Abstract

The reconstruction of an object with a method using a dual exposure single inverse Fourier transform is investigated. The method calculates phase information in the Fourier plane to perform the inverse Fourier transform. The phase information in the Fourier plane is calculated from the intensity distributions formed by an object with and without a reference electric field. The method successfully reconstructs an object in a simple and fast manner. For the practical use of the method, the effects of the intensity digitization and the noise in the intensity distributions are examined in reconstructing an object.

© 2009 Optical Society of America

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References

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  1. G. R. Ayers and J. C. Dainty, “Iterative blind deconvolution method and its applications,” Opt. Lett. 13, 547-549 (1988).
    [CrossRef] [PubMed]
  2. R. B. Holmes, K. Hughes, P. Fairchild, B. Spivey, and A. Smith, “Description and simulation of an active imaging technique utilizing two speckle fields: iterative reconstructors,” J. Opt. Soc. Am. A 19, 458-471 (2002).
    [CrossRef]
  3. A. Diaspro, Confocal and Two-Photon Microscopy: Foundations, Applications, and Advances (Wiley-Liss, 2002), Chap. 12.
  4. J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized noncrystalline specimens,” Nature 400, 342-344 (1999).
    [CrossRef]
  5. J. Miao, T. Ishikawa, B. Johnson, E. Anderson, B. Lai, and K. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
    [CrossRef] [PubMed]
  6. 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]
  7. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Weimar) 35, 237-246 (1972).
  8. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758-2769 (1982).
    [CrossRef] [PubMed]
  9. 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]
  10. T. M. Jeong, D.-K. Ko, and J. Lee, “Method of reconstructing wavefront aberrations from the intensity measurement,” Opt. Lett. 32, 3507-3509 (2007).
    [CrossRef] [PubMed]
  11. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Chap. 9.
  12. S. H. Lee, P. Naulleau, K. A. Goldberg, C. H. Cho, S. T. Jeong, and J. Boker, “Extreme-ultraviolet lensless Fourier-transform holography,” Appl. Opt. 40, 2655-2661 (2001).
    [CrossRef]
  13. 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]
  14. 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]
  15. M. Guizar-Sicairos and J. R. Fienup, “Holography with extended reference by autocorrelation linear differential operation,” Opt. Express 15, 17592-17612 (2007).
    [CrossRef] [PubMed]
  16. “Methods for reporting optical aberrations of eyes,” ANSI Z80.28-2004 (American National Standards Institute, 2004).
  17. T. M. Jeong, D.-K. Ko, and J. Lee, “Generalized ray-transfer matrix for an optical element having an arbitrary wavefront aberration,” Opt. Lett. 30, 3009-3011 (2005).
    [CrossRef] [PubMed]
  18. C. Song, D. Ramunno-Johnson, Y. Nishino, Y. Kohnura, T. Ishikawa, C.-C. Chen, T.-K. Lee, and J. Miao, “Phase retrieval from exactly oversampled diffraction intensity through deconvolution,” Phys. Rev. B 75, 012102(2007).
    [CrossRef]

2007 (4)

2005 (1)

2002 (2)

2001 (1)

1999 (1)

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

1998 (1)

1988 (1)

1987 (1)

1982 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Weimar) 35, 237-246 (1972).

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]

Anderson, E.

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

Ayers, G. R.

Boker, J.

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]

Chapman, H. N.

Charalambous, P.

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

Chen, C.-C.

C. Song, D. Ramunno-Johnson, Y. Nishino, Y. Kohnura, T. Ishikawa, C.-C. Chen, T.-K. Lee, and J. Miao, “Phase retrieval from exactly oversampled diffraction intensity through deconvolution,” Phys. Rev. B 75, 012102(2007).
[CrossRef]

Cho, C. H.

Dainty, J. C.

Diaspro, A.

A. Diaspro, Confocal and Two-Photon Microscopy: Foundations, Applications, and Advances (Wiley-Liss, 2002), Chap. 12.

Fairchild, P.

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]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Weimar) 35, 237-246 (1972).

Goldberg, K. A.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Chap. 9.

Guizar-Sicairos, M.

Hodgson, K.

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

Holmes, R. B.

Hughes, K.

Ishikawa, T.

C. Song, D. Ramunno-Johnson, Y. Nishino, Y. Kohnura, T. Ishikawa, C.-C. Chen, T.-K. Lee, and J. Miao, “Phase retrieval from exactly oversampled diffraction intensity through deconvolution,” Phys. Rev. B 75, 012102(2007).
[CrossRef]

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

Jeong, S. T.

Jeong, T. M.

Johnson, B.

J. Miao, T. Ishikawa, B. Johnson, E. Anderson, B. Lai, and K. 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 micrometer-sized noncrystalline specimens,” Nature 400, 342-344 (1999).
[CrossRef]

Ko, D.-K.

Kohnura, Y.

C. Song, D. Ramunno-Johnson, Y. Nishino, Y. Kohnura, T. Ishikawa, C.-C. Chen, T.-K. Lee, and J. Miao, “Phase retrieval from exactly oversampled diffraction intensity through deconvolution,” Phys. Rev. B 75, 012102(2007).
[CrossRef]

Lai, B.

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

Lee, J.

Lee, S. H.

Lee, T.-K.

C. Song, D. Ramunno-Johnson, Y. Nishino, Y. Kohnura, T. Ishikawa, C.-C. Chen, T.-K. Lee, and J. Miao, “Phase retrieval from exactly oversampled diffraction intensity through deconvolution,” Phys. Rev. B 75, 012102(2007).
[CrossRef]

Miao, J.

C. Song, D. Ramunno-Johnson, Y. Nishino, Y. Kohnura, T. Ishikawa, C.-C. Chen, T.-K. Lee, and J. Miao, “Phase retrieval from exactly oversampled diffraction intensity through deconvolution,” Phys. Rev. B 75, 012102(2007).
[CrossRef]

J. Miao, T. Ishikawa, B. Johnson, E. Anderson, B. Lai, and K. 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 micrometer-sized noncrystalline 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]

Naulleau, P.

Nishino, Y.

C. Song, D. Ramunno-Johnson, Y. Nishino, Y. Kohnura, T. Ishikawa, C.-C. Chen, T.-K. Lee, and J. Miao, “Phase retrieval from exactly oversampled diffraction intensity through deconvolution,” Phys. Rev. B 75, 012102(2007).
[CrossRef]

Paganin, D. M.

Pavlov, K. M.

Podorov, S. G.

Ramunno-Johnson, D.

C. Song, D. Ramunno-Johnson, Y. Nishino, Y. Kohnura, T. Ishikawa, C.-C. Chen, T.-K. Lee, and J. Miao, “Phase retrieval from exactly oversampled diffraction intensity through deconvolution,” Phys. Rev. B 75, 012102(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]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Weimar) 35, 237-246 (1972).

Sayre, D.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized noncrystalline 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]

Smith, A.

Song, C.

C. Song, D. Ramunno-Johnson, Y. Nishino, Y. Kohnura, T. Ishikawa, C.-C. Chen, T.-K. Lee, and J. Miao, “Phase retrieval from exactly oversampled diffraction intensity through deconvolution,” Phys. Rev. B 75, 012102(2007).
[CrossRef]

Spivey, B.

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]

Appl. Opt. (2)

J. Opt. Soc. Am. A (3)

Nature (1)

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

Opt. Express (2)

Opt. Lett. (3)

Optik (Weimar) (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Weimar) 35, 237-246 (1972).

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 (1)

C. Song, D. Ramunno-Johnson, Y. Nishino, Y. Kohnura, T. Ishikawa, C.-C. Chen, T.-K. Lee, and J. Miao, “Phase retrieval from exactly oversampled diffraction intensity through deconvolution,” Phys. Rev. B 75, 012102(2007).
[CrossRef]

Phys. Rev. Lett. (1)

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

Other (3)

A. Diaspro, Confocal and Two-Photon Microscopy: Foundations, Applications, and Advances (Wiley-Liss, 2002), Chap. 12.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Chap. 9.

“Methods for reporting optical aberrations of eyes,” ANSI Z80.28-2004 (American National Standards Institute, 2004).

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Figures (8)

Fig. 1
Fig. 1

Optical layout for reconstructing a complex-valued object from the intensity measurement.

Fig. 2
Fig. 2

Test object used for the reconstruction. r i is the inner radius defining the size of the test object, and r o is the outer radius defining the size of the reference electric field.

Fig. 3
Fig. 3

Intensity distributions calculated from the test object, the test object with the reference, and the reference. Intensity distributions at a certain position from the Fourier plane were calculated using a focus-shift method. The reconstructed images were calculated from the intensity distributions. Reconstructed wavefronts in the bottom were the added defocus for the focus shift and calculated only for the test object because only the test object had transmission.

Fig. 4
Fig. 4

(a) Test object and reconstructed image. In this figure, the reconstructed image was calculated from the intensity distributions at a position of 37 mm from the Fourier plane. (b) Vertical line-out data from the test object and the reconstructed image. The vertical line-out data were taken from the line-cuts in Fig. 4a.

Fig. 5
Fig. 5

Dependence of the wrong determination of the inner and the outer radii of the reference on the reconstructed image.

Fig. 6
Fig. 6

(a) One vertical line from the calculated correction factor calculated from the intensity distributions in the range of [0, π]. (b) One vertical line from the phase in the Fourier plane by the reference electric field. (c) Calculated phase information in the Fourier plane for the test object. The calculated phase information was used in the inverse Fourier transform to reconstruct the test object. (d) Deviation of the phase information from the phase directly calculated from the test object.

Fig. 7
Fig. 7

Reconstructed images obtaining by assuming the use of CCD cameras having different dynamic ranges and sampling rates.

Fig. 8
Fig. 8

Reconstructed images from the intensity distributions with the increase of noise intensity. The use of a 12   bit CCD camera is assumed, and the variance of the noise intensity is normalized to the maximum intensity (4096 counts) of the 12   bit CCD camera. The standard deviation of the noise intensity in the figure shows the intensity level with the 12   bit CCD camera.

Equations (7)

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E 2 ( x 2 , y 2 ) E 1 ( x 1 , y 1 ) P ( x 1 , y 1 ) exp [ i k W ( x 1 , y 1 ) ] exp [ i k f ( x 1 x 2 + y 1 y 2 ) ] d x 1 d y 1 .
E 2 ( x 2 , y 2 ) r = 0 r = r i P ( x 1 , y 1 ) E 1 ( x 1 , y 1 ) exp [ i k W ( x 1 , y 1 ) ] exp [ i k f ( x 1 x 2 + y 1 y 2 ) ] d x 1 d y 1 + r = r i r = r o E 1 ( x 1 , y 1 ) exp [ i k W ( x 1 , y 1 ) ] exp [ i k f ( x 1 x 2 + y 1 y 2 ) ] d x 1 d y 1 = | A ( x 2 , y 2 ) | exp [ i ϕ ( x 2 , y 2 ) ] + | R ( x 2 , y 2 ) | exp [ i { ϕ r ( x 2 , y 2 ) + ϕ 0 } ] = | A ( x 2 , y 2 ) | exp [ i ϕ ( x 2 , y 2 ) ] + | R ( x 2 , y 2 ) | exp [ i { ϕ r ( x 2 , y 2 ) + ϕ 0 } ] .
I 2 ( x 2 , y 2 ) = | A ( x 2 , y 2 ) | 2 + | R ( x 2 , y 2 ) | 2 + 2 | A ( x 2 , y 2 ) | | R ( x 2 , y 2 ) | cos [ ϕ ( x 2 , y 2 ) ϕ r ( x 2 , y 2 ) ] .
ϕ ( x 2 , y 2 ) = ϕ r ( x 2 , y 2 ) + cos 1 [ D ( x 2 , y 2 ) ] ,
D ( x 2 , y 2 ) = I 2 ( x 2 , y 2 ) | A ( x 2 , y 2 ) | 2 | R ( x 2 , y 2 ) | 2 2 | A ( x 2 , y 2 ) | | R ( x 2 , y 2 ) | .
E 1 ( x 1 , y 1 ) P ( x 1 , y 1 ) exp [ i k W ( x 1 , y 1 ) ] exp [ i k f ( x 1 x 2 + y 1 y 2 ) ] d x 1 d y 1 = | A ( x 2 , y 2 ) | exp [ i { ϕ r ( x 2 , y 2 ) + cos 1 D ( x 2 , y 2 ) } ] .
E 1 ( x 1 , y 1 ) P ( x 1 , y 1 ) exp [ i k W ( x 1 , y 1 ) ] = F 1 [ | A ( x 2 , y 2 ) | exp [ i { ϕ r ( x 2 , y 2 ) + cos 1 D ( x 2 , y 2 ) } ] ] = | B ( x 1 , y 1 ) | exp [ i ϕ i n v ( x 1 , y 1 ) ] .

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