Abstract

A noniterative method of retrieving the two-dimensional phase of a wave field from two intensity measurements is proposed. In the measurements, one records two far-field intensities of the wave field modulated and unmodulated with an exponential filter. The phase retrieval method is based on the solution of the simultaneous equations with unknown coefficients of the two-dimensional discrete Fourier transform for the phase. Then there is no need for the information about the wave field, which is used in iterative phase-retrieval methods. The usefulness of this method is shown in computer-simulated examples of the reconstruction of two-dimensional complex amplitude objects.

© 2003 Optical Society of America

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References

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    [CrossRef]
  2. J. Miao, P. Charalambous, J. Kirz, D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
    [CrossRef]
  3. S. Bajt, A. Barty, K. A. Nugent, M. MaCartney, M. Wall, D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
    [CrossRef] [PubMed]
  4. B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, S. A. Werner, “Phase radiography with neutrons,” Nature 408, 158–159 (2000).
    [CrossRef]
  5. H. A. Ferwerda, “The phase reconstruction problem for wave amplitudes and coherence functions,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 13–39.
    [CrossRef]
  6. W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).
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    [CrossRef]
  8. M. H. Hayes, “The unique reconstruction of multidimensional sequences from Fourier transform magnitude or phase,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 195–230.
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    [CrossRef]
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2002 (1)

2000 (2)

S. Bajt, A. Barty, K. A. Nugent, M. MaCartney, M. Wall, D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, S. A. Werner, “Phase radiography with neutrons,” Nature 408, 158–159 (2000).
[CrossRef]

1999 (1)

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

1996 (1)

1995 (1)

1989 (1)

1988 (1)

1987 (2)

1983 (1)

1982 (3)

1972 (1)

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

Allman, B. E.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, S. A. Werner, “Phase radiography with neutrons,” Nature 408, 158–159 (2000).
[CrossRef]

Arif, M.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, S. A. Werner, “Phase radiography with neutrons,” Nature 408, 158–159 (2000).
[CrossRef]

Bajt, S.

S. Bajt, A. Barty, K. A. Nugent, M. MaCartney, M. Wall, D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

Barty, A.

S. Bajt, A. Barty, K. A. Nugent, M. MaCartney, M. Wall, D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

Charalambous, P.

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

Dainty, J. C.

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

Ferwerda, H. A.

H. A. Ferwerda, “The phase reconstruction problem for wave amplitudes and coherence functions,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 13–39.
[CrossRef]

Fiddy, M. A.

G. Ross, M. A. Fiddy, M. Nieto-Vesperinas, “The inverse scattering problem in structural determinations,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), pp. 15–71.
[CrossRef]

M. A. Fiddy, “The role of analyticity in image recovery,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 499–529.

Fienup, J. R.

Gerchberg, R. W.

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

Gureyev, T. E.

Hayes, M. H.

M. H. Hayes, “The unique reconstruction of multidimensional sequences from Fourier transform magnitude or phase,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 195–230.

Jacobson, D. L.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, S. A. Werner, “Phase radiography with neutrons,” Nature 408, 158–159 (2000).
[CrossRef]

Kirz, J.

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

Levi, A.

A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 277–320.

MaCartney, M.

S. Bajt, A. Barty, K. A. Nugent, M. MaCartney, M. Wall, D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

McMahon, P. J.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, S. A. Werner, “Phase radiography with neutrons,” Nature 408, 158–159 (2000).
[CrossRef]

Miao, J.

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

Nakajima, N.

Nieto-Vesperinas, M.

G. Ross, M. A. Fiddy, M. Nieto-Vesperinas, “The inverse scattering problem in structural determinations,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), pp. 15–71.
[CrossRef]

Nugent, K. A.

Paganin, D.

D. Paganin, K. A. Nugent, “Phase measurement of waves that obey nonlinear equations,” Opt. Lett. 27, 622–624 (2002).
[CrossRef]

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, S. A. Werner, “Phase radiography with neutrons,” Nature 408, 158–159 (2000).
[CrossRef]

S. Bajt, A. Barty, K. A. Nugent, M. MaCartney, M. Wall, D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

Roberts, A.

Ross, G.

G. Ross, M. A. Fiddy, M. Nieto-Vesperinas, “The inverse scattering problem in structural determinations,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), pp. 15–71.
[CrossRef]

Saxton, W. O.

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

W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).

Sayre, D.

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

Stark, H.

A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 277–320.

Teague, M. R.

Walker, J. G.

Wall, M.

S. Bajt, A. Barty, K. A. Nugent, M. MaCartney, M. Wall, D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

Werner, S. A.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, S. A. Werner, “Phase radiography with neutrons,” Nature 408, 158–159 (2000).
[CrossRef]

Appl. Opt. (3)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

Nature (2)

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

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, S. A. Werner, “Phase radiography with neutrons,” Nature 408, 158–159 (2000).
[CrossRef]

Opt. Lett. (1)

Optik (Stuttgart) (1)

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

Ultramicroscopy (1)

S. Bajt, A. Barty, K. A. Nugent, M. MaCartney, M. Wall, D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

Other (8)

H. A. Ferwerda, “The phase reconstruction problem for wave amplitudes and coherence functions,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 13–39.
[CrossRef]

W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).

G. Ross, M. A. Fiddy, M. Nieto-Vesperinas, “The inverse scattering problem in structural determinations,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), pp. 15–71.
[CrossRef]

M. H. Hayes, “The unique reconstruction of multidimensional sequences from Fourier transform magnitude or phase,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 195–230.

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 277–320.

M. A. Fiddy, “The role of analyticity in image recovery,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 499–529.

N. Nakajima, “Phase retrieval using the properties of entire functions,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed., Vol. 93 (Academic, New York, 1995), pp. 109–171.
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Axes of coordinates used in the formulation of the method: the rotational angle between the object function f(u, v) and the exponential filter exp(-2πcU) is represented by θ.

Fig. 2
Fig. 2

Examples of the reconstruction by the present method: (a) modulus and (b) phase of the original object function, (c) and (d) [or (e) and (f)] are the modulus and the phase, respectively, of the reconstructed object function from noiseless [or noisy (SNR = 210)] Fourier intensities. The units of the coordinates u and v are arbitrary.

Fig. 3
Fig. 3

Same as Fig. 2 except for the value of noisy SNR (SNR = 690).

Equations (9)

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

F1x, y=σ fu, vexp-i 2πdλxu+yvdudv,
F2x, y=σ fu, vexp-2πcu cos θ+v sin θexp-i 2πdλxu+yvdudv,
F2x, y=F1x-ic1, y-ic2,
F1x, y=Mx, yexpiϕx, y.
|F1x-ic1, y-ic2|=|Mx-ic1, y-ic2|×exp-Imϕx-ic1, y-ic2,
ln|F2x, y||Mx-ic1, y-ic2|=-Imϕx-ic1, y-ic2,
ϕx, yn=-N/2+1N/2-1m=-M/2+1M/2-1 anm exp2πinK x+mL y,
Imϕx-ic1, y-ic2n=-N/2+1N/2-1m=-M/2+1M/2-1×Im anm cos2πnK x+mL y+Re anm sin2πnK x+mL y×exp2πc1nK+c2mL,
ln|F2x, y||Mx-ic1, y-ic2|2πc1a-n=-N/2+1N/2-1m=-M/2+1M/2-1×Im anm cos2πnK x+mL y+Re anm sin2πnK x+mL y×exp2πc1nK+c2mL,

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