Abstract

In this paper, we propose effective methods to detect focal planes and recover objects in the noninterferometric phase imaging. A strategy using different aberration coefficients is developed, and a series of diffraction intensity maps is sequentially recorded by a charge-coupled device. During numerical reconstruction, a phase retrieval algorithm is applied to extract object wavefront from diffraction intensity maps. Subsequently, numerical methods are proposed to detect focal planes, and high-quality phase maps are recovered by using the detected focal distances. Theoretical results are presented to demonstrate feasibility and effectiveness of the proposed methods.

© 2012 Optical Society of America

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2011

W. Chen and X. Chen, “Optical asymmetric cryptography using a three-dimensional space-based model,” J. Opt. 13, 075404 (2011).
[CrossRef]

W. Chen and X. Chen, “Quantitative phase retrieval of complex-valued specimens based on noninterferometric imaging,” Appl. Opt. 50, 2008–2015 (2011).
[CrossRef]

2010

2009

W. Chen, C. Quan, C. J. Tay, and Y. Fu, “Quantitative detection and compensation of phase-shifting error in two-step phase-shifting digital holography,” Opt. Commun. 282, 2800–2805 (2009).
[CrossRef]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[CrossRef]

C. A. Henderson, G. J. Williams, A. G. Peele, H. M. Quiney, and K. A. Nugent, “Astigmatic phase retrieval: an experimental demonstration,” Opt. Express 17, 11905–11915 (2009).
[CrossRef]

A. N. Simonov and M. C. Rombach, “Sharp-focus image restoration from defocused images,” Opt. Lett. 34, 2111–2113 (2009).
[CrossRef]

K. Matsushima and T. Shimobaba, “Band-limited angular spectrum method for numerical simulation of free-space propagation in far and near fields,” Opt. Express 17, 19662–19673 (2009).
[CrossRef]

2008

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100, 155503 (2008).
[CrossRef]

2007

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
[CrossRef]

P. Almoro, G. Pedrini, and W. Osten, “Aperture synthesis in phase retrieval using a volume-speckle field,” Opt. Lett. 32, 733–735 (2007).
[CrossRef]

2006

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. H. Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23, 1179–1200 (2006).
[CrossRef]

F. Dubois, C. Schockaert, N. Callens, and C. Yourassowsky, “Focus plane detection criteria in digital holography microscopy by amplitude analysis,” Opt. Express 14, 5895–5908 (2006).
[CrossRef]

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97, 025506 (2006).
[CrossRef]

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442, 63–66 (2006).
[CrossRef]

2005

W. McBride, N. L. O. Leary, K. A. Nugent, and L. J. Allen, “Astigmatic electron diffraction imaging: a novel mode for structure determination,” Acta Crystallogr. Sect. A 61, 321–324 (2005).

2004

2003

V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20, 40–55 (2003).
[CrossRef]

J. Miao, T. Ishikawa, E. H. Anderson, and K. O. Hodgson, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67, 174104 (2003).
[CrossRef]

J. M. Zou, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419–1421 (2003).
[CrossRef]

1999

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

1998

J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662–1669 (1998).
[CrossRef]

M. Bravo-Zanoguera, B. V. Massenbach, A. L. Kellner, and J. H. Price, “High-performance autofocus circuit for biological microscopy,” Rev. Sci. Instrum. 69, 3966–3977 (1998).
[CrossRef]

1997

1996

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]

1993

1986

1985

F. C. A. Groen, I. T. Young, and G. Ligthart, “A comparison of different focus functions for use in autofocus algorithms,” Cytometry 6, 81–91 (1985).
[CrossRef]

1983

1982

1972

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

1948

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[CrossRef]

Allen, L. J.

W. McBride, N. L. O. Leary, K. A. Nugent, and L. J. Allen, “Astigmatic electron diffraction imaging: a novel mode for structure determination,” Acta Crystallogr. Sect. A 61, 321–324 (2005).

Almoro, P.

Anderson, E. H.

J. Miao, T. Ishikawa, E. H. Anderson, and K. O. Hodgson, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67, 174104 (2003).
[CrossRef]

Barty, A.

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]

Beetz, T.

Bravo-Zanoguera, M.

M. Bravo-Zanoguera, B. V. Massenbach, A. L. Kellner, and J. H. Price, “High-performance autofocus circuit for biological microscopy,” Rev. Sci. Instrum. 69, 3966–3977 (1998).
[CrossRef]

Bunk, O.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[CrossRef]

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100, 155503 (2008).
[CrossRef]

Callens, N.

Chapman, H. N.

Charalambous, P.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

Chen, W.

Chen, X.

Cui, C.

David, C.

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100, 155503 (2008).
[CrossRef]

de Jonge, M. D.

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97, 025506 (2006).
[CrossRef]

Dhal, B. B.

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97, 025506 (2006).
[CrossRef]

Dierolf, M.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[CrossRef]

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100, 155503 (2008).
[CrossRef]

Dubois, F.

Elser, V.

Engelberg, Y. M.

Fienup, J. R.

Fu, Y.

W. Chen, C. Quan, C. J. Tay, and Y. Fu, “Quantitative detection and compensation of phase-shifting error in two-step phase-shifting digital holography,” Opt. Commun. 282, 2800–2805 (2009).
[CrossRef]

Furtak, T. E.

M. V. Klein and T. E. Furtak, Optics, 2nd ed. (Wiley, 1986).

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[CrossRef]

Gao, M.

J. M. Zou, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419–1421 (2003).
[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 35, 237–246 (1972).

Ghiglia, D. C.

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

Gohara, K.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Gray, J.

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100, 155503 (2008).
[CrossRef]

Groen, F. C. A.

F. C. A. Groen, I. T. Young, and G. Ligthart, “A comparison of different focus functions for use in autofocus algorithms,” Cytometry 6, 81–91 (1985).
[CrossRef]

Gureyev, T. E.

Harder, R.

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442, 63–66 (2006).
[CrossRef]

He, H.

Henderson, C. A.

Hodgson, K. O.

J. Miao, T. Ishikawa, E. H. Anderson, and K. O. Hodgson, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67, 174104 (2003).
[CrossRef]

Howells, M. R.

Hua, J.

Ishikawa, T.

J. Miao, T. Ishikawa, E. H. Anderson, and K. O. Hodgson, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67, 174104 (2003).
[CrossRef]

Jacobsen, C.

Jefimovs, K.

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100, 155503 (2008).
[CrossRef]

Johnson, I.

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100, 155503 (2008).
[CrossRef]

Kellner, A. L.

M. Bravo-Zanoguera, B. V. Massenbach, A. L. Kellner, and J. H. Price, “High-performance autofocus circuit for biological microscopy,” Rev. Sci. Instrum. 69, 3966–3977 (1998).
[CrossRef]

Kirz, J.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

Klein, M. V.

M. V. Klein and T. E. Furtak, Optics, 2nd ed. (Wiley, 1986).

Kreis, T.

T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley-VCH, 2005).

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform With Applications in Optics and Signal Processing (Wiley, 2001).

Leary, N. L. O.

W. McBride, N. L. O. Leary, K. A. Nugent, and L. J. Allen, “Astigmatic electron diffraction imaging: a novel mode for structure determination,” Acta Crystallogr. Sect. A 61, 321–324 (2005).

Li, G.

Ligthart, G.

F. C. A. Groen, I. T. Young, and G. Ligthart, “A comparison of different focus functions for use in autofocus algorithms,” Cytometry 6, 81–91 (1985).
[CrossRef]

Liu, L.

Lohmann, A. W.

Maehara, Y.

Marchesini, S.

Massenbach, B. V.

M. Bravo-Zanoguera, B. V. Massenbach, A. L. Kellner, and J. H. Price, “High-performance autofocus circuit for biological microscopy,” Rev. Sci. Instrum. 69, 3966–3977 (1998).
[CrossRef]

Matsushima, K.

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]

McBride, W.

W. McBride, N. L. O. Leary, K. A. Nugent, and L. J. Allen, “Astigmatic electron diffraction imaging: a novel mode for structure determination,” Acta Crystallogr. Sect. A 61, 321–324 (2005).

Menzel, A.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[CrossRef]

Miao, J.

J. Miao, T. Ishikawa, E. H. Anderson, and K. O. Hodgson, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67, 174104 (2003).
[CrossRef]

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662–1669 (1998).
[CrossRef]

Nagahara, L. A.

J. M. Zou, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419–1421 (2003).
[CrossRef]

Neild, A.

Ng, T. W.

Noy, A.

Nugent, K. A.

G. J. Williams, H. M. Quiney, A. G. Peele, and K. A. Nugent, “Fresnel coherent diffractive imaging: treatment and analysis of data,” New J. Phys. 12, 035020 (2010).
[CrossRef]

C. A. Henderson, G. J. Williams, A. G. Peele, H. M. Quiney, and K. A. Nugent, “Astigmatic phase retrieval: an experimental demonstration,” Opt. Express 17, 11905–11915 (2009).
[CrossRef]

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97, 025506 (2006).
[CrossRef]

W. McBride, N. L. O. Leary, K. A. Nugent, and L. J. Allen, “Astigmatic electron diffraction imaging: a novel mode for structure determination,” Acta Crystallogr. Sect. A 61, 321–324 (2005).

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]

Osten, W.

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
[CrossRef]

P. Almoro, G. Pedrini, and W. Osten, “Aperture synthesis in phase retrieval using a volume-speckle field,” Opt. Lett. 32, 733–735 (2007).
[CrossRef]

Ozaktas, H. M.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform With Applications in Optics and Signal Processing (Wiley, 2001).

Paterson, D.

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97, 025506 (2006).
[CrossRef]

Pedrini, G.

P. Almoro, G. Pedrini, and W. Osten, “Aperture synthesis in phase retrieval using a volume-speckle field,” Opt. Lett. 32, 733–735 (2007).
[CrossRef]

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G. J. Williams, H. M. Quiney, A. G. Peele, and K. A. Nugent, “Fresnel coherent diffractive imaging: treatment and analysis of data,” New J. Phys. 12, 035020 (2010).
[CrossRef]

C. A. Henderson, G. J. Williams, A. G. Peele, H. M. Quiney, and K. A. Nugent, “Astigmatic phase retrieval: an experimental demonstration,” Opt. Express 17, 11905–11915 (2009).
[CrossRef]

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97, 025506 (2006).
[CrossRef]

Pfeifer, M. A.

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442, 63–66 (2006).
[CrossRef]

Pfeiffer, F.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[CrossRef]

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100, 155503 (2008).
[CrossRef]

Price, J. H.

M. Bravo-Zanoguera, B. V. Massenbach, A. L. Kellner, and J. H. Price, “High-performance autofocus circuit for biological microscopy,” Rev. Sci. Instrum. 69, 3966–3977 (1998).
[CrossRef]

Pritt, M. D.

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

Quan, C.

W. Chen, C. Quan, C. J. Tay, and Y. Fu, “Quantitative detection and compensation of phase-shifting error in two-step phase-shifting digital holography,” Opt. Commun. 282, 2800–2805 (2009).
[CrossRef]

Quiney, H. M.

G. J. Williams, H. M. Quiney, A. G. Peele, and K. A. Nugent, “Fresnel coherent diffractive imaging: treatment and analysis of data,” New J. Phys. 12, 035020 (2010).
[CrossRef]

C. A. Henderson, G. J. Williams, A. G. Peele, H. M. Quiney, and K. A. Nugent, “Astigmatic phase retrieval: an experimental demonstration,” Opt. Express 17, 11905–11915 (2009).
[CrossRef]

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97, 025506 (2006).
[CrossRef]

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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]

Renker, D.

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100, 155503 (2008).
[CrossRef]

Riege, S. P. H.

Robinson, I. K.

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442, 63–66 (2006).
[CrossRef]

Rombach, M. C.

Rosen, R.

Ruschin, S.

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 35, 237–246 (1972).

Sayre, D.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662–1669 (1998).
[CrossRef]

Schockaert, C.

Shapiro, D.

Sheppard, C. J. R.

Shimobaba, T.

Shioya, H.

Simonov, A. N.

Spence, J. C. H.

Tay, C. J.

W. Chen, C. Quan, C. J. Tay, and Y. Fu, “Quantitative detection and compensation of phase-shifting error in two-step phase-shifting digital holography,” Opt. Commun. 282, 2800–2805 (2009).
[CrossRef]

Teague, M. R.

Thibault, P.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[CrossRef]

Tran, C. Q.

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97, 025506 (2006).
[CrossRef]

Vartanyants, I.

J. M. Zou, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419–1421 (2003).
[CrossRef]

Vartanyants, I. A.

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442, 63–66 (2006).
[CrossRef]

Wackerman, C. C.

Weierstall, U.

Williams, G. J.

G. J. Williams, H. M. Quiney, A. G. Peele, and K. A. Nugent, “Fresnel coherent diffractive imaging: treatment and analysis of data,” New J. Phys. 12, 035020 (2010).
[CrossRef]

C. A. Henderson, G. J. Williams, A. G. Peele, H. M. Quiney, and K. A. Nugent, “Astigmatic phase retrieval: an experimental demonstration,” Opt. Express 17, 11905–11915 (2009).
[CrossRef]

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442, 63–66 (2006).
[CrossRef]

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97, 025506 (2006).
[CrossRef]

Wu, C.

Yamaguchi, I.

Young, I. T.

F. C. A. Groen, I. T. Young, and G. Ligthart, “A comparison of different focus functions for use in autofocus algorithms,” Cytometry 6, 81–91 (1985).
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Yourassowsky, C.

Zalevsky, Z.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform With Applications in Optics and Signal Processing (Wiley, 2001).

Zhang, F.

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
[CrossRef]

Zhang, R.

J. M. Zou, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419–1421 (2003).
[CrossRef]

Zhang, T.

Zou, J. M.

J. M. Zou, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419–1421 (2003).
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Acta Crystallogr. Sect. A

W. McBride, N. L. O. Leary, K. A. Nugent, and L. J. Allen, “Astigmatic electron diffraction imaging: a novel mode for structure determination,” Acta Crystallogr. Sect. A 61, 321–324 (2005).

Appl. Opt.

Cytometry

F. C. A. Groen, I. T. Young, and G. Ligthart, “A comparison of different focus functions for use in autofocus algorithms,” Cytometry 6, 81–91 (1985).
[CrossRef]

J. Opt.

W. Chen and X. Chen, “Optical asymmetric cryptography using a three-dimensional space-based model,” J. Opt. 13, 075404 (2011).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

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]

V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20, 40–55 (2003).
[CrossRef]

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[CrossRef]

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. H. Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23, 1179–1200 (2006).
[CrossRef]

J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662–1669 (1998).
[CrossRef]

J. Hua, L. Liu, and G. Li, “Extended fractional Fourier transforms,” J. Opt. Soc. Am. A 14, 3316–3322 (1997).
[CrossRef]

J. R. Fienup, “Phase retrieval using boundary conditions,” J. Opt. Soc. Am. A 3, 284–288 (1986).
[CrossRef]

J. R. Fienup and C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
[CrossRef]

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
[CrossRef]

A. N. Simonov and M. C. Rombach, “Asymptotic behavior of the spatial frequency response of an optical system with defocus and spherical aberration,” J. Opt. Soc. Am. A 27, 2563–2573 (2010).
[CrossRef]

H. Shioya, Y. Maehara, and K. Gohara, “Spherical shell structure of distribution of images reconstructed by diffractive imaging,” J. Opt. Soc. Am. A 27, 1214–1218 (2010).
[CrossRef]

Nature

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[CrossRef]

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442, 63–66 (2006).
[CrossRef]

New J. Phys.

G. J. Williams, H. M. Quiney, A. G. Peele, and K. A. Nugent, “Fresnel coherent diffractive imaging: treatment and analysis of data,” New J. Phys. 12, 035020 (2010).
[CrossRef]

Opt. Commun.

W. Chen, C. Quan, C. J. Tay, and Y. Fu, “Quantitative detection and compensation of phase-shifting error in two-step phase-shifting digital holography,” Opt. Commun. 282, 2800–2805 (2009).
[CrossRef]

Opt. Express

Opt. Lett.

Optik

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

Phys. Rev. A

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
[CrossRef]

Phys. Rev. B

J. Miao, T. Ishikawa, E. H. Anderson, and K. O. Hodgson, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67, 174104 (2003).
[CrossRef]

Phys. Rev. Lett.

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]

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97, 025506 (2006).
[CrossRef]

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100, 155503 (2008).
[CrossRef]

Rev. Sci. Instrum.

M. Bravo-Zanoguera, B. V. Massenbach, A. L. Kellner, and J. H. Price, “High-performance autofocus circuit for biological microscopy,” Rev. Sci. Instrum. 69, 3966–3977 (1998).
[CrossRef]

Science

J. M. Zou, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419–1421 (2003).
[CrossRef]

Ultramicroscopy

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[CrossRef]

Other

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

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley-VCH, 2005).

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A. Hermerschmidt, S. Osten, J. Frank, and S. Krüger, “New liquid crystal microdisplays permit phase-only light modulation,” SPIE Newsroom (2006), http://spie.org/x8714.xml?highlight=x2408&ArticleID=x8714 .
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform With Applications in Optics and Signal Processing (Wiley, 2001).

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Figures (6)

Fig. 1.
Fig. 1.

Schematic experimental setup for the proposed methods: SLM, spatial light modulator; d, z1, and z2 axial distances; k, wave number; CCD, charge-coupled device; NN, UU, and SS are phase objects placed at different axial locations (i.e., 17.76 mm, 11.88 mm, and 6.0 mm).

Fig. 2.
Fig. 2.

Flow chart of the procedure to determine focal distances using the proposed methods.

Fig. 3.
Fig. 3.

Original phase maps of test phase objects (a) NN, (b) UU, and (c) SS. The recorded diffraction intensity maps (with logarithm scale) using aberration coefficients of (d) 60, (e) 100, and (f) 140. (g) A relationship between the number of iterations and iterative errors (with logarithm scale). (h) Real and (i) imaginary parts of the extracted object wavefront Tn(3)(ξ,η)¯ just before the SLM. Dashed lines are added in (a)–(c) for the illustration of reconstruction quality in Figs. 4 and 6.

Fig. 4.
Fig. 4.

Iterative step of 1 mm is selected. (a) Spectral analysis detection method. (b) Gradient operator detection method. Recovered phase maps of test objects (c) NN, (d) UU and (e) SS. (f)–(h) Cross sections along the dashed lines [respectively indicated in Figs. 3(a)–3(c)] for (c)–(e). When an error of 3 mm exists in the reconstruction distances: recovered phase maps of test objects (i) NN, (j) UU and (k) SS.

Fig. 5.
Fig. 5.

Iterative step of 0.01 mm is selected: (a) spectral analysis detection method and (b) gradient operator detection method.

Fig. 6.
Fig. 6.

Noise contamination (SNR of 10): (a) Spectral analysis detection method. (b) Gradient operator detection method. Recovered phase maps of test objects (c) NN, (d) UU, and (e) SS. (f)–(h) Cross sections along the dashed lines [respectively indicated in Figs. 3(a)3(c)] for (c)–(e). When an error of 3 mm exists in the reconstruction distances: recovered phase maps of test objects (i) NN, (j) UU and (k) SS.

Equations (18)

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

O(ξ,η)=jλ++O(x,y)1ρexp(j2πλρ)dxdy,
ρ=z1+(ξx)22z1+(ηy)22z118[(ξx)2+(ηy)2]2z13+.
ρ=z1+(ξx)22z1+(ηy)22z1.
O(ξ,η)=jλz1exp(j2πλz1)exp[jπλz1(ξ2+η2)]×++O(x,y)exp[jπλz1(x2+y2)]×exp[j2πλz1(xξ+yη)]dxdy.
O(m,n)=jλz1exp(j2πλz1)exp[jπλz1(m2M2Δx2+n2N2Δy2)]k=0M1l=0N1O(k,l)exp[jπλz1(k2Δx2+l2Δy2)]exp[j2π(mkM+nlN)],
Δξ=λz1MΔx,Δη=λz1NΔy.
O(ξ,η)=FT1(I(kx,ky;z1){FT[O(x,y)]}),
I(kx,ky;z1)=exp(jz1kz)=exp(jz1k2kx2ky2),
I(fx,fy;z1)={exp{jkz1[1(λfx)2(λfy)2]1/2}if(fx2+fy2)1/2<(1/λ)0Otherwise,
I(m,n;z1)=exp(jkz1{1[λ(mM2)MΔx]2[λ(nN2)NΔy]2}1/2),
Δξ=Δx,Δη=Δy.
S(h)(ξ,η)=exp[j1013B(h)(ξ2+η2)2],
I(h)(μ,ν)=|WPz2[(r=1,2,3{WPz1,z1+d,z1+2d[Or(x,y)]})S(h)(ξ,η)]|2,
Noise=({Mn[I(h)(μ,ν)]}SNR)×Rd,
Op(x,y)=WPz(p)[Tn(3)(ξ,η)¯],
L1=fxfyln{1+|FT[Ap(x,y)]|}dfxdfy,
L2=xy[Ap(x,y)/x]2+[Ap(x,y)/y]2dxdy.
L2=k=2Ml=2N[Ap(k,l)Ap(k1,l)]2+[Ap(k,l)Ap(k,l1)]2.

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