Abstract

We report on a noniterative method for coherent diffractive imaging based on a specially designed multi pinhole (MP) plate. We demonstrated that the complex amplitude of the wavefront penetrating through an MP plate inserted between the specimen and the detector plane can be directly extracted from the Fourier transform of a single far-field diffraction intensity pattern without need of any iterative algorithm. A form of scanning diffractive imaging based on a rotatable MP plate is also demonstrated, which provides a feasible approach for lensless diffractive imaging of complex-valued objects.

© 2009 Optical Society of America

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  1. J. Miao, P. Charalambous, J. Kirz, and D. Sayre, Nature 400, 342 (1999).
    [CrossRef]
  2. J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, Phys. Rev. Lett. 89, 088303 (2002).
    [CrossRef] [PubMed]
  3. J. Miao, T. Ohsuna, O. Terasaki, K. O. Hodgson, and M. A. O. Keefe, Phys. Rev. Lett. 89, 155502 (2002).
    [CrossRef] [PubMed]
  4. G. J. Williams, M. A. Pfeifer, I. A. Vartanyants, and I. K. Robinson, Phys. Rev. Lett. 90, 175501 (2003).
    [CrossRef] [PubMed]
  5. J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, Science 300, 1419 (2003).
    [CrossRef] [PubMed]
  6. S. Eisebitt, J. Luning, W. F. Schlotter, M. Lorgen, O. Hellwig, W. Eberhardt, and J. Stohr, Nature 432, 885 (2004).
    [CrossRef] [PubMed]
  7. D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, Proc. Natl. Acad. Sci. USA 102, 15343 (2005).
    [CrossRef] [PubMed]
  8. R. W. Gerchberg and W. O. Saxton, Optik (Jena) 35, 237 (1972).
  9. J. R. Fienup, Appl. Opt. 21, 2758 (1982).
    [CrossRef] [PubMed]
  10. V. Elser, J. Opt. Soc. Am. A 20, 40 (2003).
    [CrossRef]
  11. J. Miao, D. Sayre, and H. N. Chapman, J. Opt. Soc. Am. A 15, 1662 (1998).
    [CrossRef]
  12. J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
    [CrossRef] [PubMed]
  13. P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, Science 321, 379 (2008).
    [CrossRef] [PubMed]
  14. I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, Phys. Rev. Lett. 100, 155503 (2008).
    [CrossRef] [PubMed]

2008 (2)

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, Science 321, 379 (2008).
[CrossRef] [PubMed]

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, Phys. Rev. Lett. 100, 155503 (2008).
[CrossRef] [PubMed]

2007 (1)

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

2005 (1)

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, Proc. Natl. Acad. Sci. USA 102, 15343 (2005).
[CrossRef] [PubMed]

2004 (1)

S. Eisebitt, J. Luning, W. F. Schlotter, M. Lorgen, O. Hellwig, W. Eberhardt, and J. Stohr, Nature 432, 885 (2004).
[CrossRef] [PubMed]

2003 (3)

G. J. Williams, M. A. Pfeifer, I. A. Vartanyants, and I. K. Robinson, Phys. Rev. Lett. 90, 175501 (2003).
[CrossRef] [PubMed]

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, Science 300, 1419 (2003).
[CrossRef] [PubMed]

V. Elser, J. Opt. Soc. Am. A 20, 40 (2003).
[CrossRef]

2002 (2)

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

J. Miao, T. Ohsuna, O. Terasaki, K. O. Hodgson, and M. A. O. Keefe, Phys. Rev. Lett. 89, 155502 (2002).
[CrossRef] [PubMed]

1999 (1)

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, Nature 400, 342 (1999).
[CrossRef]

1998 (1)

1982 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, Optik (Jena) 35, 237 (1972).

Anderson, E. H.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

Beetz, T.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, Proc. Natl. Acad. Sci. USA 102, 15343 (2005).
[CrossRef] [PubMed]

Bunk, O.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, Science 321, 379 (2008).
[CrossRef] [PubMed]

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, Phys. Rev. Lett. 100, 155503 (2008).
[CrossRef] [PubMed]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Chapman, H. N.

Charalambous, P.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, Nature 400, 342 (1999).
[CrossRef]

Cullis, A. G.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

David, C.

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, Phys. Rev. Lett. 100, 155503 (2008).
[CrossRef] [PubMed]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, Science 321, 379 (2008).
[CrossRef] [PubMed]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Dierolf, M.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, Science 321, 379 (2008).
[CrossRef] [PubMed]

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, Phys. Rev. Lett. 100, 155503 (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, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Eberhardt, W.

S. Eisebitt, J. Luning, W. F. Schlotter, M. Lorgen, O. Hellwig, W. Eberhardt, and J. Stohr, Nature 432, 885 (2004).
[CrossRef] [PubMed]

Eisebitt, S.

S. Eisebitt, J. Luning, W. F. Schlotter, M. Lorgen, O. Hellwig, W. Eberhardt, and J. Stohr, Nature 432, 885 (2004).
[CrossRef] [PubMed]

Elser, V.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, Proc. Natl. Acad. Sci. USA 102, 15343 (2005).
[CrossRef] [PubMed]

V. Elser, J. Opt. Soc. Am. A 20, 40 (2003).
[CrossRef]

Fienup, J. R.

Gao, M.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, Science 300, 1419 (2003).
[CrossRef] [PubMed]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, Optik (Jena) 35, 237 (1972).

Gray, J.

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, Phys. Rev. Lett. 100, 155503 (2008).
[CrossRef] [PubMed]

Hellwig, O.

S. Eisebitt, J. Luning, W. F. Schlotter, M. Lorgen, O. Hellwig, W. Eberhardt, and J. Stohr, Nature 432, 885 (2004).
[CrossRef] [PubMed]

Hodgson, K. O.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

J. Miao, T. Ohsuna, O. Terasaki, K. O. Hodgson, and M. A. O. Keefe, Phys. Rev. Lett. 89, 155502 (2002).
[CrossRef] [PubMed]

Howells, M.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, Proc. Natl. Acad. Sci. USA 102, 15343 (2005).
[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, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Ishikawa, T.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

Jacobsen, C.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, Proc. Natl. Acad. Sci. USA 102, 15343 (2005).
[CrossRef] [PubMed]

Jefimovs, K.

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, Phys. Rev. Lett. 100, 155503 (2008).
[CrossRef] [PubMed]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Johnson, B.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

Johnson, I.

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, Phys. Rev. Lett. 100, 155503 (2008).
[CrossRef] [PubMed]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Keefe, M. A. O.

J. Miao, T. Ohsuna, O. Terasaki, K. O. Hodgson, and M. A. O. Keefe, Phys. Rev. Lett. 89, 155502 (2002).
[CrossRef] [PubMed]

Kirz, J.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, Proc. Natl. Acad. Sci. USA 102, 15343 (2005).
[CrossRef] [PubMed]

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, Nature 400, 342 (1999).
[CrossRef]

Lai, B.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

Lima, E.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, Proc. Natl. Acad. Sci. USA 102, 15343 (2005).
[CrossRef] [PubMed]

Lorgen, M.

S. Eisebitt, J. Luning, W. F. Schlotter, M. Lorgen, O. Hellwig, W. Eberhardt, and J. Stohr, Nature 432, 885 (2004).
[CrossRef] [PubMed]

Luning, J.

S. Eisebitt, J. Luning, W. F. Schlotter, M. Lorgen, O. Hellwig, W. Eberhardt, and J. Stohr, Nature 432, 885 (2004).
[CrossRef] [PubMed]

Menzel, A.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, Science 321, 379 (2008).
[CrossRef] [PubMed]

Miao, H.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, Proc. Natl. Acad. Sci. USA 102, 15343 (2005).
[CrossRef] [PubMed]

Miao, J.

J. Miao, T. Ohsuna, O. Terasaki, K. O. Hodgson, and M. A. O. Keefe, Phys. Rev. Lett. 89, 155502 (2002).
[CrossRef] [PubMed]

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, Nature 400, 342 (1999).
[CrossRef]

J. Miao, D. Sayre, and H. N. Chapman, J. Opt. Soc. Am. A 15, 1662 (1998).
[CrossRef]

Nagahara, L. A.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, Science 300, 1419 (2003).
[CrossRef] [PubMed]

Neiman, A. M.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, Proc. Natl. Acad. Sci. USA 102, 15343 (2005).
[CrossRef] [PubMed]

Ohsuna, T.

J. Miao, T. Ohsuna, O. Terasaki, K. O. Hodgson, and M. A. O. Keefe, Phys. Rev. Lett. 89, 155502 (2002).
[CrossRef] [PubMed]

Pfeifer, M. A.

G. J. Williams, M. A. Pfeifer, I. A. Vartanyants, and I. K. Robinson, Phys. Rev. Lett. 90, 175501 (2003).
[CrossRef] [PubMed]

Pfeiffer, F.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, Science 321, 379 (2008).
[CrossRef] [PubMed]

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, Phys. Rev. Lett. 100, 155503 (2008).
[CrossRef] [PubMed]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Renker, D.

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, Phys. Rev. Lett. 100, 155503 (2008).
[CrossRef] [PubMed]

Robinson, I. K.

G. J. Williams, M. A. Pfeifer, I. A. Vartanyants, and I. K. Robinson, Phys. Rev. Lett. 90, 175501 (2003).
[CrossRef] [PubMed]

Rodenburg, J. M.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, Optik (Jena) 35, 237 (1972).

Sayre, D.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, Proc. Natl. Acad. Sci. USA 102, 15343 (2005).
[CrossRef] [PubMed]

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, Nature 400, 342 (1999).
[CrossRef]

J. Miao, D. Sayre, and H. N. Chapman, J. Opt. Soc. Am. A 15, 1662 (1998).
[CrossRef]

Schlotter, W. F.

S. Eisebitt, J. Luning, W. F. Schlotter, M. Lorgen, O. Hellwig, W. Eberhardt, and J. Stohr, Nature 432, 885 (2004).
[CrossRef] [PubMed]

Shapiro, D.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, Proc. Natl. Acad. Sci. USA 102, 15343 (2005).
[CrossRef] [PubMed]

Stohr, J.

S. Eisebitt, J. Luning, W. F. Schlotter, M. Lorgen, O. Hellwig, W. Eberhardt, and J. Stohr, Nature 432, 885 (2004).
[CrossRef] [PubMed]

Terasaki, O.

J. Miao, T. Ohsuna, O. Terasaki, K. O. Hodgson, and M. A. O. Keefe, Phys. Rev. Lett. 89, 155502 (2002).
[CrossRef] [PubMed]

Thibault, P.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, Science 321, 379 (2008).
[CrossRef] [PubMed]

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, Proc. Natl. Acad. Sci. USA 102, 15343 (2005).
[CrossRef] [PubMed]

Vartanyants, I.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, Science 300, 1419 (2003).
[CrossRef] [PubMed]

Vartanyants, I. A.

G. J. Williams, M. A. Pfeifer, I. A. Vartanyants, and I. K. Robinson, Phys. Rev. Lett. 90, 175501 (2003).
[CrossRef] [PubMed]

Williams, G. J.

G. J. Williams, M. A. Pfeifer, I. A. Vartanyants, and I. K. Robinson, Phys. Rev. Lett. 90, 175501 (2003).
[CrossRef] [PubMed]

Zhang, R.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, Science 300, 1419 (2003).
[CrossRef] [PubMed]

Zuo, J. M.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, Science 300, 1419 (2003).
[CrossRef] [PubMed]

Appl. Opt. (1)

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

Nature (2)

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, Nature 400, 342 (1999).
[CrossRef]

S. Eisebitt, J. Luning, W. F. Schlotter, M. Lorgen, O. Hellwig, W. Eberhardt, and J. Stohr, Nature 432, 885 (2004).
[CrossRef] [PubMed]

Optik (Jena) (1)

R. W. Gerchberg and W. O. Saxton, Optik (Jena) 35, 237 (1972).

Phys. Rev. Lett. (5)

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

J. Miao, T. Ohsuna, O. Terasaki, K. O. Hodgson, and M. A. O. Keefe, Phys. Rev. Lett. 89, 155502 (2002).
[CrossRef] [PubMed]

G. J. Williams, M. A. Pfeifer, I. A. Vartanyants, and I. K. Robinson, Phys. Rev. Lett. 90, 175501 (2003).
[CrossRef] [PubMed]

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, Phys. Rev. Lett. 100, 155503 (2008).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. USA (1)

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, Proc. Natl. Acad. Sci. USA 102, 15343 (2005).
[CrossRef] [PubMed]

Science (2)

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, Science 321, 379 (2008).
[CrossRef] [PubMed]

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, Science 300, 1419 (2003).
[CrossRef] [PubMed]

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Figures (3)

Fig. 1
Fig. 1

Schematic of the experimental setup for diffractive imaging based on an MP plate.

Fig. 2
Fig. 2

(a) Example of the specially designed MP plate used in the MP-SDI. (b) Example of the Fourier transform of the far-field diffraction intensity pattern of the wave penetrating through the MP plate as shown in (a).

Fig. 3
Fig. 3

(a) One example of the far-field diffraction intensity patterns recorded in experiments. (b) Measured amplitude distribution of the wavefront on the MP plate. (c) Measured phase distribution of the wavefront on the MP plate. (d) Image reconstructed by the measured complex amplitude shown in (b) and (c) after backpropagating to the specimen plane.

Equations (3)

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

I P ( ρ ) = I 0 m = 0 M A m exp ( j ϕ m ) F { circ [ ( r r m ) R 0 ] } 2 ,
g ( r ) = F 1 { I P ( ρ ) } = m = 0 M n = 0 M P m n ( r ) A m A n exp [ j ( ϕ m ϕ n ) ] ,
g ( r m ) = P 0 A 0 A m exp [ i ( ϕ m ϕ 0 ) ] ( m = 1 , 2 , 3 N ) ,

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