Abstract

We introduce a noniterative image-reconstruction technique for coherent diffractive imaging. Through the application of differential and integral operators, an extended reference can be used to recover the complex-valued transmissivity of an object, in closed form, from a measurement of its far-field (Fraunhofer) diffraction intensity. We demonstrate the feasibility of this approach, using a reference of a pair of crossed wires and slits, through numerical simulations and a soft x-ray coherent diffractive imaging experiment.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. R. Fienup, Appl. Opt. 21, 2758 (1982).
    [CrossRef] [PubMed]
  2. J. Miao, P. Charalambous, J. Kirz, and D. Sayre, Nature 400, 342 (1999).
    [CrossRef]
  3. I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, Science 256, 1009 (1992).
    [CrossRef] [PubMed]
  4. S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, Nature 432, 885 (2004).
    [CrossRef] [PubMed]
  5. M. Guizar-Sicairos and J. R. Fienup, Opt. Express 15, 17592 (2007).
    [CrossRef] [PubMed]
  6. X. Zhao, Y. Liu, S. Inoue, T. Suzuki, R. O. Jones, and Y. Ando, Phys. Rev. Lett. 92, 125502 (2004).
    [CrossRef] [PubMed]
  7. Ancient Chinese character “elephant,” which later inherited the meaning of “image” from their phonetic connection, http://en.wikipedia.org/wiki/Oracle_bone_script.
  8. R. Rick, A. Scherz, W. F. Schlotter, D. Zhu, J. Lüning, and J. Stöhr, Opt. Lett. 34, 650 (2009).
    [CrossRef] [PubMed]
  9. S. G. Podorov, K. M. Pavlov, and D. M. Paganin, Opt. Express 15, 9954 (2007).
    [CrossRef] [PubMed]
  10. M. Guizar-Sicairos and J. R. Fienup, Opt. Lett. 33, 2668 (2008).
    [CrossRef] [PubMed]
  11. B. Enders, K. Giewekemeyer, T. Kurz, S. Podorov, and T. Salditt, New J. Phys. 11, 043021 (2009).
    [CrossRef]
  12. H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini, and H. N. Chapman, Appl. Phys. Lett. 85, 2454 (2004).
    [CrossRef]

2009 (2)

B. Enders, K. Giewekemeyer, T. Kurz, S. Podorov, and T. Salditt, New J. Phys. 11, 043021 (2009).
[CrossRef]

R. Rick, A. Scherz, W. F. Schlotter, D. Zhu, J. Lüning, and J. Stöhr, Opt. Lett. 34, 650 (2009).
[CrossRef] [PubMed]

2008 (1)

2007 (2)

2004 (3)

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini, and H. N. Chapman, Appl. Phys. Lett. 85, 2454 (2004).
[CrossRef]

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, Nature 432, 885 (2004).
[CrossRef] [PubMed]

X. Zhao, Y. Liu, S. Inoue, T. Suzuki, R. O. Jones, and Y. Ando, Phys. Rev. Lett. 92, 125502 (2004).
[CrossRef] [PubMed]

1999 (1)

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

1992 (1)

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, Science 256, 1009 (1992).
[CrossRef] [PubMed]

1982 (1)

Anderson, E. H.

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, Science 256, 1009 (1992).
[CrossRef] [PubMed]

Ando, Y.

X. Zhao, Y. Liu, S. Inoue, T. Suzuki, R. O. Jones, and Y. Ando, Phys. Rev. Lett. 92, 125502 (2004).
[CrossRef] [PubMed]

Chapman, H. N.

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini, and H. N. Chapman, Appl. Phys. Lett. 85, 2454 (2004).
[CrossRef]

Charalambous, P.

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

Eberhardt, W.

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, Nature 432, 885 (2004).
[CrossRef] [PubMed]

Eisebitt, S.

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, Nature 432, 885 (2004).
[CrossRef] [PubMed]

Enders, B.

B. Enders, K. Giewekemeyer, T. Kurz, S. Podorov, and T. Salditt, New J. Phys. 11, 043021 (2009).
[CrossRef]

Fienup, J. R.

Giewekemeyer, K.

B. Enders, K. Giewekemeyer, T. Kurz, S. Podorov, and T. Salditt, New J. Phys. 11, 043021 (2009).
[CrossRef]

Guizar-Sicairos, M.

He, H.

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini, and H. N. Chapman, Appl. Phys. Lett. 85, 2454 (2004).
[CrossRef]

Hellwig, O.

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, Nature 432, 885 (2004).
[CrossRef] [PubMed]

Howells, M.

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini, and H. N. Chapman, Appl. Phys. Lett. 85, 2454 (2004).
[CrossRef]

Howells, M. R.

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, Science 256, 1009 (1992).
[CrossRef] [PubMed]

Inoue, S.

X. Zhao, Y. Liu, S. Inoue, T. Suzuki, R. O. Jones, and Y. Ando, Phys. Rev. Lett. 92, 125502 (2004).
[CrossRef] [PubMed]

Jacobsen, C.

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, Science 256, 1009 (1992).
[CrossRef] [PubMed]

Jones, R. O.

X. Zhao, Y. Liu, S. Inoue, T. Suzuki, R. O. Jones, and Y. Ando, Phys. Rev. Lett. 92, 125502 (2004).
[CrossRef] [PubMed]

Kern, D. P.

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, Science 256, 1009 (1992).
[CrossRef] [PubMed]

Kirz, J.

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

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, Science 256, 1009 (1992).
[CrossRef] [PubMed]

Kurz, T.

B. Enders, K. Giewekemeyer, T. Kurz, S. Podorov, and T. Salditt, New J. Phys. 11, 043021 (2009).
[CrossRef]

Liu, Y.

X. Zhao, Y. Liu, S. Inoue, T. Suzuki, R. O. Jones, and Y. Ando, Phys. Rev. Lett. 92, 125502 (2004).
[CrossRef] [PubMed]

Lörgen, M.

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, Nature 432, 885 (2004).
[CrossRef] [PubMed]

Lüning, J.

R. Rick, A. Scherz, W. F. Schlotter, D. Zhu, J. Lüning, and J. Stöhr, Opt. Lett. 34, 650 (2009).
[CrossRef] [PubMed]

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, Nature 432, 885 (2004).
[CrossRef] [PubMed]

Marchesini, S.

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini, and H. N. Chapman, Appl. Phys. Lett. 85, 2454 (2004).
[CrossRef]

McNulty, I.

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, Science 256, 1009 (1992).
[CrossRef] [PubMed]

Miao, J.

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

Padmore, H. A.

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini, and H. N. Chapman, Appl. Phys. Lett. 85, 2454 (2004).
[CrossRef]

Paganin, D. M.

Pavlov, K. M.

Podorov, S.

B. Enders, K. Giewekemeyer, T. Kurz, S. Podorov, and T. Salditt, New J. Phys. 11, 043021 (2009).
[CrossRef]

Podorov, S. G.

Rick, R.

Salditt, T.

B. Enders, K. Giewekemeyer, T. Kurz, S. Podorov, and T. Salditt, New J. Phys. 11, 043021 (2009).
[CrossRef]

Sayre, D.

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

Scherz, A.

Schlotter, W. F.

R. Rick, A. Scherz, W. F. Schlotter, D. Zhu, J. Lüning, and J. Stöhr, Opt. Lett. 34, 650 (2009).
[CrossRef] [PubMed]

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, Nature 432, 885 (2004).
[CrossRef] [PubMed]

Spence, J. C. H.

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini, and H. N. Chapman, Appl. Phys. Lett. 85, 2454 (2004).
[CrossRef]

Stöhr, J.

R. Rick, A. Scherz, W. F. Schlotter, D. Zhu, J. Lüning, and J. Stöhr, Opt. Lett. 34, 650 (2009).
[CrossRef] [PubMed]

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, Nature 432, 885 (2004).
[CrossRef] [PubMed]

Suzuki, T.

X. Zhao, Y. Liu, S. Inoue, T. Suzuki, R. O. Jones, and Y. Ando, Phys. Rev. Lett. 92, 125502 (2004).
[CrossRef] [PubMed]

Weierstall, U.

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini, and H. N. Chapman, Appl. Phys. Lett. 85, 2454 (2004).
[CrossRef]

Zhao, X.

X. Zhao, Y. Liu, S. Inoue, T. Suzuki, R. O. Jones, and Y. Ando, Phys. Rev. Lett. 92, 125502 (2004).
[CrossRef] [PubMed]

Zhu, D.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini, and H. N. Chapman, Appl. Phys. Lett. 85, 2454 (2004).
[CrossRef]

Nature (2)

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

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, Nature 432, 885 (2004).
[CrossRef] [PubMed]

New J. Phys. (1)

B. Enders, K. Giewekemeyer, T. Kurz, S. Podorov, and T. Salditt, New J. Phys. 11, 043021 (2009).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

X. Zhao, Y. Liu, S. Inoue, T. Suzuki, R. O. Jones, and Y. Ando, Phys. Rev. Lett. 92, 125502 (2004).
[CrossRef] [PubMed]

Science (1)

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, Science 256, 1009 (1992).
[CrossRef] [PubMed]

Other (1)

Ancient Chinese character “elephant,” which later inherited the meaning of “image” from their phonetic connection, http://en.wikipedia.org/wiki/Oracle_bone_script.

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 (3)

Fig. 1
Fig. 1

Object with crossed-wires reference.

Fig. 2
Fig. 2

(a) 512 × 512 object space field amplitude, f ( x , y ) . (b) Field autocorrelation, f f ; an overall bias was subtracted for better visualization. (c) Cross correlation of object and crossed wires extracted from (b). (d) Object transmissivity reconstructed from (c).

Fig. 3
Fig. 3

(a) Experimental setup and measured diffraction pattern ( 600 × 600 inset, number of photons, logarithmic scale). (b) SEM image of the sample. (c) Autocorrelation computed from an inverse FT of the measured data. (d) Region extracted from (c). (e) Direct reconstruction from (d). Real part is shown in (c), (d), and (e).

Equations (7)

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

f f = o o + r r + o r + r o ,
L ( n ) { r ( x , y ) } = A D ( m ) { d ( x x 0 , y y 0 ) } + g ( x , y ) ,
L ( n ) { f f } = L ( n ) { o o } + L ( n ) { r r } + ( 1 ) n o g + g o + ( 1 ) n A * × D ( m ) { o ( x + x 0 , y + y 0 ) d * ( x , y ) } + A D ( m ) { o * ( x 0 x , y 0 y ) d ( x , y ) } ,
r ( x + x 0 , y + y 0 ) = 1 ( 1 B ) [ rect ( y cos α x sin α w ) + rect ( y cos β x sin β w ) ] + A d ( x , y ) ,
A d ( x , y ) = ( 1 B ) 2 rect ( y cos α x sin α w ) × rect ( y cos β x sin β w )
b m , n = a m , n ( a 1 , n + a M , n ) 2 .
o m , n = b m , n ( b m , 1 + b m , N ) 2 .

Metrics