Abstract

A scanning imaging system is proposed that is capable of reconstructing not only the modulus but the phase of an object without a holographic reference wave. The reconstruction involves phase retrieval from two intensity measurements: the intensity of the correlation function of the object and a probe beam and the intensity of the correlation function of the object and the probe beam that is modulated with an exponential filter. A computer simulation shows that the reconstruction of a complex-valued object can be done from a relatively small data set.

© 1996 Optical Society of America

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References

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  1. N. E. Hurt, Phase Retrieval and Zero Crossings (Kluwer, Dordrecht, The Netherlands, 1989).
  2. A. Levi, H. Stark, in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 277–320.
  3. J. C. Dainty, J. R. Fienup, in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.
  4. N. Nakajima, in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, New York, 1995), Vol. 93, pp. 109–171.
    [CrossRef]
  5. R. H. T. Bates, J. M. Rodenburg, Ultramicroscopy 31, 303 (1989).
    [CrossRef]
  6. J. M. Rodenburg, R. H. T. Bates, Philos. Trans. R. Soc. London Ser. A 339, 521 (1992).
    [CrossRef]
  7. N. Nakajima, J. Opt. Soc. Am. A 4, 154 (1987).
    [CrossRef]
  8. N. Nakajima, J. Opt. Soc. Am. A 5, 257 (1988).
    [CrossRef]

1992 (1)

J. M. Rodenburg, R. H. T. Bates, Philos. Trans. R. Soc. London Ser. A 339, 521 (1992).
[CrossRef]

1989 (1)

R. H. T. Bates, J. M. Rodenburg, Ultramicroscopy 31, 303 (1989).
[CrossRef]

1988 (1)

1987 (1)

Bates, R. H. T.

J. M. Rodenburg, R. H. T. Bates, Philos. Trans. R. Soc. London Ser. A 339, 521 (1992).
[CrossRef]

R. H. T. Bates, J. M. Rodenburg, Ultramicroscopy 31, 303 (1989).
[CrossRef]

Dainty, J. C.

J. C. Dainty, J. R. Fienup, in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

Fienup, J. R.

J. C. Dainty, J. R. Fienup, in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

Hurt, N. E.

N. E. Hurt, Phase Retrieval and Zero Crossings (Kluwer, Dordrecht, The Netherlands, 1989).

Levi, A.

A. Levi, H. Stark, in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 277–320.

Nakajima, N.

N. Nakajima, J. Opt. Soc. Am. A 5, 257 (1988).
[CrossRef]

N. Nakajima, J. Opt. Soc. Am. A 4, 154 (1987).
[CrossRef]

N. Nakajima, in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, New York, 1995), Vol. 93, pp. 109–171.
[CrossRef]

Rodenburg, J. M.

J. M. Rodenburg, R. H. T. Bates, Philos. Trans. R. Soc. London Ser. A 339, 521 (1992).
[CrossRef]

R. H. T. Bates, J. M. Rodenburg, Ultramicroscopy 31, 303 (1989).
[CrossRef]

Stark, H.

A. Levi, H. Stark, in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 277–320.

J. Opt. Soc. Am. A (2)

Philos. Trans. R. Soc. London Ser. A (1)

J. M. Rodenburg, R. H. T. Bates, Philos. Trans. R. Soc. London Ser. A 339, 521 (1992).
[CrossRef]

Ultramicroscopy (1)

R. H. T. Bates, J. M. Rodenburg, Ultramicroscopy 31, 303 (1989).
[CrossRef]

Other (4)

N. E. Hurt, Phase Retrieval and Zero Crossings (Kluwer, Dordrecht, The Netherlands, 1989).

A. Levi, H. Stark, in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 277–320.

J. C. Dainty, J. R. Fienup, in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

N. Nakajima, in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, New York, 1995), Vol. 93, pp. 109–171.
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Schematic diagram of the geometry of the scanning phase-retrieval system. The object is scanned with respect to each Fourier transform of two probes that are modulated with and without an exponential filter. The mask in front of the detector selects the zero-frequency component to yield a spatial integration of the amplitude in the object plane.

Fig. 2
Fig. 2

Original object and probe functions used in the computer simulation: (a) modulus (dashed curve) and (b) phase of the object function; (c) Gaussian function (solid curve) used as a known probe function in the x coordinate’s plane of Fig. 1 and (dashed curve) the exponential function by which the probe is modulated. The Fourier modulus of the probe function in (c) is represented by the dashed curve in (a).

Fig. 3
Fig. 3

Measurable intensities in the Fourier plane with the mask shown in Fig. 1. (a), (b) Intensity of the correlation of the object and the probe and intensity of the correlation of the object and the exponential-filtered probe, respectively.

Fig. 4
Fig. 4

Reconstructed object: (a) modulus, (b) phase.

Equations (11)

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p ( u ) = - P ( x ) exp ( - 2 π i u x ) d x ,
p ( u - i c ) = - P ( x ) exp ( - 2 π c x ) exp ( - 2 π i u x ) d x ,
g ( τ ) 2 = | - p ( u ) f ( u - τ ) d u | 2 ,
g ^ ( τ ) 2 = | - p ( u - i c ) f ( u - τ ) d u | 2 ,
g ^ ( τ ) 2 = g ( τ - i c ) 2 .
g ( τ ) = m ( τ ) exp [ i ϕ ( τ ) ] .
g ( τ ) 2 = m 2 ( τ ) ,
g ^ ( τ ) 2 = m ( τ - i c ) 2 exp [ - 2 Im ϕ ( τ - i c ) ] ,
ln  g ^ ( τ ) m ( τ - i c ) = - Im  ϕ ( τ - i c ) .
G ( x ) = P ( x ) F ( - x ) ,
F ( - x ) = G ( x ) P * ( x ) P ( x ) 2 + ɛ ,

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