Abstract

A new computational imaging technique, termed Fourier ptychographic microscopy (FPM), uses a sequence of low-resolution images captured under varied illumination to iteratively converge upon a high-resolution complex sample estimate. Here, we propose a mathematical model of FPM that explicitly connects its operation to conventional ptychography, a common procedure applied to electron and X-ray diffractive imaging. Our mathematical framework demonstrates that under ideal illumination conditions, conventional ptychography and FPM both produce datasets that are mathematically linked by a linear transformation. We hope this finding encourages the future cross-pollination of ideas between two otherwise unconnected experimental imaging procedures. In addition, the coherence state of the illumination source used by each imaging platform is critical to successful operation, yet currently not well understood. We apply our mathematical framework to demonstrate that partial coherence uniquely alters both conventional ptychography’s and FPM’s captured data, but up to a certain threshold can still lead to accurate resolution-enhanced imaging through appropriate computational post-processing. We verify this theoretical finding through simulation and experiment.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. D. Nellist, B. C. McCallum, J. M. Rodenburg, “Resolution beyond the ‘infromation limit’ in transmission electron microscopy,” Nature 374, 630–632 (1995).
    [CrossRef]
  2. F. Hue, J. M. Rodenburg, A. M. Maiden, F. Sweeney, P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B 82, 121415 (2010).
    [CrossRef]
  3. J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, I. Johnson, “Hard-X-ray lensless imaging of extended objects,” PRL 98, 034801 (2007).
    [CrossRef]
  4. P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, F. Pheiffer, “High-resolution scanning X-ray diffraction microscopy,” Science 321, 379–382 (2008).
    [CrossRef] [PubMed]
  5. M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, F. Pheiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature 467, 437–439 (2010).
    [CrossRef]
  6. A. M. Maiden, J. M. Rodenburg, M. J. Humphry, “Optical ptychography: a practical implementation with useful resolution,” Opt. Lett. 35(15), 2585–2587 (2010).
    [CrossRef] [PubMed]
  7. A. M. Maiden, M. J. Humphry, F. Zhang, J. M. Rodenburg, “Superresolution imaging via ptychography,” J. Opt. Soc. Am. A. 28(4), 604–612 (2011).
    [CrossRef]
  8. G. Zheng, R. Horstmeyer, C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nature Photon. 7, 739–745 (2013).
    [CrossRef]
  9. J. M. Rodenburg, R. H. T. Bates, “The theory of super-resolution electron microscopy via Wigner-distribution deconvolution,” Phil. Trans. R. Soc. Lond. A 339, 521–553 (1992).
    [CrossRef]
  10. H. N. Chapman, “Phase retrieval x-ray microscopy by Wigner distribution deconvolution,” Ultramicroscopy 66, 153 (1996).
    [CrossRef]
  11. J. N. Clark, X. Huang, R. Harder, I. K. Robinsion, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3, 993 (2012).
    [CrossRef] [PubMed]
  12. P. Thibault, A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013).
    [CrossRef] [PubMed]
  13. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  14. K. Nugent, “Coherent methods in the X-ray sciences,” Adv. Phys. 59(1), 1–99 (2010).
    [CrossRef]
  15. M. Testorf, B. M. Hennelly, J. Ojeda-Castaneda, Phase-Space Optics: Fundamentals and Applications (McGraw-Hill, 2010).
  16. M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” JOSA A 3(8), 1227–1238 (1986).
    [CrossRef]
  17. R. Horstmeyer, S. B. Oh, R. Raskar, “Iterative aperture mask design in phase space using a rank constraint,” Opt. Express 18(21), 22545–22555 (2010).
    [CrossRef] [PubMed]
  18. H. M. L. Faulkner, J. M. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
    [CrossRef] [PubMed]
  19. A. M. Maiden, J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1562 (2009).
    [CrossRef] [PubMed]
  20. A. M. Maiden, M. J. Humphry, M. C. Sarahan, B. Kraus, J. M. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy 120, 64–72 (2012).
    [CrossRef] [PubMed]
  21. M. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
    [CrossRef]
  22. C. Teale, D. Adams, M. Murnane, H. Kapteyn, D. J. Kane, “Imaging by integrating stitched spectrograms,” Opt. Express 21(6), 6783–6793 (2012).
    [CrossRef]
  23. G. Zheng, X. Ou, R. Horstmeyer, C. Yang, “Characterization of spatially varying aberrations for wide field-of-view microscopy,” Opt. Express 21(13), 15131–15143 (2013).
    [CrossRef] [PubMed]
  24. D. Brady, Optical Imaging and Spectroscopy (John Wiley & Sons, 2009).
    [CrossRef]
  25. R. G. Brown, P. Y. C. Hwang, Introduction to Random Signals and Applied Kalman Filtering (John Wiley & Sons, 1996).
  26. X. Ou, R. Horstmeyer, G. Zheng, C. Yang, “Quantitative phase imaging via Fourier ptychographic microscopy,” Opt. Lett. 38(2), 4845–4848 (2013).
    [CrossRef] [PubMed]

2013

2012

C. Teale, D. Adams, M. Murnane, H. Kapteyn, D. J. Kane, “Imaging by integrating stitched spectrograms,” Opt. Express 21(6), 6783–6793 (2012).
[CrossRef]

J. N. Clark, X. Huang, R. Harder, I. K. Robinsion, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3, 993 (2012).
[CrossRef] [PubMed]

A. M. Maiden, M. J. Humphry, M. C. Sarahan, B. Kraus, J. M. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy 120, 64–72 (2012).
[CrossRef] [PubMed]

2011

A. M. Maiden, M. J. Humphry, F. Zhang, J. M. Rodenburg, “Superresolution imaging via ptychography,” J. Opt. Soc. Am. A. 28(4), 604–612 (2011).
[CrossRef]

2010

K. Nugent, “Coherent methods in the X-ray sciences,” Adv. Phys. 59(1), 1–99 (2010).
[CrossRef]

F. Hue, J. M. Rodenburg, A. M. Maiden, F. Sweeney, P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B 82, 121415 (2010).
[CrossRef]

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, F. Pheiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature 467, 437–439 (2010).
[CrossRef]

A. M. Maiden, J. M. Rodenburg, M. J. Humphry, “Optical ptychography: a practical implementation with useful resolution,” Opt. Lett. 35(15), 2585–2587 (2010).
[CrossRef] [PubMed]

R. Horstmeyer, S. B. Oh, R. Raskar, “Iterative aperture mask design in phase space using a rank constraint,” Opt. Express 18(21), 22545–22555 (2010).
[CrossRef] [PubMed]

2009

A. M. Maiden, J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1562 (2009).
[CrossRef] [PubMed]

2008

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, F. Pheiffer, “High-resolution scanning X-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

M. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[CrossRef]

2007

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, I. Johnson, “Hard-X-ray lensless imaging of extended objects,” PRL 98, 034801 (2007).
[CrossRef]

2004

H. M. L. Faulkner, J. M. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef] [PubMed]

1996

H. N. Chapman, “Phase retrieval x-ray microscopy by Wigner distribution deconvolution,” Ultramicroscopy 66, 153 (1996).
[CrossRef]

1995

P. D. Nellist, B. C. McCallum, J. M. Rodenburg, “Resolution beyond the ‘infromation limit’ in transmission electron microscopy,” Nature 374, 630–632 (1995).
[CrossRef]

1992

J. M. Rodenburg, R. H. T. Bates, “The theory of super-resolution electron microscopy via Wigner-distribution deconvolution,” Phil. Trans. R. Soc. Lond. A 339, 521–553 (1992).
[CrossRef]

1986

M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” JOSA A 3(8), 1227–1238 (1986).
[CrossRef]

Adams, D.

Bastiaans, M. J.

M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” JOSA A 3(8), 1227–1238 (1986).
[CrossRef]

Bates, R. H. T.

J. M. Rodenburg, R. H. T. Bates, “The theory of super-resolution electron microscopy via Wigner-distribution deconvolution,” Phil. Trans. R. Soc. Lond. A 339, 521–553 (1992).
[CrossRef]

Brady, D.

D. Brady, Optical Imaging and Spectroscopy (John Wiley & Sons, 2009).
[CrossRef]

Brown, R. G.

R. G. Brown, P. Y. C. Hwang, Introduction to Random Signals and Applied Kalman Filtering (John Wiley & Sons, 1996).

Bunk, M.

M. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[CrossRef]

Bunk, O.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, F. Pheiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature 467, 437–439 (2010).
[CrossRef]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, F. Pheiffer, “High-resolution scanning X-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, I. Johnson, “Hard-X-ray lensless imaging of extended objects,” PRL 98, 034801 (2007).
[CrossRef]

Chapman, H. N.

H. N. Chapman, “Phase retrieval x-ray microscopy by Wigner distribution deconvolution,” Ultramicroscopy 66, 153 (1996).
[CrossRef]

Clark, J. N.

J. N. Clark, X. Huang, R. Harder, I. K. Robinsion, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3, 993 (2012).
[CrossRef] [PubMed]

Cullis, A. G.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, I. Johnson, “Hard-X-ray lensless imaging of extended objects,” PRL 98, 034801 (2007).
[CrossRef]

David, C.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, F. Pheiffer, “High-resolution scanning X-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, I. Johnson, “Hard-X-ray lensless imaging of extended objects,” PRL 98, 034801 (2007).
[CrossRef]

Dierolf, M.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, F. Pheiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature 467, 437–439 (2010).
[CrossRef]

M. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[CrossRef]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, F. Pheiffer, “High-resolution scanning X-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

Dobson, B. R.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, I. Johnson, “Hard-X-ray lensless imaging of extended objects,” PRL 98, 034801 (2007).
[CrossRef]

Faulkner, H. M. L.

H. M. L. Faulkner, J. M. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef] [PubMed]

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Harder, R.

J. N. Clark, X. Huang, R. Harder, I. K. Robinsion, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3, 993 (2012).
[CrossRef] [PubMed]

Hennelly, B. M.

M. Testorf, B. M. Hennelly, J. Ojeda-Castaneda, Phase-Space Optics: Fundamentals and Applications (McGraw-Hill, 2010).

Horstmeyer, R.

Huang, X.

J. N. Clark, X. Huang, R. Harder, I. K. Robinsion, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3, 993 (2012).
[CrossRef] [PubMed]

Hue, F.

F. Hue, J. M. Rodenburg, A. M. Maiden, F. Sweeney, P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B 82, 121415 (2010).
[CrossRef]

Humphry, M. J.

A. M. Maiden, M. J. Humphry, M. C. Sarahan, B. Kraus, J. M. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy 120, 64–72 (2012).
[CrossRef] [PubMed]

A. M. Maiden, M. J. Humphry, F. Zhang, J. M. Rodenburg, “Superresolution imaging via ptychography,” J. Opt. Soc. Am. A. 28(4), 604–612 (2011).
[CrossRef]

A. M. Maiden, J. M. Rodenburg, M. J. Humphry, “Optical ptychography: a practical implementation with useful resolution,” Opt. Lett. 35(15), 2585–2587 (2010).
[CrossRef] [PubMed]

Hurst, A. C.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, I. Johnson, “Hard-X-ray lensless imaging of extended objects,” PRL 98, 034801 (2007).
[CrossRef]

Hwang, P. Y. C.

R. G. Brown, P. Y. C. Hwang, Introduction to Random Signals and Applied Kalman Filtering (John Wiley & Sons, 1996).

Jefimovs, K.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, I. Johnson, “Hard-X-ray lensless imaging of extended objects,” PRL 98, 034801 (2007).
[CrossRef]

Johnson, I.

M. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[CrossRef]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, I. Johnson, “Hard-X-ray lensless imaging of extended objects,” PRL 98, 034801 (2007).
[CrossRef]

Kane, D. J.

Kapteyn, H.

Kewish, C. M.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, F. Pheiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature 467, 437–439 (2010).
[CrossRef]

Kraus, B.

A. M. Maiden, M. J. Humphry, M. C. Sarahan, B. Kraus, J. M. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy 120, 64–72 (2012).
[CrossRef] [PubMed]

Kynde, S.

M. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[CrossRef]

Maiden, A. M.

A. M. Maiden, M. J. Humphry, M. C. Sarahan, B. Kraus, J. M. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy 120, 64–72 (2012).
[CrossRef] [PubMed]

A. M. Maiden, M. J. Humphry, F. Zhang, J. M. Rodenburg, “Superresolution imaging via ptychography,” J. Opt. Soc. Am. A. 28(4), 604–612 (2011).
[CrossRef]

F. Hue, J. M. Rodenburg, A. M. Maiden, F. Sweeney, P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B 82, 121415 (2010).
[CrossRef]

A. M. Maiden, J. M. Rodenburg, M. J. Humphry, “Optical ptychography: a practical implementation with useful resolution,” Opt. Lett. 35(15), 2585–2587 (2010).
[CrossRef] [PubMed]

A. M. Maiden, J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1562 (2009).
[CrossRef] [PubMed]

Marti, O.

M. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[CrossRef]

McCallum, B. C.

P. D. Nellist, B. C. McCallum, J. M. Rodenburg, “Resolution beyond the ‘infromation limit’ in transmission electron microscopy,” Nature 374, 630–632 (1995).
[CrossRef]

Menzel, A.

P. Thibault, A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013).
[CrossRef] [PubMed]

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, F. Pheiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature 467, 437–439 (2010).
[CrossRef]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, F. Pheiffer, “High-resolution scanning X-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

Midgley, P. A.

F. Hue, J. M. Rodenburg, A. M. Maiden, F. Sweeney, P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B 82, 121415 (2010).
[CrossRef]

Murnane, M.

Nellist, P. D.

P. D. Nellist, B. C. McCallum, J. M. Rodenburg, “Resolution beyond the ‘infromation limit’ in transmission electron microscopy,” Nature 374, 630–632 (1995).
[CrossRef]

Nugent, K.

K. Nugent, “Coherent methods in the X-ray sciences,” Adv. Phys. 59(1), 1–99 (2010).
[CrossRef]

Oh, S. B.

Ojeda-Castaneda, J.

M. Testorf, B. M. Hennelly, J. Ojeda-Castaneda, Phase-Space Optics: Fundamentals and Applications (McGraw-Hill, 2010).

Ou, X.

Pfeiffer, F.

M. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[CrossRef]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, I. Johnson, “Hard-X-ray lensless imaging of extended objects,” PRL 98, 034801 (2007).
[CrossRef]

Pheiffer, F.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, F. Pheiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature 467, 437–439 (2010).
[CrossRef]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, F. Pheiffer, “High-resolution scanning X-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

Raskar, R.

Robinsion, I. K.

J. N. Clark, X. Huang, R. Harder, I. K. Robinsion, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3, 993 (2012).
[CrossRef] [PubMed]

Rodenburg, J. M.

A. M. Maiden, M. J. Humphry, M. C. Sarahan, B. Kraus, J. M. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy 120, 64–72 (2012).
[CrossRef] [PubMed]

A. M. Maiden, M. J. Humphry, F. Zhang, J. M. Rodenburg, “Superresolution imaging via ptychography,” J. Opt. Soc. Am. A. 28(4), 604–612 (2011).
[CrossRef]

A. M. Maiden, J. M. Rodenburg, M. J. Humphry, “Optical ptychography: a practical implementation with useful resolution,” Opt. Lett. 35(15), 2585–2587 (2010).
[CrossRef] [PubMed]

F. Hue, J. M. Rodenburg, A. M. Maiden, F. Sweeney, P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B 82, 121415 (2010).
[CrossRef]

A. M. Maiden, J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1562 (2009).
[CrossRef] [PubMed]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, I. Johnson, “Hard-X-ray lensless imaging of extended objects,” PRL 98, 034801 (2007).
[CrossRef]

H. M. L. Faulkner, J. M. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef] [PubMed]

P. D. Nellist, B. C. McCallum, J. M. Rodenburg, “Resolution beyond the ‘infromation limit’ in transmission electron microscopy,” Nature 374, 630–632 (1995).
[CrossRef]

J. M. Rodenburg, R. H. T. Bates, “The theory of super-resolution electron microscopy via Wigner-distribution deconvolution,” Phil. Trans. R. Soc. Lond. A 339, 521–553 (1992).
[CrossRef]

Sarahan, M. C.

A. M. Maiden, M. J. Humphry, M. C. Sarahan, B. Kraus, J. M. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy 120, 64–72 (2012).
[CrossRef] [PubMed]

Schneider, P.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, F. Pheiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature 467, 437–439 (2010).
[CrossRef]

Sweeney, F.

F. Hue, J. M. Rodenburg, A. M. Maiden, F. Sweeney, P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B 82, 121415 (2010).
[CrossRef]

Teale, C.

Testorf, M.

M. Testorf, B. M. Hennelly, J. Ojeda-Castaneda, Phase-Space Optics: Fundamentals and Applications (McGraw-Hill, 2010).

Thibault, P.

P. Thibault, A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013).
[CrossRef] [PubMed]

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, F. Pheiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature 467, 437–439 (2010).
[CrossRef]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, F. Pheiffer, “High-resolution scanning X-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

Wepf, R.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, F. Pheiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature 467, 437–439 (2010).
[CrossRef]

Yang, C.

Zhang, F.

A. M. Maiden, M. J. Humphry, F. Zhang, J. M. Rodenburg, “Superresolution imaging via ptychography,” J. Opt. Soc. Am. A. 28(4), 604–612 (2011).
[CrossRef]

Zheng, G.

Adv. Phys.

K. Nugent, “Coherent methods in the X-ray sciences,” Adv. Phys. 59(1), 1–99 (2010).
[CrossRef]

J. Opt. Soc. Am. A.

A. M. Maiden, M. J. Humphry, F. Zhang, J. M. Rodenburg, “Superresolution imaging via ptychography,” J. Opt. Soc. Am. A. 28(4), 604–612 (2011).
[CrossRef]

JOSA A

M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” JOSA A 3(8), 1227–1238 (1986).
[CrossRef]

Nat. Commun.

J. N. Clark, X. Huang, R. Harder, I. K. Robinsion, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3, 993 (2012).
[CrossRef] [PubMed]

Nature

P. Thibault, A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013).
[CrossRef] [PubMed]

P. D. Nellist, B. C. McCallum, J. M. Rodenburg, “Resolution beyond the ‘infromation limit’ in transmission electron microscopy,” Nature 374, 630–632 (1995).
[CrossRef]

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, F. Pheiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature 467, 437–439 (2010).
[CrossRef]

Nature Photon.

G. Zheng, R. Horstmeyer, C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nature Photon. 7, 739–745 (2013).
[CrossRef]

Opt. Express

Opt. Lett.

Phil. Trans. R. Soc. Lond. A

J. M. Rodenburg, R. H. T. Bates, “The theory of super-resolution electron microscopy via Wigner-distribution deconvolution,” Phil. Trans. R. Soc. Lond. A 339, 521–553 (1992).
[CrossRef]

Phys. Rev. B

F. Hue, J. M. Rodenburg, A. M. Maiden, F. Sweeney, P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B 82, 121415 (2010).
[CrossRef]

Phys. Rev. Lett.

H. M. L. Faulkner, J. M. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef] [PubMed]

PRL

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, I. Johnson, “Hard-X-ray lensless imaging of extended objects,” PRL 98, 034801 (2007).
[CrossRef]

Science

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, F. Pheiffer, “High-resolution scanning X-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

Ultramicroscopy

A. M. Maiden, J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1562 (2009).
[CrossRef] [PubMed]

A. M. Maiden, M. J. Humphry, M. C. Sarahan, B. Kraus, J. M. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy 120, 64–72 (2012).
[CrossRef] [PubMed]

M. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[CrossRef]

H. N. Chapman, “Phase retrieval x-ray microscopy by Wigner distribution deconvolution,” Ultramicroscopy 66, 153 (1996).
[CrossRef]

Other

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

M. Testorf, B. M. Hennelly, J. Ojeda-Castaneda, Phase-Space Optics: Fundamentals and Applications (McGraw-Hill, 2010).

D. Brady, Optical Imaging and Spectroscopy (John Wiley & Sons, 2009).
[CrossRef]

R. G. Brown, P. Y. C. Hwang, Introduction to Random Signals and Applied Kalman Filtering (John Wiley & Sons, 1996).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Conventional ptychography’s optical setup. A sample ψ (in green) is shifted through many positions as the intensity of the probe light it diffracts is recorded at a far-field detector. In a typical visible light setup, the lens at A(r′) is a multi-element system containing the aperture stop a(r′) at some intermediate plane, as diagrammed.

Fig. 2
Fig. 2

Conventional ptychography (CP) data acquisition. A chirped amplitude grating (400 μm wide, 4μm minimum pitch) serves as our sample ψ(r). It is shifted and illuminated by a probe function ã(r), here a sinc function from a rectangular-shaped focusing element. At detector plane D, the diffracted light’s intensity is recorded. (Bottom) Corresponding probe and sample Wigner functions, whose two-dimensional convolution creates CP’s data matrix m(x, r′). Note specific parameters used for this simulation are listed in Section 3.

Fig. 3
Fig. 3

Fourier ptychographic microscopy’s (FPM) optical setup. An LED array replaces CP’s single illumination source in Fig. 1, and planes S(r′) and A(r) have switched places along the optical axis. Each LED sequentially illuminates the sample from a different angle.

Fig. 4
Fig. 4

FPM data acquisition diagram. (Top) The same grating sample ψ(r) used in Fig. 2 is sequentially illuminated by tilted plane waves, adding a different linear phase ∝ x to each image (tilted green line). At plane A(r), the aperture a(r) limits the extent of the field before the sample is imaged to detector plane D(r′) at low resolution. (Bottom) Corresponding WDF’s and their convolution, representing FPM’s data matrix. Color maps here follow those included in Fig. 2.

Fig. 5
Fig. 5

The experimental factors influencing CP and FPM data matrices. (top) Geometrical factors define the data matrix scaling and sampling, while (bottom) parameters specific to the focusing/imaging lens define data matrix blurring for both setups.

Fig. 6
Fig. 6

Partially coherent light manifests itself as an additional convolution along the data matrix scan dimension x for both (a) CP and (b) FPM. The convolution is one-dimensional, as indicated by the vertical bar. With matrices rotated by 90° with respect to one another, this convolution will mix the data from each respective setup in a unique manner. For this simulation, we used the same setup parameters as for Fig. 2 and Fig. 4, but assumed each illumination source C(x) (i.e., LED) is a rectangle 200 μm in diameter.

Fig. 7
Fig. 7

Simulation of partially coherent effects produce blurred (a) CP and (b) FPM data matrices of an example grating. A Wiener filter can approximately recover the coherent data matrix for each setup, from which an accurate sample reconstruction is direct. (c) Reconstruction error as a function of LED diameter (i.e., blur kernel width) increases for both CP and FPM, although FPM’s error is consistently lower. (d) The chirped grating sample and its coherent CP data matrix, for comparison.

Fig. 8
Fig. 8

(a) Simulated and (b) experimental FPM data matrices with varying degrees of partially coherent illumination. The experimental sample closely matches the distribution of ψ(r) in Fig. 7(d). C at top indicates the LED active area diameter used in each experiment.

Equations (47)

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

S + ( r ) = exp ( j k 2 f r 2 ) j λ f a ( r ) exp ( j k f r r ) d r [ a ( r ) ] = a ˜ ( r ) ,
S ( r ) = a ˜ ( r ) ψ ( r x ) .
m ( x , r ) = | [ a ˜ ( r ) ψ ( r x ) ] | 2 .
m ( x , r ) = a ˜ ( r 1 ) a ˜ * ( r 2 ) ψ ( r 1 x ) ψ * ( r 2 x ) exp [ j k r ( r 1 r 2 ) ] d r 1 d r 2 ,
m ( x , r ) = W ψ ( r x , u ) W a ˜ ( r , r u ) d r d u ,
W ψ ( r , u ) = ψ ( r + y 2 ) ψ * ( r y 2 ) exp ( j k y u ) d y
S ( r ) = ψ ( r ) e j k x r .
[ S ( r ) ] a ( r ) = S ˜ ( r ) a ( r ) = ψ ˜ ( r x ) a ( r ) .
m F ( x , r ) = | [ ψ ˜ ( r x ) a ( r ) ] | 2
m F ( x , r ) = ψ ˜ ( r 1 x ) ψ ˜ * ( r 2 x ) a ( r 1 ) a * ( r 2 ) exp ( j k r ( r 1 r 2 ) ) d r 1 d r 2 ,
m F ( x , r ) = W ψ ( u x , r ) W a ˜ ( u , r r ) d u d r .
m F ( x , r ) = m ( r , x ) .
Γ I ( r 1 , r 2 , ω ) = γ 2 C ( r 1 , ω ) δ ( r 1 r 2 ) ,
Γ z ( Δ r ) = e j k q 2 z C ( r ) e j k z ( r Δ r ) d r C ˜ ( r ) ,
Γ S a ˜ ( r 1 , r 2 ) = C ( p ) a ˜ ( r 1 p ) a ˜ * ( r 2 p ) d p ,
m ( x , r ) = Γ S a ˜ ( r 1 , r 2 ) ψ ( r 1 x ) ψ * ( r 2 x ) exp [ j k r ( r 1 r 2 ) ] d r 1 d r 2 .
m ( x , r ) = C ( p ) W ψ ( r x , u ) W a ˜ ( r p , r u ) d r d u d p ,
Γ S ( ρ 1 ρ 2 ) = C ( r x ) e j k r ( ρ 1 ρ 2 ) d r = C ˜ ( ρ 1 ρ 2 ) exp ( j k x ( ρ 1 ρ 2 ) ) ,
Γ D ( r 1 , r 2 ) = Γ S ( ρ 1 ρ 2 ) ψ ( ρ 1 ) ψ * ( ρ 2 ) a ˜ ( ρ 1 r 1 ) a ˜ * ( ρ 2 r 2 ) d ρ 1 d ρ 2 .
m F ( x , r ) = C ˜ ( ρ 1 ρ 2 ) ψ ( ρ 1 ) ψ * ( ρ 2 ) a ˜ ( ρ 1 r ) a ˜ * ( ρ 2 r ) exp ( j k x ( ρ 1 ρ 2 ) ) d ρ 1 d ρ 2 .
m F ( x , r ) = C ( p ) W ψ ( p u x , r ) W a ˜ ( u , r r ) d r d u d p .
m ( x , r ) = | r , r / λ d [ a ˜ ( r / λ f ) ψ ( r x ) ] | 2 ,
m ( x , r ) = a ˜ ( r 1 λ f ) a ˜ * ( r 2 λ f ) ψ ( r 1 x ) ψ * ( r 2 x ) exp [ j k d r ( r 1 r 2 ) ] d r 1 d r 2
m ( x , r ) = W ψ λ f ( r x λ f , u ) W a ˜ ( r , λ f d r u ) d r d u .
m F ( x , r ) = ψ ˜ ( r 1 λ d o x λ ) ψ ˜ * ( r 2 λ d o x λ ) a ( r 1 ) a * ( r 2 ) exp ( j k r λ d i ( r 1 r 2 ) ) d r 1 d r 2 .
m F ( x , r ) = W ψ λ d o ( u d o x , r ) W a ˜ ( u , r λ d o r d i ) d u d r .
m F ( x , r ) = ψ ˜ ( r + y 2 x ) ψ ˜ * ( r y 2 x ) a ( r + y 2 ) a * ( r y 2 ) × exp ( j k r y ) d y d r ,
W ψ ˜ ( r , u ) = ψ ˜ ( r + y 2 ) ψ ˜ * ( r y 2 ) exp ( j k y u ) d y .
ψ ˜ ( r + y 2 ) ψ ˜ * ( r y 2 ) = k 2 π W ψ ˜ ( r , u ) exp ( j k y u ) d u
W a ( r , r u ) = a ( r + y 2 ) a * ( r y 2 ) exp ( j k y ( r u ) ) d y
a ( r + y 2 ) a * ( r y 2 ) = k 2 π W a ( r , r u ) exp ( j k y ( r u ) ) d ( r u )
m F ( x , r ) = W ψ ˜ ( r x , u ) W a ( r , r u ) d r d u ,
W ψ ˜ ( r , u ) = W ψ ( u , r ) .
m F ( x , r ) = W ψ ( u x , r ) W a ˜ ( u , r r ) d u d r ,
m ( x , r ) = C ( p ) a ˜ ( r 1 p ) a ˜ * ( r 2 p ) ψ ( r 1 x ) ψ * ( r 2 x ) × exp [ j k r ( r 1 r 2 ) ] d r 1 d r 2 d p .
m ( x , r ) = C ( p ) a ˜ ( r + y 2 p ) a ˜ * ( r y 2 p ) ψ ( r + y 2 x ) ψ * ( r y 2 x ) × exp [ j k r y ] d y d r d p .
m ( x , r ) = k 2 π C ( p ) a ˜ ( r + y 2 p ) a ˜ * ( r y 2 p ) W ψ ( r x , u ) × exp [ j k y ( u r ) ] d y d r d p d u
a ˜ ( r + y 2 p ) a ˜ * ( r y 2 p ) = k 2 π W a ˜ ( r p , r u ) exp ( j k y ( r u ) ) d ( r u )
m ( x , r ) = C ( p ) W a ˜ ( r p , r u ) W ψ ( r x , u ) × exp ( j k y ( u r + r u ) ) d y d r d p d u d ( r u ) .
m ( x , r ) = C ( p ) W a ˜ ( r p , r u ) W ψ ( r x , u ) d r d u d p ,
m F ( r , x ) = C ( p ) ψ ( ρ 1 ) ψ * ( ρ 2 ) a ˜ ( ρ 1 r ) a ˜ * ( ρ 2 r ) × exp ( j k ( x + p ) ( ρ 1 ρ 2 ) ) d ρ 1 d ρ 2 d p .
m F ( r , x ) = C ( t x ) ψ ( ρ 1 ) ψ * ( ρ 2 ) a ˜ ( ρ 1 r ) a ˜ * ( ρ 2 r ) × exp ( j k t ( ρ 1 ρ 2 ) ) d ρ 1 d ρ 2 d t
m F ( r , x ) = C ( t x ) ψ ( r + y 2 ) ψ * ( r y 2 ) a ˜ ( r + y 2 r ) a ˜ * ( r y 2 r ) × exp ( j k t y ) d r d y d t ,
a ˜ ( r + y 2 r ) a ˜ * ( r y 2 r ) = k 2 π W a ˜ ( r r , u ) exp ( j k y u ) d u
ψ ( r + y 2 ) ψ * ( r y 2 ) = k 2 π W ψ ( r , t u ) exp ( j k y ( t u ) ) d ( t u )
m F ( r , x ) = C ( t x ) W ψ ( r , t u ) W a ˜ ( r r , u ) d r d t d u .
m F ( x , r ) = C ( p ) W ψ ( x + p u , r ) W a ˜ ( u , r r ) d u d r d p ,

Metrics