Abstract

When a very intense beam is used for illuminating an object in coherent x-ray diffraction imaging, the intensities at the center of the diffraction pattern for the object are cut off by a beam stop that is utilized to block the intense beam. Until now, only iterative phase-retrieval methods have been applied to object reconstruction from a single diffraction pattern with a deficiency of central data due to a beam stop. As an alternative method, I present a noniterative solution in which an interpolation method based on the sampling theorem for the missing data is used for object reconstruction with our previously proposed phase-retrieval method using an aperture-array filter. Computer simulations demonstrate the reconstruction of a complex-amplitude object from a single diffraction pattern with a missing data area, which is generally difficult to treat with the iterative methods because a nonnegativity constraint cannot be used for such an object.

© 2010 Optical Society of America

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References

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  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]
  2. J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
    [CrossRef] [PubMed]
  3. S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstrall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101(R) (2003).
    [CrossRef]
  4. 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]
  5. I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87, 195505 (2001).
    [CrossRef] [PubMed]
  6. K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, “Unique phase recovery for nonperiodic objects,” Phys. Rev. Lett. 91, 203902 (2003).
    [CrossRef] [PubMed]
  7. J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. H. Risbud, and T. Ishikawa, “Quantitative image reconstruction of GaN quantum dots from oversampled diffraction intensities alone,” Phys. Rev. Lett. 95, 085503 (2005).
    [CrossRef] [PubMed]
  8. J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
    [CrossRef]
  9. J. R. Fienup, “Lensless coherent imaging by phase retrieval with an illumination pattern constraint,” Opt. Express 14, 498–508 (2006).
    [CrossRef] [PubMed]
  10. J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662–1669 (1998).
    [CrossRef]
  11. 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]
  12. J. L. Harris, “Diffraction and resolving power,” J. Opt. Soc. Am. 54, 931–936 (1964).
    [CrossRef]
  13. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), pp. 66–74.
  14. R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, “Potential for biomolecular imaging with femtosecond x-ray pulses,” Nature 406, 752–757 (2000).
    [CrossRef] [PubMed]

2008 (1)

2006 (2)

2005 (1)

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. H. Risbud, and T. Ishikawa, “Quantitative image reconstruction of GaN quantum dots from oversampled diffraction intensities alone,” Phys. Rev. Lett. 95, 085503 (2005).
[CrossRef] [PubMed]

2003 (2)

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

K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, “Unique phase recovery for nonperiodic objects,” Phys. Rev. Lett. 91, 203902 (2003).
[CrossRef] [PubMed]

2002 (1)

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

2001 (1)

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87, 195505 (2001).
[CrossRef] [PubMed]

2000 (1)

R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, “Potential for biomolecular imaging with femtosecond x-ray pulses,” Nature 406, 752–757 (2000).
[CrossRef] [PubMed]

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]

1998 (1)

1987 (1)

1964 (1)

Anderson, E. H.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

Barty, A.

Beetz, T.

Chapman, H. N.

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]

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

K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, “Unique phase recovery for nonperiodic objects,” Phys. Rev. Lett. 91, 203902 (2003).
[CrossRef] [PubMed]

J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662–1669 (1998).
[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]

Cui, C.

Fienup, J. R.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), pp. 66–74.

Hajdu, J.

R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, “Potential for biomolecular imaging with femtosecond x-ray pulses,” Nature 406, 752–757 (2000).
[CrossRef] [PubMed]

Harris, J. L.

Hau-Riege, S. P.

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]

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

He, H.

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]

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

Hodgson, K. O.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

Howells, M. R.

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]

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

Ishikawa, T.

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. H. Risbud, and T. Ishikawa, “Quantitative image reconstruction of GaN quantum dots from oversampled diffraction intensities alone,” Phys. Rev. Lett. 95, 085503 (2005).
[CrossRef] [PubMed]

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

Jacobsen, C.

Johnson, B.

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. H. Risbud, and T. Ishikawa, “Quantitative image reconstruction of GaN quantum dots from oversampled diffraction intensities alone,” Phys. Rev. Lett. 95, 085503 (2005).
[CrossRef] [PubMed]

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[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]

Kohmura, Y.

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. H. Risbud, and T. Ishikawa, “Quantitative image reconstruction of GaN quantum dots from oversampled diffraction intensities alone,” Phys. Rev. Lett. 95, 085503 (2005).
[CrossRef] [PubMed]

Lai, B.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

Mancuso, A. P.

K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, “Unique phase recovery for nonperiodic objects,” Phys. Rev. Lett. 91, 203902 (2003).
[CrossRef] [PubMed]

Marchesini, S.

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]

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

Miao, J.

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. H. Risbud, and T. Ishikawa, “Quantitative image reconstruction of GaN quantum dots from oversampled diffraction intensities alone,” Phys. Rev. Lett. 95, 085503 (2005).
[CrossRef] [PubMed]

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

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]

J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662–1669 (1998).
[CrossRef]

Nakajima, N.

Neutze, R.

R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, “Potential for biomolecular imaging with femtosecond x-ray pulses,” Nature 406, 752–757 (2000).
[CrossRef] [PubMed]

Nishino, Y.

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. H. Risbud, and T. Ishikawa, “Quantitative image reconstruction of GaN quantum dots from oversampled diffraction intensities alone,” Phys. Rev. Lett. 95, 085503 (2005).
[CrossRef] [PubMed]

Noy, A.

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]

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

Nugent, K. A.

K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, “Unique phase recovery for nonperiodic objects,” Phys. Rev. Lett. 91, 203902 (2003).
[CrossRef] [PubMed]

Peele, A. G.

K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, “Unique phase recovery for nonperiodic objects,” Phys. Rev. Lett. 91, 203902 (2003).
[CrossRef] [PubMed]

Pfeifer, M. A.

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87, 195505 (2001).
[CrossRef] [PubMed]

Pitney, J. A.

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87, 195505 (2001).
[CrossRef] [PubMed]

Risbud, S. H.

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. H. Risbud, and T. Ishikawa, “Quantitative image reconstruction of GaN quantum dots from oversampled diffraction intensities alone,” Phys. Rev. Lett. 95, 085503 (2005).
[CrossRef] [PubMed]

Robinson, I. K.

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87, 195505 (2001).
[CrossRef] [PubMed]

Rosen, R.

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]

J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662–1669 (1998).
[CrossRef]

Shapiro, D.

Song, C.

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. H. Risbud, and T. Ishikawa, “Quantitative image reconstruction of GaN quantum dots from oversampled diffraction intensities alone,” Phys. Rev. Lett. 95, 085503 (2005).
[CrossRef] [PubMed]

Spence, J. C. H.

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]

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

van der Spoel, D.

R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, “Potential for biomolecular imaging with femtosecond x-ray pulses,” Nature 406, 752–757 (2000).
[CrossRef] [PubMed]

Vartanyants, I. A.

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87, 195505 (2001).
[CrossRef] [PubMed]

Weckert, E.

R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, “Potential for biomolecular imaging with femtosecond x-ray pulses,” Nature 406, 752–757 (2000).
[CrossRef] [PubMed]

Weierstall, U.

Weierstrall, U.

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

Williams, G. J.

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87, 195505 (2001).
[CrossRef] [PubMed]

Wouts, R.

R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, “Potential for biomolecular imaging with femtosecond x-ray pulses,” Nature 406, 752–757 (2000).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

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

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]

R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, “Potential for biomolecular imaging with femtosecond x-ray pulses,” Nature 406, 752–757 (2000).
[CrossRef] [PubMed]

Opt. Express (1)

Phys. Rev. B (1)

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

Phys. Rev. Lett. (4)

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87, 195505 (2001).
[CrossRef] [PubMed]

K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, “Unique phase recovery for nonperiodic objects,” Phys. Rev. Lett. 91, 203902 (2003).
[CrossRef] [PubMed]

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. H. Risbud, and T. Ishikawa, “Quantitative image reconstruction of GaN quantum dots from oversampled diffraction intensities alone,” Phys. Rev. Lett. 95, 085503 (2005).
[CrossRef] [PubMed]

Other (1)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), pp. 66–74.

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of the measurement system. The filled apertures around the center of the array filter work as a beam stop that blocks the direct beam. The object function is reconstructed from a single intensity pattern of a diffracted wave through an array of square apertures by using the phase-retrieval method with the interpolation for the missing data due to the beam stop.

Fig. 2
Fig. 2

Original object function used in the simulations: (a) modulus and (b) phase of an object with a rectangular extent of 0.844 μm × 0.844 μm (where the values of the phase are in the range 2.05 to 1.16 rad ), and (c) intensity distribution in the detector plane of Fig. 1, where the central intensities are lost by a beam stop of 3 × 3 filled apertures and the direct beam is set to be zero. Only (c) is represented on a base-10 logarithmic grey scale of a normalized intensity truncated to 10 3 for display purposes.

Fig. 3
Fig. 3

Reconstruction of the object function shown in Fig. 2 from a noiseless intensity distribution in the detector plane: (a) and (b) [(c) and (d)] are the modulus and phase, respectively, of a reconstructed object from the noiseless intensity in Fig. 2c without (or with) interpolation for the missing data. (e) and (f) are the modulus and phase, respectively, of a reconstructed object from an intensity distribution obtained without the beam stop and the direct beam.

Fig. 4
Fig. 4

Dependence of the NRMS error (ER) on the normalized deviations L / 2 σ j from the true extent 2 σ j ( j = u , v ) of the autocorrelation of the object, where the same deviation in the directions of u and v is used at each point of the abscissa. The circles and squares indicate the NRMS errors in the cases where 3 × 3 and 5 × 5 filled apertures, respectively, are used as a beam stop.

Fig. 5
Fig. 5

Reconstruction of the object function shown in Fig. 2 from a noisy intensity distribution in the case where an intense direct beam with a Gaussian modulus is added to the object function: (a) modulus of the wave field immediately in front of the array filter with the beam stop of 3 × 3 filled apertures, which is represented on a base-10 logarithmic grey scale of a normalized modulus truncated to 10 5 for display purposes. (b) and (c) [(d) and (e)] are the modulus and phase, respectively, of an object reconstructed from the noisy modulus for SNR = 184 (22.6 dB) [ SNR = 46 (16.6 dB)] with interpolation for the missing data.

Equations (14)

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

P ( x , y ) = exp [ i π λ z ( x 2 + y 2 ) ] [ b ( u , v ) + f ( u , v ) ] exp [ i 2 π λ z ( x u + y v ) ] d u d v .
G ( ξ , η ) = P ( x , y ) A ( x , y ) exp { i π λ l [ ( x ξ ) 2 + ( y η ) 2 ] } d x d y ,
A ( x , y ) = n = N / 2 N / 2 1 m = M / 2 M / 2 1 R ( x x n , y y m ) ,
P ( x , y ) = exp [ i π λ z ( x 2 + y 2 ) ] [ B ( x , y ) + F ( x , y ) ] ,
G ( ξ , η ) = [ B ( x , y ) + F ( x , y ) ] A ( x , y ) exp { i π α λ l [ ( x ξ α ) 2 + ( y η α ) 2 + l ( ξ 2 + η 2 ) z α 2 ] } d x d y ,
| G ( α x n , α y m ) | 2 = | [ B ( x , y ) + F ( x , y ) ] R ( x x n , y y m ) d x d y | 2 ,
R ( x , y ) = R ( x , y ) exp [ i π α λ l ( x 2 + y 2 ) ] .
| G ( α x n ± τ , α y m ) | 2 = | [ B ( x , y ) + F ( x , y ) ] R ( x x n , y y m ) exp ( i 2 π λ l x τ ) d x d y | 2 ,
| G ( α x n , α y m ) | 2 = | F ( x , y ) R ( x x n , y y m ) d x d y | 2 .
I 1 [ | G ( α x n , α y m ) | 2 ] = g ( u , v ) g ( u , v ) ,
g ( u , v ) = f ( u , v ) r ( u , v ) ,
| G ( α x n , C ) | 2 k = K / 2 K / 2 1 | G ( α k L , C ) | 2 sinc [ L ( x n k L ) ] ,
| G ( α x n , C ) | 2 = k = K / 2 K / 2 1 | G ( α k L , C ) | 2 sinc [ L ( x n k L ) ] , ( n = K + 2 n c 2 , , n c 1 , n c + 1 , , K + 2 n c 2 ) .
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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