Abstract

Transport of intensity equation (TIE) has been a popular and convenient phase imaging method that retrieves phase profile from the measurement of intensity differentials. Conventional 2-shot uniform illumination TIE can give reliable inversion of the phase from intensity in many situations of practical interest; however, it has a null space consisting of fields with non–zero circulation of the Poynting vector. Here, we propose the hybrid illumination TIE method to disambiguate such objects. By comparing the diffraction signals using uniform and structured (sinusoidal) illumination patterns, we obtain a modulation-induced signal that depends solely on the phase gradient. In this way, we also increase signal sensitivity in the low spatial frequency region.

© 2014 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
  3. P. B. Bach, J. R. Jett, U. Pastorino, M. S. Tockman, S. J. Swensen, and C. B. Begg, “Computed tomography screening and lung cancer outcomes,” Jama 297, 953–961 (2007).
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    [Crossref]
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    [Crossref]
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    [Crossref]
  9. D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586 (1998).
    [Crossref]
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    [Crossref]
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    [Crossref]
  17. A. Lubk, G. Guzzinati, F. Börrnert, and J. Verbeeck, “Transport of intensity phase retrieval of arbitrary wave fields including vortices,” Phys. Rev. Lett. 111, 173902 (2013).
    [Crossref] [PubMed]
  18. A. Shanker, L. Tian, M. Sczyrba, B. Connolly, A. Neureuther, and L. Waller, “Transport of intensity phase imaging in the presence of curl effects induced by strongly absorbing photomasks,” Appl. Opt. 53, J1–J6 (2014).
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    [Crossref]
  22. Y. Zhu, A. Shanker, L. Tian, L. Waller, and G. Barbastathis, “Low-noise phase imaging by hybrid uniform and structured illumination transport of intensity equation,” Opt. Express 22, 26696–26711 (2014).
    [Crossref]
  23. M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Progress in Optics 53, 293–363 (2009).
    [Crossref]
  24. L. Allen, H. Faulkner, K. Nugent, M. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  30. P. Pérez, M. Gangnet, and A. Blake, “Poisson image editing,” in “ACM Transactions on Graphics (TOG),”, vol. 22 (ACM, 2003), vol. 22, pp. 313–318.
    [Crossref]
  31. M. R. Hestenes and E. Stiefel, Methods of Conjugate Gradients for Solving Linear Systems, (National Bureau of Standards, 1952, vol. 49).

2014 (2)

2013 (1)

A. Lubk, G. Guzzinati, F. Börrnert, and J. Verbeeck, “Transport of intensity phase retrieval of arbitrary wave fields including vortices,” Phys. Rev. Lett. 111, 173902 (2013).
[Crossref] [PubMed]

2012 (1)

2011 (2)

L. Waller, M. Tsang, S. Ponda, S. Y. Yang, and G. Barbastathis, “Phase and amplitude imaging from noisy images by kalman filtering,” Opt. Express 19, 2805–2814 (2011).
[Crossref] [PubMed]

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]

2010 (2)

2009 (1)

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Progress in Optics 53, 293–363 (2009).
[Crossref]

2007 (1)

P. B. Bach, J. R. Jett, U. Pastorino, M. S. Tockman, S. J. Swensen, and C. B. Begg, “Computed tomography screening and lung cancer outcomes,” Jama 297, 953–961 (2007).
[Crossref] [PubMed]

2006 (2)

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]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Physics 2, 258–261 (2006).
[Crossref]

2004 (3)

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

L. Turner, B. Dhal, J. Hayes, A. Mancuso, K. Nugent, D. Paterson, R. Scholten, C. Tran, and A. Peele, “X-ray phase imaging: Demonstration of extended conditions for homogeneous objects,” Opt. Express 12, 2960–2965 (2004).
[Crossref] [PubMed]

R. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49, 3573 (2004).
[Crossref] [PubMed]

2001 (3)

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

K. A. Nugent, D. Paganin, and T. E. Gureyev, “A phase odyssey,” Phys. Today 54, 27–32 (2001).
[Crossref]

L. Allen, H. Faulkner, K. Nugent, M. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
[Crossref]

1998 (1)

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

1995 (1)

1988 (1)

1983 (1)

M. Reed Teague, “Deterministic phase retrieval: a green’s function solution,” J. Opt. Soc. Am. A 73, 1434–1441 (1983).
[Crossref]

1982 (1)

W. Hoppe, “Trace structure analysis, ptychography, phase tomography,” Ultramicroscopy 10, 187–198 (1982).
[Crossref]

1978 (1)

1971 (1)

RW Gerchber and W. Saxton, “Phase determination from image and diffraction plane pictures in electron-microscope,” Optik 34, 275 (1971).

1942 (1)

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9, 686–698 (1942).
[Crossref]

Allen, L.

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

L. Allen, H. Faulkner, K. Nugent, M. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
[Crossref]

Bach, P. B.

P. B. Bach, J. R. Jett, U. Pastorino, M. S. Tockman, S. J. Swensen, and C. B. Begg, “Computed tomography screening and lung cancer outcomes,” Jama 297, 953–961 (2007).
[Crossref] [PubMed]

Barbastathis, G.

Barty, A.

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

Begg, C. B.

P. B. Bach, J. R. Jett, U. Pastorino, M. S. Tockman, S. J. Swensen, and C. B. Begg, “Computed tomography screening and lung cancer outcomes,” Jama 297, 953–961 (2007).
[Crossref] [PubMed]

Blake, A.

P. Pérez, M. Gangnet, and A. Blake, “Poisson image editing,” in “ACM Transactions on Graphics (TOG),”, vol. 22 (ACM, 2003), vol. 22, pp. 313–318.
[Crossref]

Börrnert, F.

A. Lubk, G. Guzzinati, F. Börrnert, and J. Verbeeck, “Transport of intensity phase retrieval of arbitrary wave fields including vortices,” Phys. Rev. Lett. 111, 173902 (2013).
[Crossref] [PubMed]

Bunk, O.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Physics 2, 258–261 (2006).
[Crossref]

Connolly, B.

A. Shanker, L. Tian, M. Sczyrba, B. Connolly, A. Neureuther, and L. Waller, “Transport of intensity phase imaging in the presence of curl effects induced by strongly absorbing photomasks,” Appl. Opt. 53, J1–J6 (2014).

A. Shanker, M. Sczyrba, B. Connolly, F. Kalk, A. Neureuther, and L. Waller, “Critical assessment of the transport of intensity equation as a phase recovery technique in optical lithography,” in “SPIE Advanced Lithography,” (International Society for Optics and Photonics, 2014), pp. 90521D.

David, C.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Physics 2, 258–261 (2006).
[Crossref]

Dennis, M. R.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Progress in Optics 53, 293–363 (2009).
[Crossref]

Dhal, B.

Faulkner, H.

L. Allen, H. Faulkner, K. Nugent, M. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
[Crossref]

Fienup, J. R.

Gangnet, M.

P. Pérez, M. Gangnet, and A. Blake, “Poisson image editing,” in “ACM Transactions on Graphics (TOG),”, vol. 22 (ACM, 2003), vol. 22, pp. 313–318.
[Crossref]

Gerchber, RW

RW Gerchber and W. Saxton, “Phase determination from image and diffraction plane pictures in electron-microscope,” Optik 34, 275 (1971).

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]

T. Gureyev, A. Roberts, and K. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12, 1942–1946 (1995).
[Crossref]

Gureyev, T. E.

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]

K. A. Nugent, D. Paganin, and T. E. Gureyev, “A phase odyssey,” Phys. Today 54, 27–32 (2001).
[Crossref]

Guzzinati, G.

A. Lubk, G. Guzzinati, F. Börrnert, and J. Verbeeck, “Transport of intensity phase retrieval of arbitrary wave fields including vortices,” Phys. Rev. Lett. 111, 173902 (2013).
[Crossref] [PubMed]

Hayes, J.

Hestenes, M. R.

M. R. Hestenes and E. Stiefel, Methods of Conjugate Gradients for Solving Linear Systems, (National Bureau of Standards, 1952, vol. 49).

Hoppe, W.

W. Hoppe, “Trace structure analysis, ptychography, phase tomography,” Ultramicroscopy 10, 187–198 (1982).
[Crossref]

Ichikawa, K.

Jett, J. R.

P. B. Bach, J. R. Jett, U. Pastorino, M. S. Tockman, S. J. Swensen, and C. B. Begg, “Computed tomography screening and lung cancer outcomes,” Jama 297, 953–961 (2007).
[Crossref] [PubMed]

Kalk, F.

A. Shanker, M. Sczyrba, B. Connolly, F. Kalk, A. Neureuther, and L. Waller, “Critical assessment of the transport of intensity equation as a phase recovery technique in optical lithography,” in “SPIE Advanced Lithography,” (International Society for Optics and Photonics, 2014), pp. 90521D.

Kou, S. S.

Lewis, R.

R. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49, 3573 (2004).
[Crossref] [PubMed]

Lohmann, A. W.

Lubk, A.

A. Lubk, G. Guzzinati, F. Börrnert, and J. Verbeeck, “Transport of intensity phase retrieval of arbitrary wave fields including vortices,” Phys. Rev. Lett. 111, 173902 (2013).
[Crossref] [PubMed]

Mancuso, A.

McMahon, P.

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

Nesterets, Y. I.

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.

A. Shanker, L. Tian, M. Sczyrba, B. Connolly, A. Neureuther, and L. Waller, “Transport of intensity phase imaging in the presence of curl effects induced by strongly absorbing photomasks,” Appl. Opt. 53, J1–J6 (2014).

A. Shanker, M. Sczyrba, B. Connolly, F. Kalk, A. Neureuther, and L. Waller, “Critical assessment of the transport of intensity equation as a phase recovery technique in optical lithography,” in “SPIE Advanced Lithography,” (International Society for Optics and Photonics, 2014), pp. 90521D.

Nugent, K.

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

L. Turner, B. Dhal, J. Hayes, A. Mancuso, K. Nugent, D. Paterson, R. Scholten, C. Tran, and A. Peele, “X-ray phase imaging: Demonstration of extended conditions for homogeneous objects,” Opt. Express 12, 2960–2965 (2004).
[Crossref] [PubMed]

L. Allen, H. Faulkner, K. Nugent, M. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
[Crossref]

T. Gureyev, A. Roberts, and K. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12, 1942–1946 (1995).
[Crossref]

Nugent, K. A.

K. A. Nugent, D. Paganin, and T. E. Gureyev, “A phase odyssey,” Phys. Today 54, 27–32 (2001).
[Crossref]

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

O’Holleran, K.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Progress in Optics 53, 293–363 (2009).
[Crossref]

Oxley, M.

L. Allen, H. Faulkner, K. Nugent, M. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
[Crossref]

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

Padgett, M. J.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Progress in Optics 53, 293–363 (2009).
[Crossref]

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. McMahon, and K. Nugent, “Quantitative phase-amplitude microscopy. iii. the effects of noise,” J Microsc. 214, 51–61 (2004).
[Crossref] [PubMed]

L. Allen, H. Faulkner, K. Nugent, M. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
[Crossref]

K. A. Nugent, D. Paganin, and T. E. Gureyev, “A phase odyssey,” Phys. Today 54, 27–32 (2001).
[Crossref]

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

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]

Pan, A.

Y. Zhu, A. Pan, and G. Barbastathis, “Low-noise tie phase imaging by structured illumination,” in “Computational Optical Sensing and Imaging,” (Optical Society of America, 2014), pp. CTh3C–5.
[Crossref]

Pastorino, U.

P. B. Bach, J. R. Jett, U. Pastorino, M. S. Tockman, S. J. Swensen, and C. B. Begg, “Computed tomography screening and lung cancer outcomes,” Jama 297, 953–961 (2007).
[Crossref] [PubMed]

Paterson, D.

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]

Peele, A.

Pérez, P.

P. Pérez, M. Gangnet, and A. Blake, “Poisson image editing,” in “ACM Transactions on Graphics (TOG),”, vol. 22 (ACM, 2003), vol. 22, pp. 313–318.
[Crossref]

Petruccelli, J. C.

Pfeiffer, F.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Physics 2, 258–261 (2006).
[Crossref]

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]

Ponda, S.

Reed Teague, M.

M. Reed Teague, “Deterministic phase retrieval: a green’s function solution,” J. Opt. Soc. Am. A 73, 1434–1441 (1983).
[Crossref]

Roberts, A.

Saxton, W.

RW Gerchber and W. Saxton, “Phase determination from image and diffraction plane pictures in electron-microscope,” Optik 34, 275 (1971).

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]

Scholten, R.

Sczyrba, M.

A. Shanker, L. Tian, M. Sczyrba, B. Connolly, A. Neureuther, and L. Waller, “Transport of intensity phase imaging in the presence of curl effects induced by strongly absorbing photomasks,” Appl. Opt. 53, J1–J6 (2014).

A. Shanker, M. Sczyrba, B. Connolly, F. Kalk, A. Neureuther, and L. Waller, “Critical assessment of the transport of intensity equation as a phase recovery technique in optical lithography,” in “SPIE Advanced Lithography,” (International Society for Optics and Photonics, 2014), pp. 90521D.

Shanker, A.

A. Shanker, L. Tian, M. Sczyrba, B. Connolly, A. Neureuther, and L. Waller, “Transport of intensity phase imaging in the presence of curl effects induced by strongly absorbing photomasks,” Appl. Opt. 53, J1–J6 (2014).

Y. Zhu, A. Shanker, L. Tian, L. Waller, and G. Barbastathis, “Low-noise phase imaging by hybrid uniform and structured illumination transport of intensity equation,” Opt. Express 22, 26696–26711 (2014).
[Crossref]

A. Shanker, M. Sczyrba, B. Connolly, F. Kalk, A. Neureuther, and L. Waller, “Critical assessment of the transport of intensity equation as a phase recovery technique in optical lithography,” in “SPIE Advanced Lithography,” (International Society for Optics and Photonics, 2014), pp. 90521D.

A. Shanker, L. Tian, and L. Waller, “Defocus-based quantitative phase imaging by coded illumination,” in “SPIE BiOS,” (International Society for Optics and Photonics, 2014), pp. 89490R.

Sheppard, C. J.

Stiefel, E.

M. R. Hestenes and E. Stiefel, Methods of Conjugate Gradients for Solving Linear Systems, (National Bureau of Standards, 1952, vol. 49).

Swensen, S. J.

P. B. Bach, J. R. Jett, U. Pastorino, M. S. Tockman, S. J. Swensen, and C. B. Begg, “Computed tomography screening and lung cancer outcomes,” Jama 297, 953–961 (2007).
[Crossref] [PubMed]

Takeda, M.

Tian, L.

Tockman, M. S.

P. B. Bach, J. R. Jett, U. Pastorino, M. S. Tockman, S. J. Swensen, and C. B. Begg, “Computed tomography screening and lung cancer outcomes,” Jama 297, 953–961 (2007).
[Crossref] [PubMed]

Tran, C.

Tsang, M.

Turner, L.

Verbeeck, J.

A. Lubk, G. Guzzinati, F. Börrnert, and J. Verbeeck, “Transport of intensity phase retrieval of arbitrary wave fields including vortices,” Phys. Rev. Lett. 111, 173902 (2013).
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Waller, L.

Weitkamp, T.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Physics 2, 258–261 (2006).
[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).
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[Crossref]

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Y. Zhu, A. Shanker, L. Tian, L. Waller, and G. Barbastathis, “Low-noise phase imaging by hybrid uniform and structured illumination transport of intensity equation,” Opt. Express 22, 26696–26711 (2014).
[Crossref]

Y. Zhu, A. Pan, and G. Barbastathis, “Low-noise tie phase imaging by structured illumination,” in “Computational Optical Sensing and Imaging,” (Optical Society of America, 2014), pp. CTh3C–5.
[Crossref]

Appl. Opt. (2)

J Microsc. (1)

D. Paganin, A. Barty, P. McMahon, and K. Nugent, “Quantitative phase-amplitude microscopy. iii. the effects of noise,” J Microsc. 214, 51–61 (2004).
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Jama (1)

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Nat. Physics (1)

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Physics 2, 258–261 (2006).
[Crossref]

Opt. Commun. (2)

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]

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[Crossref]

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RW Gerchber and W. Saxton, “Phase determination from image and diffraction plane pictures in electron-microscope,” Optik 34, 275 (1971).

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R. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49, 3573 (2004).
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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]

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L. Allen, H. Faulkner, K. Nugent, M. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
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A. Lubk, G. Guzzinati, F. Börrnert, and J. Verbeeck, “Transport of intensity phase retrieval of arbitrary wave fields including vortices,” Phys. Rev. Lett. 111, 173902 (2013).
[Crossref] [PubMed]

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[Crossref]

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K. A. Nugent, D. Paganin, and T. E. Gureyev, “A phase odyssey,” Phys. Today 54, 27–32 (2001).
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Physica (1)

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A. Shanker, L. Tian, and L. Waller, “Defocus-based quantitative phase imaging by coded illumination,” in “SPIE BiOS,” (International Society for Optics and Photonics, 2014), pp. 89490R.

Y. Zhu, A. Pan, and G. Barbastathis, “Low-noise tie phase imaging by structured illumination,” in “Computational Optical Sensing and Imaging,” (Optical Society of America, 2014), pp. CTh3C–5.
[Crossref]

P. Pérez, M. Gangnet, and A. Blake, “Poisson image editing,” in “ACM Transactions on Graphics (TOG),”, vol. 22 (ACM, 2003), vol. 22, pp. 313–318.
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Figures (5)

Fig. 1
Fig. 1 Aberration in uniform illumination TIE phase retrieval with a 2-ball object. The absorbance of the smaller ball is twice as strong as the bigger ball, while the refractive index of the smaller ball is half as much as that of the bigger ball. Figure (a) and (b) show the phase profile ϕ (in rad) and the transmission T respectively. Figure (c) shows the gradient vectors of the transmission and phase. Misalignment is seen in the overlapping areas. Figure (d) shows the phase retrieval result ϕu using Eq. (6). Aberration is observed in the overlapping area where circulation effect occurs.
Fig. 2
Fig. 2 Conceptual setup of the conventional 2-shot TIE (a) and the hybrid illumination TIE (b). Intensity differentials are obtained via displacement of CCD and via changing the illumination pattern, respectively.
Fig. 3
Fig. 3 Experiment setup of the hybrid structured illumination TIE.
Fig. 4
Fig. 4 Simulated TIE phase imaging results of a half-spherical lens with gradient absorption background. (a), in-focus intensity profile I(0) (normalized); (b), calibrated phase profile ϕ; (c) uniform illumination TIE phase retrieval without noise ϕ u noiseless ; (d) uniform illumination TIE phase retrieval with noise ϕ u noisy and (e) hybrid illumination TIE phase retrieval with noise ϕ h noisy . (d) and (e) both use a noise power of −25 dB in the intensity simulation. Other parameters used in the simulation include the modulation attenuation a = 0.5, modulation depth η = 0.4 and the modulation frequency fm = 0.0103 μm−1.
Fig. 5
Fig. 5 Experimental TIE phase imaging results of a half-spherical lens with gradient absorption background. (a), experimental measurement of in-focus (normalized) intensity profile I(0); (b) expected phase profile ϕ; (c), highly exposed (10× exposure time) uniform illumination TIE phase retrieval ϕ h less noisy , (d) normally exposed (1× exposure time) uniform illumination TIE phase retrieval ϕ y noisy , and (e) hybrid illumination TIE phase retrieval ϕ h noisy (1× exposure time). In the experiment, the modulation attenuation a = 0.31, modulation depth η = 0.4 and the modulation frequency fm = 0.0103 μm−1.

Equations (15)

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k I ( x , y ; z ) z | z = 0 = ( I ( x , y ; 0 ) ϕ ( x , y ) ) ,
S = ω 4 π I ϕ .
1 k ( U 0 T ϕ ) = I z I ( d ) U 0 T d Δ ( d ) d ,
T ϕ = × C ( x , y ) + D ( x , y ) ,
2 D ( x , y ) = k d U 0 Δ ( d ) ,
ϕ u ( x , y ) = k 2 { [ 1 T D ( x , y ) ] } ,
× ( T ϕ ) = T × ϕ = 0 .
1 k ( U s T ϕ ) = I z I s ( d ) U s T d ,
1 k [ U s ( × C ( x , y ) + D ( x , y ) ) + U s 2 D ( x , y ) ] = I s ( d ) U s T d .
1 k [ ( U s T ϕ ) U s ( T ϕ ) ] = = I s ( d ) U s T U s / U 0 Δ ( d ) d = I s ( d ) U s / U 0 I ( d ) d ,
U s ϕ = k d T ( I s ( d ) U s U 0 I ( d ) ) k d s ( d ) .
U s , X ( x , y ) = a U 0 ( 1 + η sin ( 2 π f m x ) ) , U s , Y ( x , y ) = a U 0 ( 1 + η sin ( 2 π f m y ) ) ;
U q , X ( x , y ) = a U 0 ( 1 + η cos ( 2 π f m x ) ) , U q , Y ( x , y ) = a U 0 ( 1 + η cos ( 2 π f m y ) ) ,
ϕ x = k d 1 a U 0 η 2 π f m ( s X ( d ) i s q , X ( d ) ) e i 2 π f m x ϕ y = k d 1 a U 0 η 2 π f m ( s Y d i s q , Y ( d ) ) e i 2 π f m y .
ϕ x = 1 [ i k d 1 π f m a η U 0 ( s X ( d ) ) ( u + f m , v ) H x ( u , v ) ] ϕ y = 1 [ i k d 1 π f m a η U 0 ( s Y ( d ) ) ( u , v + f m ) H y ( u , v ) ] ,

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