Abstract

We introduce a generalization of Fourier transform holography that allows the use of the boundary waves of an extended object to act as a holographic-like reference. By applying a linear differential operator on the field autocorrelation, we use a sharp feature on the extended reference to reconstruct a complex-valued image of the object of interest in a single-step computation. We generalize the approach of Podorov et al. [Opt. Express 15, 9954 (2007)] to a much wider class of extended reference objects. Effects of apertures in Fourier domain and imperfections in the reference object are analyzed. Realistic numerical simulations show the feasibility of our approach and its robustness against noise.

© 2007 Optical Society of America

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  19. O. Hellwig, S. Eisebitt, W. Eberhardt, W. F. Schlotter, J. Lüning and J. Stöhr, "Magnetic imaging with soft x-ray spectroholography," J. Appl. Phys. 99, 08H307 (2006).
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  24. M. R. Howells, C. J. Jacobsen, S. Marchesini, S. Miller, J. C. H. Spence and U. Weirstall, "Toward a practical X-ray Fourier holography at high resolution," Nucl. Instrum. Methods Phys. Res. A 467, 864-867 (2001).
    [CrossRef]
  25. R. N. Bracewell, K.-Y. Chang, A. K. Jha and Y.-H. Wang, "Affine theorem for two-dimensional Fourier transform," Electron. Lett. 29, 304 (1993).
    [CrossRef]
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    [CrossRef]

2007 (2)

L.M. Stadler, R. Harder, I. K. Robinson, C. Rentenberger, H. P. Karnthaler, B. Sepiol and G. Vogl, "Coherent x-ray diffraction imaging of grown-in antiphase boundaries in Fe65Al35," Phys. Rev. B 76, 014204 (2007).
[CrossRef]

S. G. Podorov, K. M. Pavlov and D. M. Paganin, "A non-iterative reconstruction method for direct and unambiguous coherent diffractive imaging," Opt. Express 15, 9954-9962 (2007).
[CrossRef] [PubMed]

2006 (4)

H. N. Chapman et al., "Femtosecond diffractive imaging with a soft-x-ray free-electron laser," Nat. Phys. 2, 839-843 (2006).
[CrossRef]

W. F. Schlotter et al., "Multiple reference Fourier transform holography with soft x rays," Appl. Phys. Lett. 89, 163112 (2006).
[CrossRef]

H. N. Chapman et al., "High-resolution ab initio three-dimensional x-ray diffraction microscopy," J. Opt. Soc. Am. A 23, 1179-1200 (2006).
[CrossRef]

J. R. Fienup, "Lensless coherent imaging by phase retrieval with an illumination pattern constraint," Opt. Express 14, 498-508 (2006).
[CrossRef] [PubMed]

2005 (1)

D. Shapiro et al., "Biological imaging by soft x-ray diffraction microscopy," Proc. Natl. Acad. Sci. U.S.A. 102, 15343-15346 (2005).
[CrossRef] [PubMed]

2004 (2)

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt and J. Stöhr, "Lensless imaging of magnetic nanostructures by x-ray spectro-holography," Nature 432, 885-888 (2004).
[CrossRef] [PubMed]

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini and H. N. Chapman, "Use of extended and prepared reference objects in experimental Fourier transform x-ray holography," Appl. Phys. Lett. 85, 2454-2456 (2004).
[CrossRef]

2003 (1)

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101 (2003).
[CrossRef]

2001 (1)

M. R. Howells, C. J. Jacobsen, S. Marchesini, S. Miller, J. C. H. Spence and U. Weirstall, "Toward a practical X-ray Fourier holography at high resolution," Nucl. Instrum. Methods Phys. Res. A 467, 864-867 (2001).
[CrossRef]

1999 (1)

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

1993 (1)

R. N. Bracewell, K.-Y. Chang, A. K. Jha and Y.-H. Wang, "Affine theorem for two-dimensional Fourier transform," Electron. Lett. 29, 304 (1993).
[CrossRef]

1992 (1)

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells and D. P. Kern, "High-Resolution Imaging by Fourier Transform X-ray Holography," Science 256, 1009-1012 (1992).
[CrossRef] [PubMed]

1990 (1)

1988 (1)

1987 (2)

1986 (1)

1982 (2)

1969 (1)

J. D. Gaskill and J. W. Goodman, "Use of multiple reference sources to increase the effective field of view in lensless Fourier-transform holography," Proc. IEEE 57, 823-825 (1969).
[CrossRef]

1965 (1)

G.W. Stroke, R. Restrick, A. Funkhouser and D. Brumm, "Resolution-retrieving compensation of source effects by correlative reconstruction in high-resolution holography," Phys. Lett. 18, 274-275 (1965).
[CrossRef]

1962 (1)

Anderson, E. H.

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells and D. P. Kern, "High-Resolution Imaging by Fourier Transform X-ray Holography," Science 256, 1009-1012 (1992).
[CrossRef] [PubMed]

Bracewell, R. N.

R. N. Bracewell, K.-Y. Chang, A. K. Jha and Y.-H. Wang, "Affine theorem for two-dimensional Fourier transform," Electron. Lett. 29, 304 (1993).
[CrossRef]

Brumm, D.

G.W. Stroke, R. Restrick, A. Funkhouser and D. Brumm, "Resolution-retrieving compensation of source effects by correlative reconstruction in high-resolution holography," Phys. Lett. 18, 274-275 (1965).
[CrossRef]

Cederquist, J. N.

Chang, K.-Y.

R. N. Bracewell, K.-Y. Chang, A. K. Jha and Y.-H. Wang, "Affine theorem for two-dimensional Fourier transform," Electron. Lett. 29, 304 (1993).
[CrossRef]

Chapman, H. N.

H. N. Chapman et al., "Femtosecond diffractive imaging with a soft-x-ray free-electron laser," Nat. Phys. 2, 839-843 (2006).
[CrossRef]

H. N. Chapman et al., "High-resolution ab initio three-dimensional x-ray diffraction microscopy," J. Opt. Soc. Am. A 23, 1179-1200 (2006).
[CrossRef]

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini and H. N. Chapman, "Use of extended and prepared reference objects in experimental Fourier transform x-ray holography," Appl. Phys. Lett. 85, 2454-2456 (2004).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Charalambous, P.

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

Crimmins, T. R.

Eberhardt, W.

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt and J. Stöhr, "Lensless imaging of magnetic nanostructures by x-ray spectro-holography," Nature 432, 885-888 (2004).
[CrossRef] [PubMed]

Eisebitt, S.

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt and J. Stöhr, "Lensless imaging of magnetic nanostructures by x-ray spectro-holography," Nature 432, 885-888 (2004).
[CrossRef] [PubMed]

Fienup, J. R.

Funkhouser, A.

G.W. Stroke, R. Restrick, A. Funkhouser and D. Brumm, "Resolution-retrieving compensation of source effects by correlative reconstruction in high-resolution holography," Phys. Lett. 18, 274-275 (1965).
[CrossRef]

Gaskill, J. D.

J. D. Gaskill and J. W. Goodman, "Use of multiple reference sources to increase the effective field of view in lensless Fourier-transform holography," Proc. IEEE 57, 823-825 (1969).
[CrossRef]

Goodman, J. W.

J. D. Gaskill and J. W. Goodman, "Use of multiple reference sources to increase the effective field of view in lensless Fourier-transform holography," Proc. IEEE 57, 823-825 (1969).
[CrossRef]

Goodman, R. S.

Harder, R.

L.M. Stadler, R. Harder, I. K. Robinson, C. Rentenberger, H. P. Karnthaler, B. Sepiol and G. Vogl, "Coherent x-ray diffraction imaging of grown-in antiphase boundaries in Fe65Al35," Phys. Rev. B 76, 014204 (2007).
[CrossRef]

Hau-Riege, S. P.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101 (2003).
[CrossRef]

He, H.

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini and H. N. Chapman, "Use of extended and prepared reference objects in experimental Fourier transform x-ray holography," Appl. Phys. Lett. 85, 2454-2456 (2004).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Hellwig, O.

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt and J. Stöhr, "Lensless imaging of magnetic nanostructures by x-ray spectro-holography," Nature 432, 885-888 (2004).
[CrossRef] [PubMed]

Holsztynski, W.

Howells, M.

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini and H. N. Chapman, "Use of extended and prepared reference objects in experimental Fourier transform x-ray holography," Appl. Phys. Lett. 85, 2454-2456 (2004).
[CrossRef]

Howells, M. R.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101 (2003).
[CrossRef]

M. R. Howells, C. J. Jacobsen, S. Marchesini, S. Miller, J. C. H. Spence and U. Weirstall, "Toward a practical X-ray Fourier holography at high resolution," Nucl. Instrum. Methods Phys. Res. A 467, 864-867 (2001).
[CrossRef]

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells and D. P. Kern, "High-Resolution Imaging by Fourier Transform X-ray Holography," Science 256, 1009-1012 (1992).
[CrossRef] [PubMed]

Idell, P. S.

Jacobsen, C.

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells and D. P. Kern, "High-Resolution Imaging by Fourier Transform X-ray Holography," Science 256, 1009-1012 (1992).
[CrossRef] [PubMed]

Jacobsen, C. J.

M. R. Howells, C. J. Jacobsen, S. Marchesini, S. Miller, J. C. H. Spence and U. Weirstall, "Toward a practical X-ray Fourier holography at high resolution," Nucl. Instrum. Methods Phys. Res. A 467, 864-867 (2001).
[CrossRef]

Jha, A. K.

R. N. Bracewell, K.-Y. Chang, A. K. Jha and Y.-H. Wang, "Affine theorem for two-dimensional Fourier transform," Electron. Lett. 29, 304 (1993).
[CrossRef]

Karnthaler, H. P.

L.M. Stadler, R. Harder, I. K. Robinson, C. Rentenberger, H. P. Karnthaler, B. Sepiol and G. Vogl, "Coherent x-ray diffraction imaging of grown-in antiphase boundaries in Fe65Al35," Phys. Rev. B 76, 014204 (2007).
[CrossRef]

Kern, D. P.

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells and D. P. Kern, "High-Resolution Imaging by Fourier Transform X-ray Holography," Science 256, 1009-1012 (1992).
[CrossRef] [PubMed]

Kirz, J.

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

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells and D. P. Kern, "High-Resolution Imaging by Fourier Transform X-ray Holography," Science 256, 1009-1012 (1992).
[CrossRef] [PubMed]

Leith, E. N.

Lörgen, M.

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt and J. Stöhr, "Lensless imaging of magnetic nanostructures by x-ray spectro-holography," Nature 432, 885-888 (2004).
[CrossRef] [PubMed]

Lüning, J.

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt and J. Stöhr, "Lensless imaging of magnetic nanostructures by x-ray spectro-holography," Nature 432, 885-888 (2004).
[CrossRef] [PubMed]

Marchesini, S.

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini and H. N. Chapman, "Use of extended and prepared reference objects in experimental Fourier transform x-ray holography," Appl. Phys. Lett. 85, 2454-2456 (2004).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101 (2003).
[CrossRef]

M. R. Howells, C. J. Jacobsen, S. Marchesini, S. Miller, J. C. H. Spence and U. Weirstall, "Toward a practical X-ray Fourier holography at high resolution," Nucl. Instrum. Methods Phys. Res. A 467, 864-867 (2001).
[CrossRef]

Marron, J. C.

McNulty, I.

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells and D. P. Kern, "High-Resolution Imaging by Fourier Transform X-ray Holography," Science 256, 1009-1012 (1992).
[CrossRef] [PubMed]

Miao, J.

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

Miller, S.

M. R. Howells, C. J. Jacobsen, S. Marchesini, S. Miller, J. C. H. Spence and U. Weirstall, "Toward a practical X-ray Fourier holography at high resolution," Nucl. Instrum. Methods Phys. Res. A 467, 864-867 (2001).
[CrossRef]

Noy, A.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Padmore, H. A.

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini and H. N. Chapman, "Use of extended and prepared reference objects in experimental Fourier transform x-ray holography," Appl. Phys. Lett. 85, 2454-2456 (2004).
[CrossRef]

Paganin, D. M.

Pavlov, K. M.

Paxman, R. G.

Podorov, S. G.

Rentenberger, C.

L.M. Stadler, R. Harder, I. K. Robinson, C. Rentenberger, H. P. Karnthaler, B. Sepiol and G. Vogl, "Coherent x-ray diffraction imaging of grown-in antiphase boundaries in Fe65Al35," Phys. Rev. B 76, 014204 (2007).
[CrossRef]

Restrick, R.

G.W. Stroke, R. Restrick, A. Funkhouser and D. Brumm, "Resolution-retrieving compensation of source effects by correlative reconstruction in high-resolution holography," Phys. Lett. 18, 274-275 (1965).
[CrossRef]

Robinson, I. K.

L.M. Stadler, R. Harder, I. K. Robinson, C. Rentenberger, H. P. Karnthaler, B. Sepiol and G. Vogl, "Coherent x-ray diffraction imaging of grown-in antiphase boundaries in Fe65Al35," Phys. Rev. B 76, 014204 (2007).
[CrossRef]

Sayre, D.

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

Schlotter, W. F.

W. F. Schlotter et al., "Multiple reference Fourier transform holography with soft x rays," Appl. Phys. Lett. 89, 163112 (2006).
[CrossRef]

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt and J. Stöhr, "Lensless imaging of magnetic nanostructures by x-ray spectro-holography," Nature 432, 885-888 (2004).
[CrossRef] [PubMed]

Sepiol, B.

L.M. Stadler, R. Harder, I. K. Robinson, C. Rentenberger, H. P. Karnthaler, B. Sepiol and G. Vogl, "Coherent x-ray diffraction imaging of grown-in antiphase boundaries in Fe65Al35," Phys. Rev. B 76, 014204 (2007).
[CrossRef]

Shapiro, D.

D. Shapiro et al., "Biological imaging by soft x-ray diffraction microscopy," Proc. Natl. Acad. Sci. U.S.A. 102, 15343-15346 (2005).
[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, "Use of extended and prepared reference objects in experimental Fourier transform x-ray holography," Appl. Phys. Lett. 85, 2454-2456 (2004).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101 (2003).
[CrossRef]

M. R. Howells, C. J. Jacobsen, S. Marchesini, S. Miller, J. C. H. Spence and U. Weirstall, "Toward a practical X-ray Fourier holography at high resolution," Nucl. Instrum. Methods Phys. Res. A 467, 864-867 (2001).
[CrossRef]

Stadler, L.M.

L.M. Stadler, R. Harder, I. K. Robinson, C. Rentenberger, H. P. Karnthaler, B. Sepiol and G. Vogl, "Coherent x-ray diffraction imaging of grown-in antiphase boundaries in Fe65Al35," Phys. Rev. B 76, 014204 (2007).
[CrossRef]

Stöhr, J.

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt and J. Stöhr, "Lensless imaging of magnetic nanostructures by x-ray spectro-holography," Nature 432, 885-888 (2004).
[CrossRef] [PubMed]

Stroke, G.W.

G.W. Stroke, R. Restrick, A. Funkhouser and D. Brumm, "Resolution-retrieving compensation of source effects by correlative reconstruction in high-resolution holography," Phys. Lett. 18, 274-275 (1965).
[CrossRef]

Thelen, B. J.

Upatnieks, J.

Vogl, G.

L.M. Stadler, R. Harder, I. K. Robinson, C. Rentenberger, H. P. Karnthaler, B. Sepiol and G. Vogl, "Coherent x-ray diffraction imaging of grown-in antiphase boundaries in Fe65Al35," Phys. Rev. B 76, 014204 (2007).
[CrossRef]

Wackerman, C. C.

Wang, Y.-H.

R. N. Bracewell, K.-Y. Chang, A. K. Jha and Y.-H. Wang, "Affine theorem for two-dimensional Fourier transform," Electron. Lett. 29, 304 (1993).
[CrossRef]

Weierstall, U.

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini and H. N. Chapman, "Use of extended and prepared reference objects in experimental Fourier transform x-ray holography," Appl. Phys. Lett. 85, 2454-2456 (2004).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Weirstall, U.

M. R. Howells, C. J. Jacobsen, S. Marchesini, S. Miller, J. C. H. Spence and U. Weirstall, "Toward a practical X-ray Fourier holography at high resolution," Nucl. Instrum. Methods Phys. Res. A 467, 864-867 (2001).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini and H. N. Chapman, "Use of extended and prepared reference objects in experimental Fourier transform x-ray holography," Appl. Phys. Lett. 85, 2454-2456 (2004).
[CrossRef]

W. F. Schlotter et al., "Multiple reference Fourier transform holography with soft x rays," Appl. Phys. Lett. 89, 163112 (2006).
[CrossRef]

Electron. Lett. (1)

R. N. Bracewell, K.-Y. Chang, A. K. Jha and Y.-H. Wang, "Affine theorem for two-dimensional Fourier transform," Electron. Lett. 29, 304 (1993).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

Nat. Phys. (1)

H. N. Chapman et al., "Femtosecond diffractive imaging with a soft-x-ray free-electron laser," Nat. Phys. 2, 839-843 (2006).
[CrossRef]

Nature (2)

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

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt and J. Stöhr, "Lensless imaging of magnetic nanostructures by x-ray spectro-holography," Nature 432, 885-888 (2004).
[CrossRef] [PubMed]

Nucl. Instrum. Methods Phys. Res. A (1)

M. R. Howells, C. J. Jacobsen, S. Marchesini, S. Miller, J. C. H. Spence and U. Weirstall, "Toward a practical X-ray Fourier holography at high resolution," Nucl. Instrum. Methods Phys. Res. A 467, 864-867 (2001).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Phys. Lett. (1)

G.W. Stroke, R. Restrick, A. Funkhouser and D. Brumm, "Resolution-retrieving compensation of source effects by correlative reconstruction in high-resolution holography," Phys. Lett. 18, 274-275 (1965).
[CrossRef]

Phys. Rev. B (2)

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101 (2003).
[CrossRef]

L.M. Stadler, R. Harder, I. K. Robinson, C. Rentenberger, H. P. Karnthaler, B. Sepiol and G. Vogl, "Coherent x-ray diffraction imaging of grown-in antiphase boundaries in Fe65Al35," Phys. Rev. B 76, 014204 (2007).
[CrossRef]

Proc. IEEE (1)

J. D. Gaskill and J. W. Goodman, "Use of multiple reference sources to increase the effective field of view in lensless Fourier-transform holography," Proc. IEEE 57, 823-825 (1969).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A. (1)

D. Shapiro et al., "Biological imaging by soft x-ray diffraction microscopy," Proc. Natl. Acad. Sci. U.S.A. 102, 15343-15346 (2005).
[CrossRef] [PubMed]

Science (1)

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells and D. P. Kern, "High-Resolution Imaging by Fourier Transform X-ray Holography," Science 256, 1009-1012 (1992).
[CrossRef] [PubMed]

Other (3)

J. W. Goodman, Introduction to Fourier Optics, 3rd Ed. (Roberts & Company, Englewood, 2005).

O. Hellwig, S. Eisebitt, W. Eberhardt, W. F. Schlotter, J. Lüning and J. Stöhr, "Magnetic imaging with soft x-ray spectroholography," J. Appl. Phys. 99, 08H307 (2006).
[CrossRef]

R. N. Bracewell, The Fourier Transform and its Applications, 2nd Ed. (McGraw-Hill, New York, 1978).

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Figures (11)

Fig. 1.
Fig. 1.

(a) Experimental setup for lensless imaging. (b) Image reconstruction by phase retrieval does not require a reference wave. (c) Sample prepared with properly separated reference point source for conventional FT holography. Sample prepared for HERALDO with object (d) contained within a rectangular reference and (e) object sitting outside of the rectangular reference (both described by Podorov et al.). (f) Sample prepared for HERALDO with an obscuring tip. Notice that the farthest corner of the rectangles from the object in (d) and (e) and the tip apex in (f) satisfy the conventional holographic separation condition.

Fig. 2.
Fig. 2.

(a) Field f (x,y) with a slit reference. (b) Field autocorrelation ff (inverse FT of far-field intensity). (c) HERALDO separation conditions; the reference feature must satisfy separation conditions from o(x,y), g(x,y), g*(-x,-y) and rr. (d) Directional derivative of the autocorrelation in the direction of the slit, α̂.

Fig. 3.
Fig. 3.

(a) Finite line Dirac delta along the direction of α̂. (b) After applying the the directional derivative along α̂, we obtain two point Dirac delta functions at the line ends.

Fig. 4.
Fig. 4.

(a) Bright corner with angle (β-α), edges run along the unit vectors α̂ and β̂. (b) Example holographic setup with an obscuring tip. (c) Result of the double directional derivative on the autocorrelation (along α̂ and β̂) of the field shown in (b). The object is well separated from the tip and its transmissivity t(x,y) can be directly computed.

Fig. 5.
Fig. 5.

(a) Parallelogram r(x,y) with sides making angles α and β with respect to the x-axis. (b) Directional derivatives of r(x,y) taken along α̂ and β̂ produce four point Dirac deltas. (c) The term (2) {rr} term produces a uniform amplitude parallelogram of twice the size of the original one.

Fig. 6.
Fig. 6.

Example holographic setups with object outside of the parallelogram reference are shown in (a) and (b). (c) Example holographic setup with object inside the parallelogram reference. Directional derivatives are taken along α̂ and β̂. Reconstruction result, (2) {ff}, of fields in (a), (b) and (c) are shown in (d), (e) and (f), respectively. The support of (2) {oo} is shown as a circle in (d), (e) and (f). Since HERALDO separation Condition 2 is not satisfied by the corners of the parallelogram in (b) the reconstructions, shown in (e), overlap with one another. However, since the parallelogram sides are greater than ρ 0, and HERALDO separation Conditions 1 and 3 are satisfied, we can arrive at a full reconstruction by selectively stitching reconstruction pixels that do not overlap. For example, the object upper-left quadrant (open eye) can be obtained from the upper-left reconstruction.

Fig. 7.
Fig. 7.

(a) Original 128×128 object. (b) Object space field amplitude f (x,y). The original object is illuminated by a Gaussian beam and a 40° tip is used to partially obstruct the beam. (c) Simulated intensity measurement, including the effect of finite detector diameter. (d) Downsampled, 64×64, version of original object shown in (a); resolution was halved in Fourier domain. (e) Differential operator, Eq. (17), is applied as a polynomial product on the measured intensity. (f) Magnitude of the inverse FT of (e). Due to their large dynamic ranges, the sixth and fourth roots of the magnitude are shown in (c) and (e) respectively.

Fig. 8.
Fig. 8.

(a) Cut through (from DC to upper right corner) simulated noisy measurements for 1012, 1010 and 109 photons on the brightest pixel. Magnitude of the 64×64 reconstructions directly obtained from noisy patterns with (b) 1012, (c) 1010 and (d) 109 photons on the brightest pixel.

Fig. 9.
Fig. 9.

(a) A square beam stop (42 pixel diagonal) was added to the simulated intensity shown in Fig. 7(c). (b) After polynomial multiplication in Fourier domain. (c) 64×64 reconstruction directly obtained from the inverse FT of (b).

Fig. 10.
Fig. 10.

Original 220×220 object (a) amplitude (b) and phase. (c) Object space field amplitude f (x,y). (d) Far-field noisy intensity pattern. (e) Magnitude after multiplication by i2πu. (f) Inverse FFT of (e). (g) Amplitude (h) and phase of the 220×220 bottom left reconstructed image. (i) 40×40 pixel inset of (c). Fifth root of magnitude is shown in (d) and (e). Phase in (b) and (h) has a peak-to-valley range of 3π/2 and is shown in the range [-π,π].

Fig. 11.
Fig. 11.

Heaviside function and its directional derivative along θ ^ .

Equations (55)

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F ( u , v ) = 𝓕 { f ( x , y ) } = f ( x , y ) exp [ i 2 π ( ux + vy ) ] d x d y .
𝓕 1 { F ( u , v ) 2 } = f f = o o + r r + o r + r o ,
o r = o ( x , y ) r * ( x x , y y ) d x   d y
( n ) { r ( x , y ) } = A δ ( x x 0 ) δ ( y y 0 ) + g ( x , y ) ,
( n ) { · } k = 0 n a k n x n k y k
( n ) { f f } = ( n ) { o o } + ( n ) { r r } + ( 1 ) n [ o ( n ) { r } ] + [ ( n ) { r } o ] ,
( n ) { h g } = ( 1 ) n [ h ( n ) { g } ] = [ ( n ) { h } g ] .
( n ) { f f } = ( n ) { r r } + ( n ) { o o } + ( 1 ) n o g + g o
+ ( 1 ) n A * o ( x + x 0 , y + y 0 ) + A o * ( x 0 x , y 0 y ) ,
( n ) { f f } = 1 2 ( 1 ) n [ A * r ( x + x 0 , y + y 0 ) + r g ] + 1 2 [ A r * ( x 0 - x , y 0 - y ) + g r ]
+ ( n ) { o o } + ( 1 ) n o g +g o + ( 1 ) n A * o ( x + x 0 , y + y 0 ) + A o * ( x 0 x , y 0 y ) .
r ( x , y ) = A δ ( y ) rect ( x L ) = A δ ( y ) [ H ( x + L 2 ) H ( x L 2 ) ] ,
H ( x ) = { 0 , if x < 0 1 , if x > 0 1 2 , if x = 0 and rect ( x ) = { 1 , if x < 1 2 0 , if x > 1 2 1 2 , if x = 1 2 .
x r ( x , y ) = A δ ( y ) [ δ ( x + L 2 ) δ ( x L 2 ) ]
x H ( x ) = δ ( x ) .
x = x cos α y sin α ,
y = y cos α + x sin α .
( 1 ) { · } = x = cos α x + sin α y = α ̂ · ,
y ( x , y ) = A H ( y cos α x sin α ) H ( x sin β y cos β ) .
( 2 ) { · } = 1 sin ( β α ) [ α ̂ · ] [ β ̂ · ]
1 sin ( β α ) [ α ̂ · ] [ β ̂ · ] H ( y cos α x sin α ) H ( x sin β y cos β ) = β ( x ) δ ( y ) .
r ( x , y ) = A { 1 H [ ( y y 0 ) cos α ( x x 0 ) sin α ] H [ ( x x 0 ) sin β ( y y 0 ) cos β ] } ,
1 sin ( β α ) [ α ̂ · ] [ β ̂ · ] r ( x , y ) = A δ ( x x 0 ) δ ( y y 0 ) .
F H ( u , v ) 2 = H ( u , v ) 2 F ( U , v ) 2 ,
( n ) { f h f h } = ( n ) { o h o h } + ( n ) { r h r h } + ( 1 ) n [ o h ( n ) { r h } ]
+ [ ( n ) { r h } o h ] ,
o h ( x , y ) = o * h ( x , y ) = o ( x , y ) h ( x x , y y ) d x d y .
( n ) { r h ( x , y ) } = A h ( x x 0 , y y 0 ) + g h ( x , y ) .
( n ) { f h f h } = ( n ) { o h o h } + ( n ) { r h r h } + ( 1 ) n o h g h + g h o h
+ ( 1 ) n A * o h h ( x + x 0 , y + y 0 ) + A o h h * ( x 0 x , y 0 y ) .
o h h ( x + x 0 , y + y 0 ) = [ o * ( h h ) ] ( x + x 0 , y + y 0 ) ,
o h h ( x + x 0 , y + y 0 ) = ( o * h ) ( x + x 0 , y + y 0 ) ,
( n ) { r d ( x , y ) } = A d ( x x 0 , y y 0 ) + g d ( x , y ) ,
o ( x , y ) A d ( x x 0 , y y 0 ) = A * o ( x + x 0 , y + y 0 ) * d * ( x , y ) ,
A d ( x x 0 , y y 0 ) o ( x , y ) = A o * ( x 0 x , y o y ) * d ( x , y ) ,
( n ) { f f } = ( n ) { o o } + ( n ) { r d r d } + ( 1 ) n o g d + g d o
+ ( 1 ) n A * o ( x + x 0 , y + y 0 ) * d * ( x , y ) + A o * ( x 0 x , y 0 y ) * d ( x , y ) ,
r d ( x , y ) = ( r * d ) ( x , y ) ,
( n ) { r d ( x , y ) } = ( ( n ) { r } * d ) ( x , y ) = A d ( x x 0 , y y 0 ) + ( g * d ) ( x , y )
x [ f ( x ) * g ( x ) ] = f x ( x ) * g ( x ) = f ( x ) * g x ( x ) ,
f x ( x ) = f ( x ) x ,
x [ h g ( x , y ) ] = h g x ( x , y ) = h x g ( x , y ) .
i 2 π u [ H ( u , v ) G * ( u , v ) ] = H ( u , v ) [ i 2 π u G ( u , v ) ] * = [ i 2 π u H ( u , v ) ] G * ( u , v )
[ θ ̂ · ] H ( y ) = [ cos θ x + sin θ y ] H ( y ) = sin θ δ ( y ) .
r ( x , y ) = H ( y cos α x sin α ) H ( x sin β y cos β ) ,
[ α ̂ · ] [ β ̂ · ] r ( x , y ) = [ α ̂ · ] [ β ̂ · ] [ H ( y cos α x sin α ) ( x sin β y cos β ) ]
= [ α ̂ · ] { H ( x sin β y cos β ) [ β ̂ · ] H ( y cos α x sin α ) } ,
[ α ̂ · ] [ β ̂ · ] r ( x , y ) = [ α ̂ · ] [ H ( x sin β y cos β ) sin ( β α ) ( y cos α x sin α ) ]
= sin ( β α ) δ ( y cos α x sin α ) [ α ̂ · ] H ( x sin β y cos β ) ,
[ α ̂ · ] [ β ̂ · ] r ( x , y ) = sin 2 ( β α ) δ ( y cos α x sin α ) δ ( x sin β y cos β ) ,
x = x cos α + y sin α ,
y = y cos α x sin α .
[ α ̂ · ] [ β ̂ · ] r ( x , y ) = sin 2 ( β α ) δ ( y ) δ [ x sin ( β α ) y cos ( β α ) ]
= sin 2 ( β α ) δ ( y ) δ [ x sin ( β α ) ] = sin ( β α ) δ ( y ) δ ( x )
= sin ( β α ) δ ( y ) δ ( x ) ,

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