Abstract

Almost all noninterferometric phase-retrieval methods used in coherent diffractive imaging have been based on the measurement system with low numerical aperture, in which Fresnel or Fraunhofer approximation is valid to express the wave propagation between an object and a detector. In microscopy, which is a typical application of coherent diffractive imaging, the measurement of the diffraction intensity with high numerical aperture is required for object reconstruction at high spatial resolution. We here propose an extension procedure to apply the previous phase-retrieval method using an aperture-array filter [J. Opt. Soc. Am. A 25, 742 (2008) ] to the system with high numerical aperture, in which the first Rayleigh–Sommerfeld integral for spherical waves is utilized instead of the Fresnel integral for parabolic waves. Computer-simulated examples in the high-numerical-aperture system demonstrate object reconstruction at high lateral resolution and retrieval of information in the depth direction of an object.

© 2009 Optical Society of America

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2009 (1)

Y. Nishino, Y. Takahashi, N. Imamoto, T. Ishikawa, and K. Maeshima, “Three-dimensional visualization of a human chromosome using coherent x-ray diffraction,” Phys. Rev. Lett. 102, 018101 (2009).
[CrossRef] [PubMed]

2008 (4)

2007 (4)

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

N. Nakajima, “Noniterative phase retrieval from a single diffraction intensity pattern by use of an aperture array,” Phys. Rev. Lett. 98, 223901 (2007).
[CrossRef] [PubMed]

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]

M. Guizar-Sicairos and J. R. Fienup, “Holography with extended reference by autocorrelation linear differential operation,” Opt. Express 15, 17592-17612 (2007).
[CrossRef] [PubMed]

2006 (5)

2004 (2)

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

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöohr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature (London) 432, 885-888 (2004).
[CrossRef]

2001 (2)

K. A. Nugent, D. Paganin, and T. E. Gureyev, “A phase odyssey,” Phys. Today 54, 27-32 (2001).
[CrossRef]

T. E. Gureyev, S. Nayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative in-line phase-contrast imaging with multi-energy x rays,” Phys. Rev. Lett. 86, 5827-5830 (2001).
[CrossRef] [PubMed]

2000 (1)

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 speciments,” Nature (London) 400, 342-344 (1999).
[CrossRef]

1998 (1)

1995 (1)

1983 (1)

1982 (1)

Allman, B. E.

Bajt, S.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel X-ray holography,” Nat. Photonics 2, 560-563 (2008).
[CrossRef]

Barth, R.

Barty, A.

Beetz, T.

Bogan, M. J.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel X-ray holography,” Nat. Photonics 2, 560-563 (2008).
[CrossRef]

Boutet, S.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel X-ray holography,” Nat. Photonics 2, 560-563 (2008).
[CrossRef]

Bunk, O.

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

Cassou, K.

Caumes, J.

Chapman, H. N.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel X-ray holography,” Nat. Photonics 2, 560-563 (2008).
[CrossRef]

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-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]

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 speciments,” Nature (London) 400, 342-344 (1999).
[CrossRef]

Cui, C.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel X-ray holography,” Nat. Photonics 2, 560-563 (2008).
[CrossRef]

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-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]

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, “Hard x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

David, C.

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

Delen, N.

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, “Hard x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Eberhardt, W.

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöohr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature (London) 432, 885-888 (2004).
[CrossRef]

Eisebitt, S.

A. Rosenhahn, R. Barth, F. Staier, T. Simpson, S. Mittler, S. Eisebitt, and M. Grunze, “Digital in-line soft x-ray holography with element contrast,” J. Opt. Soc. Am. A 25, 416-422 (2008).
[CrossRef]

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöohr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature (London) 432, 885-888 (2004).
[CrossRef]

Fajardo, M.

Faulkner, H. M. L.

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

Fienup, J. R.

Frank, M.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel X-ray holography,” Nat. Photonics 2, 560-563 (2008).
[CrossRef]

Frigo, S. P.

Garcia-Sucerquia, J.

Gautier, J.

Goodman, J. W.

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

Grunze, M.

Guizar-Sicairos, M.

Gureyev, T. E.

K. A. Nugent, D. Paganin, and T. E. Gureyev, “A phase odyssey,” Phys. Today 54, 27-32 (2001).
[CrossRef]

T. E. Gureyev, S. Nayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative in-line phase-contrast imaging with multi-energy x rays,” Phys. Rev. Lett. 86, 5827-5830 (2001).
[CrossRef] [PubMed]

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. A 12, 1932-1941 (1995).
[CrossRef]

Hajdu, J.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel X-ray holography,” Nat. Photonics 2, 560-563 (2008).
[CrossRef]

Hau-riege, S. P.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel X-ray holography,” Nat. Photonics 2, 560-563 (2008).
[CrossRef]

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-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]

He, H.

Hellwig, O.

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöohr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature (London) 432, 885-888 (2004).
[CrossRef]

Hooker, B.

Howells, M. R.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel X-ray holography,” Nat. Photonics 2, 560-563 (2008).
[CrossRef]

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-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]

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, “Hard x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Imamoto, N.

Y. Nishino, Y. Takahashi, N. Imamoto, T. Ishikawa, and K. Maeshima, “Three-dimensional visualization of a human chromosome using coherent x-ray diffraction,” Phys. Rev. Lett. 102, 018101 (2009).
[CrossRef] [PubMed]

Ishikawa, T.

Y. Nishino, Y. Takahashi, N. Imamoto, T. Ishikawa, and K. Maeshima, “Three-dimensional visualization of a human chromosome using coherent x-ray diffraction,” Phys. Rev. Lett. 102, 018101 (2009).
[CrossRef] [PubMed]

H. Yamazaki, Y. Kohmura, T. Sakurai, and T. Ishikawa, “Reconstruction of complex-valued electron density with x-ray in-line holograms,” J. Opt. Soc. Am. A 23, 3171-3176 (2006).
[CrossRef]

Jacobsen, C.

Jefimovs, K.

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

Jericho, M. H.

Jericho, S. K.

Johnson, I.

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

Kazamias, S.

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 speciments,” Nature (London) 400, 342-344 (1999).
[CrossRef]

Klages, P.

Kohmura, Y.

Kos-Rosset, M.

Kreuzer, H. J.

Lee, J. Y.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel X-ray holography,” Nat. Photonics 2, 560-563 (2008).
[CrossRef]

Lörgen, M.

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöohr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature (London) 432, 885-888 (2004).
[CrossRef]

Lüning, J.

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöohr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature (London) 432, 885-888 (2004).
[CrossRef]

MacNulty, I.

Maeshima, K.

Y. Nishino, Y. Takahashi, N. Imamoto, T. Ishikawa, and K. Maeshima, “Three-dimensional visualization of a human chromosome using coherent x-ray diffraction,” Phys. Rev. Lett. 102, 018101 (2009).
[CrossRef] [PubMed]

Marchesini, S.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel X-ray holography,” Nat. Photonics 2, 560-563 (2008).
[CrossRef]

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-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]

McMahon, P. J.

Merdji, H.

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 speciments,” Nature (London) 400, 342-344 (1999).
[CrossRef]

Mittler, S.

Morlens, A.

Nakajima, N.

N. Nakajima, “Lensless coherent imaging by a deterministic phase retrieval method with an aperture-array filter,” J. Opt. Soc. Am. A 25, 742-750 (2008).
[CrossRef]

N. Nakajima, “Noniterative phase retrieval from a single diffraction intensity pattern by use of an aperture array,” Phys. Rev. Lett. 98, 223901 (2007).
[CrossRef] [PubMed]

N. Nakajima, “Improvement of resolution for phase retrieval by use of a scanning slit,” Appl. Opt. 45, 5976-5983 (2006).
[CrossRef] [PubMed]

N. Nakajima, “Phase retrieval using the properties of entire functions,” in Advances in Imaging and Electron Physics, P.W.Hawkes, ed., Vol. 93 (Academic, 1995).
[CrossRef]

Nayo, S.

T. E. Gureyev, S. Nayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative in-line phase-contrast imaging with multi-energy x rays,” Phys. Rev. Lett. 86, 5827-5830 (2001).
[CrossRef] [PubMed]

Nishino, Y.

Y. Nishino, Y. Takahashi, N. Imamoto, T. Ishikawa, and K. Maeshima, “Three-dimensional visualization of a human chromosome using coherent x-ray diffraction,” Phys. Rev. Lett. 102, 018101 (2009).
[CrossRef] [PubMed]

Noy, A.

Nugent, K. A.

Paganin, D.

K. A. Nugent, D. Paganin, and T. E. Gureyev, “A phase odyssey,” Phys. Today 54, 27-32 (2001).
[CrossRef]

T. E. Gureyev, S. Nayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative in-line phase-contrast imaging with multi-energy x rays,” Phys. Rev. Lett. 86, 5827-5830 (2001).
[CrossRef] [PubMed]

B. E. Allman, P. J. McMahon, J. B. Tiller, K. A. Nugent, D. Paganin, A. Barty, I. MacNulty, S. P. Frigo, Y. Wang, and C. C. Retsch, “Noninterferometric quantitative phase imaging with soft x rays,” J. Opt. Soc. Am. A 17, 1732-1743 (2000).
[CrossRef]

Paganin, D. M.

Pavlov, K. M.

Pfeiffer, F.

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

Podorov, S. G.

Retsch, C. C.

Rey, G.

Roberts, A.

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, “Hard x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

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

Rosen, R.

Rosenhahn, A.

Sakdinawat, A. E.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel X-ray holography,” Nat. Photonics 2, 560-563 (2008).
[CrossRef]

Sakurai, T.

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 speciments,” Nature (London) 400, 342-344 (1999).
[CrossRef]

Schlotter, W. F.

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöohr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature (London) 432, 885-888 (2004).
[CrossRef]

Seibert, M. M.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel X-ray holography,” Nat. Photonics 2, 560-563 (2008).
[CrossRef]

Shaevitz, J. W.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel X-ray holography,” Nat. Photonics 2, 560-563 (2008).
[CrossRef]

Shapiro, D.

Shapiro, D. A.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel X-ray holography,” Nat. Photonics 2, 560-563 (2008).
[CrossRef]

Simpson, T.

Spence, J. C. H.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel X-ray holography,” Nat. Photonics 2, 560-563 (2008).
[CrossRef]

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-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]

Staier, F.

Stevenson, A. W.

T. E. Gureyev, S. Nayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative in-line phase-contrast imaging with multi-energy x rays,” Phys. Rev. Lett. 86, 5827-5830 (2001).
[CrossRef] [PubMed]

Stöohr, J.

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöohr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature (London) 432, 885-888 (2004).
[CrossRef]

Szöke, A.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel X-ray holography,” Nat. Photonics 2, 560-563 (2008).
[CrossRef]

Takahashi, Y.

Y. Nishino, Y. Takahashi, N. Imamoto, T. Ishikawa, and K. Maeshima, “Three-dimensional visualization of a human chromosome using coherent x-ray diffraction,” Phys. Rev. Lett. 102, 018101 (2009).
[CrossRef] [PubMed]

Teague, M. R.

Tiller, J. B.

Wang, Y.

Weierstall, U.

Wilkins, S. W.

T. E. Gureyev, S. Nayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative in-line phase-contrast imaging with multi-energy x rays,” Phys. Rev. Lett. 86, 5827-5830 (2001).
[CrossRef] [PubMed]

Xu, W.

Yamazaki, H.

Zeitoun, P.

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (8)

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. A 12, 1932-1941 (1995).
[CrossRef]

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-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]

B. E. Allman, P. J. McMahon, J. B. Tiller, K. A. Nugent, D. Paganin, A. Barty, I. MacNulty, S. P. Frigo, Y. Wang, and C. C. Retsch, “Noninterferometric quantitative phase imaging with soft x rays,” J. Opt. Soc. Am. A 17, 1732-1743 (2000).
[CrossRef]

M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with Fourier-weighted projections,” J. Opt. Soc. Am. A 25, 701-709 (2008).
[CrossRef]

H. Yamazaki, Y. Kohmura, T. Sakurai, and T. Ishikawa, “Reconstruction of complex-valued electron density with x-ray in-line holograms,” J. Opt. Soc. Am. A 23, 3171-3176 (2006).
[CrossRef]

A. Rosenhahn, R. Barth, F. Staier, T. Simpson, S. Mittler, S. Eisebitt, and M. Grunze, “Digital in-line soft x-ray holography with element contrast,” J. Opt. Soc. Am. A 25, 416-422 (2008).
[CrossRef]

N. Delen and B. Hooker, “Free-space beam propagation between arbitrarily oriented planes based on full diffraction theory: a fast Fourier transform approach,” J. Opt. Soc. Am. A 15, 857-867 (1998).
[CrossRef]

N. Nakajima, “Lensless coherent imaging by a deterministic phase retrieval method with an aperture-array filter,” J. Opt. Soc. Am. A 25, 742-750 (2008).
[CrossRef]

Nat. Photonics (1)

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel X-ray holography,” Nat. Photonics 2, 560-563 (2008).
[CrossRef]

Nature (London) (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 speciments,” Nature (London) 400, 342-344 (1999).
[CrossRef]

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöohr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature (London) 432, 885-888 (2004).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. Lett. (5)

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

T. E. Gureyev, S. Nayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative in-line phase-contrast imaging with multi-energy x rays,” Phys. Rev. Lett. 86, 5827-5830 (2001).
[CrossRef] [PubMed]

N. Nakajima, “Noniterative phase retrieval from a single diffraction intensity pattern by use of an aperture array,” Phys. Rev. Lett. 98, 223901 (2007).
[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, “Hard x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Y. Nishino, Y. Takahashi, N. Imamoto, T. Ishikawa, and K. Maeshima, “Three-dimensional visualization of a human chromosome using coherent x-ray diffraction,” Phys. Rev. Lett. 102, 018101 (2009).
[CrossRef] [PubMed]

Phys. Today (1)

K. A. Nugent, D. Paganin, and T. E. Gureyev, “A phase odyssey,” Phys. Today 54, 27-32 (2001).
[CrossRef]

Other (2)

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

N. Nakajima, “Phase retrieval using the properties of entire functions,” in Advances in Imaging and Electron Physics, P.W.Hawkes, ed., Vol. 93 (Academic, 1995).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram of the measurement system for object reconstruction by the phase-retrieval method. The object function is reconstructed from a single intensity distribution of a diffracted wave through an array of square apertures.

Fig. 2
Fig. 2

Object reconstruction from a high-numerical-aperture (NA) diffraction intensity ( NA = 0.214 ) . (a) Modulus of an original object consisting of seven Gaussian functions. (b), (d) Moduli of reconstructed objects from noiseless and noisy intensities, respectively, in the detector plane. (c) Noiseless modulus in the detector plane. (e) Modulus of another reconstructed object that is backpropagated from the same amplitude distribution as used in (b) by using the Fresnel integral instead of the first Rayleigh–Sommerfeld integral. The only picture of (c) is represented on a base-10 logarithmic grayscale of a normalized modulus truncated to 10 4 for display purposes.

Fig. 3
Fig. 3

Examples of reconstructed objects by backpropagating with the diffraction kernels for spherical and parabolic waves in varying NA. (a), (c), (e) [or (b), (d), (f)] Moduli of reconstructed objects from the retrieved complex amplitude distribution in the diffraction plane for NA = 0.107 , 0.0852 , 0.0746 , respectively, by use of backpropagation with the spherical (or parabolic) diffraction kernel.

Fig. 4
Fig. 4

Original object function containing a depth structure. The object consists of two stops of plus and minus symbols placed at distances of 1.6 μ m and 0.64 μ m , respectively, in front of the object plane. (a), (b) Moduli of the complex amplitudes in the planes immediately behind the plus and the minus stops, respectively, when illuminating with a Gaussian beam. (c), (d) Modulus and phase, respectively, of the complex amplitude in the object plane. (e) Noiseless modulus in the detector plane represented on the same grayscale as in Fig. 2c.

Fig. 5
Fig. 5

Ideal complex amplitudes generated by the backpropagation of the original complex amplitude in Figs. 4c, 4d. (a), (b) [or (c), (d)] Modulus and phase of the backpropagated complex amplitude at the position of 0.64 μ m (or 1.6 μ m ) in front of the object plane.

Fig. 6
Fig. 6

Reconstruction of the object in Fig. 4 from noiseless modulus data. (a), (c), (e) [or (b), (d), (f)] Moduli (or phases) of reconstructed objects at the focal positions of 0.32 μ m , 0.32 μ m , and 1.28 μ m from the object plane, respectively. The complex amplitudes at three positions minimize the NRMS errors in comparison with the corresponding ideal ones in Figs. 4, 5.

Fig. 7
Fig. 7

Reconstruction of the object in Fig. 4 from noisy modulus data. (a), (c), (e) [or (b), (d), (f)] Moduli (or phases) of reconstructed objects at the focal positions of 0.16 μ m , 0.48 μ m , 1.44 μ m from the object plane, respectively. The complex amplitudes at three positions minimize the NRMS errors in comparison with the corresponding ideal ones in Figs. 4, 5.

Equations (22)

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z 3 > π 4 λ [ ( x u ) 2 + ( y v ) 2 ] 2 max ,
z = [ π ( H 2 ) 4 4 λ ] 1 3 .
NA = sin θ = H 2 [ z 2 + ( H 2 ) 2 ] 1 2 ,
δ l = λ 2 NA .
NA limit = ( 2 λ π z + 2 λ ) 1 2 .
F ( x , y ) = 1 2 π σ f ( u , v ) exp ( i k r ) r ( 1 r i k ) z r d u d v ,
r = [ z 2 + ( x u ) 2 + ( y v ) 2 ] 1 2 .
F ( x , y ) = F 1 { F [ f ( u , v ) ] exp [ i 2 π z λ 1 ( λ μ ) 2 ( λ ν ) 2 ] } ,
F [ 1 2 π exp ( i k r ) r ( 1 r i k ) z r ] = exp [ i 2 π z λ 1 ( λ μ ) 2 ( λ ν ) 2 ] ,
K ( ξ , η ) = 1 2 π n = N 2 N 2 1 m = M 2 M 2 1 F ( x , y ) R ( x x n , y y m ) exp ( i k r ) r ( 1 r i k ) l r d x d y ,
r = [ l 2 + ( ξ x ) 2 + ( η y ) 2 ] 1 2 ,
K ( ξ , η ) = e i k l i λ l n = N 2 N 2 1 m = M 2 M 2 1 F ( x , y ) R ( x x n , y y m ) exp { i π λ l [ ( ξ x ) 2 + ( η y ) 2 ] } d x d y .
K [ x n ( 1 + l z ) , y m ( 1 + l z ) ] = e i k l i λ l n = N 2 N 2 1 m = M 2 M 2 1 ρ exp [ i π λ z ( x 2 + y 2 ) ] F ( x , y ) R ( x x n , y y m ) exp { i π λ l ( 1 + l z ) [ ( x x n ) 2 + ( y y m ) 2 ] } d x d y ,
| K [ x n ( 1 + l z ) , y m ( 1 + l z ) ] | 2 | exp [ i π λ z ( x 2 + y 2 ) ] F ( x , y ) R ( x x n , y y m ) d x d y | 2 ,
R ( x , y ) = R ( x , y ) exp [ i π λ l ( 1 + l z ) ( x 2 + y 2 ) ] ,
F 1 { exp [ i π λ z ( x 2 + y 2 ) ] F ( x , y ) R ( x x n , y y m ) d x d y } = F 1 { exp [ i π λ z ( x 2 + y 2 ) ] F ( x , y ) } × F 1 [ R ( x , y ) ] ,
F 1 { exp [ i π λ z ( x 2 + y 2 ) ] F ( x , y ) } = exp [ i π λ z ( x 2 + y 2 ) ] F ( x , y ) exp [ i 2 π λ z ( x u + y v ) ] d x d y = exp [ i π λ z ( u 2 + v 2 ) ] F ( x , y ) exp { i π λ z [ ( u x ) 2 + ( v y ) 2 ] } d x d y .
F 1 { exp [ i π λ z ( x 2 + y 2 ) ] F ( x , y ) } = exp [ i π λ z ( u 2 + v 2 ) ] f b ( u , v ) .
F 1 { exp [ i π λ z ( x 2 + y 2 ) ] F ( x , y ) R ( x x n , y y m ) d x d y } = exp [ i π λ z ( u 2 + v 2 ) ] f b ( u , v ) r ( u , v ) ,
| K ( ξ n ± τ , η m ) | 2 = | K [ x n ( 1 + l z ) ± τ , y m ( 1 + l z ) ] | 2 | exp [ i π λ z ( x 2 + y 2 ) ] F ( x , y ) R ( x x n , y y m ) exp ( i 2 π x τ λ l ) d x d y | 2 ,
F 1 { exp [ i π λ z ( x 2 + y 2 ) ] F ( x , y ) R ( x x n , y y m ) exp ( i 2 π x τ λ l ) d x d y } = exp { i π λ z [ ( u z τ l ) 2 + v 2 ] } f b ( u z τ l , v ) r ( u , v ) .
ER = [ u , v σ | f ( u , v ) f r ( u , v ) | 2 u , v σ | f ( u , v ) | 2 ] 1 2 ,

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