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

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

Offerman, S. E.

Paganin, D.

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

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214, 51–61 (2004).
[Crossref] [PubMed]

B. E. Allan, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408, 158–159 (2000).
[Crossref]

A. Barty, K. 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]

K. A. Nugent, T. E. Gureyev, D. J. 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.

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter-wave fields: Validity of Teague’s method for solution of the transport-of-intensity equation,” Phys. Rev. A 84, 023808 (2011).
[Crossref]

Pavlov, K. M.

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter-wave fields: Validity of Teague’s method for solution of the transport-of-intensity equation,” Phys. Rev. A 84, 023808 (2011).
[Crossref]

Perciante, C. D.

J. A. Ferrari, G. A. Ayubi, J. L. Flores, and C. D. Perciante, “Transport of intensity equation: Validity limits of the usually accepted solution,” Opt. Commun. 318, 133–136 (2014).
[Crossref]

Petruccelli, J. C.

Pogany, A.

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

Reed Teague, M.

Roberts, A.

Scheerschmidt, K.

K. Scheerschmidt, “Retrieval of object information by inverse problems in electron diffraction,” J. Microsc. 190, 238–248 (1998).
[Crossref]

Schmalz, J. A.

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter-wave fields: Validity of Teague’s method for solution of the transport-of-intensity equation,” Phys. Rev. A 84, 023808 (2011).
[Crossref]

Schofield, M.

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

Sczyrba, M.

Shanker, A.

Sheppard, C. J. R.

Streibl, N.

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

Tian, L.

Trefethen, L. N.

L. N. Trefethen and D. Bau, Numerical Linear Algebra, vol. 50 (Siam, 1997), pp. 77–82.

Unser, M.

E. Froustey, E. Bostan, S. Lefkimmiatis, and M. Unser, “Digital phase reconstruction via iterative solutions of transport-of-intensity equation,” in the 2014 13th Workshop on Information Optics, WIO (2014) pp. 1–3.

Unwin, P. N. T.

R. Henderson and P. N. T. Unwin, “Three-dimensional model of purple membrane obtained by electron microscopy,” Nature 257, 28–32 (1975).
[Crossref] [PubMed]

van Vliet, L. J.

Volkov, V. V.

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

Waller, L.

Werner, S. A.

B. E. Allan, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408, 158–159 (2000).
[Crossref]

Wilkins, S.

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

Wilkins, S. W.

Xue, B.

Xue, W.

Yu, Y.

Yuan, X.-H.

Zhang, L.

Zhao, M.

Zheng, S.

Zhou, F.

Zhu, Y.

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

Zuo, C.

Advances in Computational Mathematics (1)

J. M. Bardsley, S. Knepper, and J. Nagy, “Structured linear algebra problems in adaptive optics imaging,” Advances in Computational Mathematics 35, 103–117 (2011).
[Crossref]

Appl. Opt. (2)

J. Electron Microsc. (1)

K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. 54, 191–197 (2005).
[Crossref]

J. Microsc. (2)

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214, 51–61 (2004).
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K. Scheerschmidt, “Retrieval of object information by inverse problems in electron diffraction,” J. Microsc. 190, 238–248 (1998).
[Crossref]

J. Opt. Soc. Am. (1)

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

Nature (2)

B. E. Allan, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408, 158–159 (2000).
[Crossref]

R. Henderson and P. N. T. Unwin, “Three-dimensional model of purple membrane obtained by electron microscopy,” Nature 257, 28–32 (1975).
[Crossref] [PubMed]

Opt. Commun. (4)

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

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

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

J. A. Ferrari, G. A. Ayubi, J. L. Flores, and C. D. Perciante, “Transport of intensity equation: Validity limits of the usually accepted solution,” Opt. Commun. 318, 133–136 (2014).
[Crossref]

Opt. Express (8)

C. Zuo, Q. Chen, Y. Yu, and A. Asundi, “Transport-of-intensity phase imaging using savitzky-golay differentiation filter-theory and applications,” Opt. Express 21, 5346–5362 (2013).
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A. Kostenko, K. J. Batenburg, A. King, S. E. Offerman, and L. J. van Vliet, “Total variation minimization approach in in-line x-ray phase-contrast tomography,” Opt. Express 21, 12185–12196 (2013).
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J. C. Petruccelli, L. Tian, and G. Barbastathis, “The transport of intensity equation for optical path length recovery using partially coherent illumination,” Opt. Express 21, 14430–14441 (2013).
[Crossref] [PubMed]

Z. Jingshan, R. A. Claus, J. Dauwels, L. Tian, and L. Waller, “Transport of intensity phase imaging by intensity spectrum fitting of exponentially spaced defocus planes,” Opt. Express 22, 10661–10674 (2014).
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G. R. Brady and J. R. Fienup, “Nonlinear optimization algorithm for retrieving the full complex pupil function,” Opt. Express 14, 474–486 (2006).
[Crossref] [PubMed]

L. Waller, L. Tian, and G. Barbastathis, “Transport of intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express 18, 12552–12561 (2010).
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S. Zheng, B. Xue, W. Xue, X. Bai, and F. Zhou, “Transport of intensity phase imaging from multiple noisy intensities measured in unequally-spaced planes,” Opt. Express 20, 972–985 (2012).
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Opt. Lett. (4)

Phys. Rev. A (1)

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter-wave fields: Validity of Teague’s method for solution of the transport-of-intensity equation,” Phys. Rev. A 84, 023808 (2011).
[Crossref]

Phys. Rev. Lett. (2)

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[Crossref]

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Ultramicroscopy (1)

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E. Froustey, E. Bostan, S. Lefkimmiatis, and M. Unser, “Digital phase reconstruction via iterative solutions of transport-of-intensity equation,” in the 2014 13th Workshop on Information Optics, WIO (2014) pp. 1–3.

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Figures (8)

Fig. 1
Fig. 1

(a) An off-axis point source one focal length away from a lens produces plane waves that travel with transverse vector direction −x/f. (b) Blur due to the source scales with defocus. A circular source of radius r illuminates a sample placed at z (shown as vertical dotted line). Rays passing through a single point on the sample will produce a circular spot in the defocused planes located at z ± Δz of diameter 2Δzr/f.

Fig. 2
Fig. 2

(a) For thermal noise dominated imaging, HTot is a metric for system performance with respect to noise in the spatial frequency domain, which is illustrated for circular sources of differing radii r. (b) For shot–noise dominated imaging, normalized total transfer function Tot is a metric for system performance with respect to noise, since the noise variance increases with source size. This is illustrated for circular sources of differing radii r. Sizes in mm refer to source sizes used in the experimental setup (see Section 3). For all instances, Δz/f = 0.0067.

Fig. 3
Fig. 3

For thermal noise dominated imaging: (a) the total transfer function HTot for a single, small circular source (r = 1.5 mm) (dashed, red) versus the combined transfer function which is the sum of the transfer functions of the r = 1.5, 2.7 and 3.5 mm sources each with 1/3 of the single–source exposure (solid, orange); (b) HTot for a single, large circular source (r = 3.5 mm) (dashed, red) versus the combined transfer function (solid, orange). For shot noise dominated imaging: (c) the normalized total transfer function Tot for an (r = 1.5 mm) (dashed, red) circular source versus the combined transfer function (solid, orange); (d) Tot for an (r = 3.5 mm) (dashed, red) circular source versus the combined transfer function (solid, orange). Poorly sampled frequency regions (aside from the one at |u| = 0) are labeled with * for the single–source and + for the multi–source cases. Notice that the * regions are nulls of the transfer function while the + regions are small, but non–null values.

Fig. 4
Fig. 4

Schematic diagram of the experimental setup used.

Fig. 5
Fig. 5

(a) Star target for simulated results, etched 50 nm deep in a piece of glass (refractive index n = 1.5). (b) Cartoon face for experimental results, etched 50 nm deep in a piece of glass (n ≈ 1.5) used in experiment. (c) Depth profile along the dashed line in (b).

Fig. 6
Fig. 6

Etch depth calculation of a 50 nm deep star target (600×600 pixels) using simulated propagation and circular sources. Sample thickness (in nm) reconstructed using single sources of the indicated size along with (a1)–(a3) deconvolution with minimal regularization ρs = 0.1, (b1)–(b3) deconvolution with moderate regularization ρs = 1000, (c1)–(c3) deconvolution with strong regularization ρs = 50000 and (d1)–(d3) no deconvolution. (e) Reconstruction using source diversity and the three indicated sources.

Fig. 7
Fig. 7

Experimental etch depth calculation of a 50 nm depth cartoon face (600×600 pixels) data using circular sources of the indicated sizes using (a1)–(a3) deconvolution with moderate regularization ρs = 1000 and (b1)–(b3) no deconvolution. (c) Illustrates reconstruction using source diversity.

Fig. 8
Fig. 8

Line plot of etch depth calculation of a 50 nm depth cartoon face (600×600 pixels) data using circular sources of the indicated sizes using (a1)–(a3) deconvolution with moderate regularization ρs = 1000 and (b1)–(b3) no deconvolution. (c) Illustrates reconstruction using source diversity. Dashed line indicates the expected, constant background value of the phase target.

Equations (21)

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i z U ( x , z ) 1 2 k x 2 U ( x , z ) = 0 ,
U ( x , z ) = I ( x , z ) exp [ i ϕ ( x , z ) ] ,
I ( x , z ) z + x [ I ( x , z ) x ϕ ( x , z ) k ] = 0 ,
1 k ϕ ( x , z ) z x 2 I ( x , z ) 2 k 2 I ( x , z ) = 1 2 k 2 | x ϕ ( x , z ) | 2 ,
I ( x , z ± Δ z ) I ( x , z ) Δ z k x [ I ( x , z ) x ϕ ( x , z ) ] ,
g ( x , z ) k I ( x , z + Δ z ) I ( x , z Δ z ) 2 Δ z = x [ I ( x , z ) x ϕ ( x , z ) ] ,
x Θ ( x , z ) = I ( x , z ) x ϕ ( x , z ) .
f ˜ ( u ) = [ f ( x ) ] = f ( x ) exp ( i 2 π x u ) d 2 x ,
f ( x ) = 1 [ f ˜ ( u ) ] = f ˜ ( u ) exp ( i 2 π x u ) d 2 u ,
Θ ˜ ( u ) = g ˜ ( u , z ) H L ( u ) ,
ϕ ( x ) = 1 [ 1 H L ( u ) ( 2 π i u { 1 [ 2 π i u Θ ˜ ( u ) ] I ( x , z ) } ) ] ,
H LT ( u ) = H L ( u ) 2 + ( 2 π ) 4 ρ 4 H L ( u ) ,
g ( x , z ) k I ( x , z + Δ z ) I ( x , z Δ z ) 2 Δ z = I s ( f x Δ z ) * { x [ I ( x , z ) x ϕ ( x , z ) ] } ,
g ( x , z ) = I s ( f x Δ z ) * x 2 Θ ( x , z ) = 1 [ H Tot ( u ) Θ ˜ ( u ) ] ,
H Tot ( u ) = H s ( Δ z f u ) H L ( u ) ,
Θ ˜ ( u ) = g ˜ ( u , z ) H Tot ( u ) .
g ˜ i ( u , z ) = H s i ( Δ z f u ) H L ( u ) Θ ˜ ( u )
Θ ˜ ( u ) = i = 1 N H s i * ( Δ z f u ) g ˜ i ( u ) H L ( u ) i = 1 N | H s i ( Δ z f u ) | 2 .
ϕ ( x ) = 1 [ 1 H L ( u ) ( 2 π i u { 1 [ 2 π i u Θ ˜ ( u ) ] I ¯ ( x , z ) } ) ] ,
I ¯ ( x , z ) = 1 N i = 1 N I i ( x , z ) H s i ( 0 ) .
Θ ˜ ( u ) = H s * ( Δ z f u ) g ˜ ( u ) H LT ( u ) [ | H s ( Δ z f u ) | 2 + ρ s ] .

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