Abstract

The reconstruction of a complex-valued object from intensity measurements in the Fresnel zone region is considered. The reconstruction is based on the phase-retrieval method from Fresnel zone intensity data obtained by modulation of the object with a known Gaussian function and with its shifted functions along the horizontal and the vertical directions in rectangular coordinates. Two types of reconstruction system are presented. In the two systems a Gaussian amplitude filter and a Gaussian illuminating beam are used as the known Gaussian functions. Computer-simulated examples in two dimensions illustrate the performance of the reconstruction in these systems.

© 1998 Optical Society of America

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References

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  1. 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]
  2. W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).
  3. 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]
  4. 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.
  5. 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.
  6. 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.
  7. N. E. Hurt, Phase Retrieval and Zero Crossings (Kluwer, Dordrecht, The Netherlands, 1989).
  8. 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).
  9. M. R. Teague, “Irradiance moments: their propagation and use for unique retrieval of phase,” J. Opt. Soc. Am. 72, 1199–1209 (1982).
    [CrossRef]
  10. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
  11. T. E. Gureyev, A. Roberts, 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]
  12. T. E. Gureyev, K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13, 1670–1682 (1996).
    [CrossRef]
  13. T. E. Gureyev, K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
    [CrossRef]
  14. N. Nakajima, “Phase-retrieval system using a shifted Gaussian filter,” J. Opt. Soc. Am. A 15, 402–406 (1998).
    [CrossRef]
  15. N. Nakajima, “Phase retrieval from two intensity measurements using the Fourier series expansion,” J. Opt. Soc. Am. A 4, 154–158 (1987).
    [CrossRef]
  16. N. Nakajima, “Phase retrieval using the logarithmic Hilbert transform and the Fourier-series expansion,” J. Opt. Soc. Am. A 5, 257–262 (1988).
    [CrossRef]

1998 (1)

1997 (1)

T. E. Gureyev, K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
[CrossRef]

1996 (1)

1995 (1)

1988 (1)

1987 (1)

1983 (1)

1982 (1)

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).

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.

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.

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.

Hurt, N. E.

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

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.

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.

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).

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.

J. Opt. Soc. Am. (2)

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

Opt. Commun. (1)

T. E. Gureyev, K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
[CrossRef]

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).

Other (7)

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]

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.

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.

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

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of the object-reconstruction system by use of phase retrieval. The object function is reconstructed from Fresnel zone intensity measurements of the object modulated with a Gaussian filter and with its horizontally and vertically shifted Gaussian filters.

Fig. 2
Fig. 2

Schematic diagram of the alternative object-reconstruction system by use of phase retrieval. The object function is reconstructed from Fresnel zone intensity measurements by use of the illumination of a Gaussian beam instead of a Gaussian filter in the same shifting procedure as in Fig. 1.

Fig. 3
Fig. 3

Original object function: (a) modulus and (b) phase of an object function of square extent with sides of 2.5 mm. Fresnel zone intensity distributions (at a distance d = 500 mm) of the object (c) modulated with a Gaussian filter of width W = 2.6 mm and (d) illuminated by a Gaussian beam with a beam radius of W p = 2.6 mm and a radius of curvature of R p = 2000 mm.

Fig. 4
Fig. 4

Reconstruction of the object shown in Figs. 3(a) and 3(b) from noisy Fresnel zone intensities when a Gaussian filter of W = 2.6 mm is used in the system of Fig. 1: (a) Modulus and (b) phase of the reconstructed object, (c) cross-sectional profile of the function shown in (a), (d) cross-sectional profile of the function shown in (b). Profiles were taken along the horizontal line that passes through the center of the reconstructed object (i.e., the u axis in Fig. 1). The dotted and solid curves represent the original and the reconstructed objects, respectively. SNR = 1197.

Fig. 5
Fig. 5

Same as in Fig. 4 except that illumination by a Gaussian beam with a beam radius of W p = 2.6 mm and a radius of curvature of R p = 2000 mm was used instead of the Gaussian filter. SNR = 1197.

Equations (29)

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g u ,   v = A   exp - u 2 + v 2 W 2 ,
U 1 x ,   y = σ   f u ,   v g u ,   v × exp i   π d λ x - u 2 + y - v 2 d u d v ,
U 2 x ,   y = σ   f u ,   v g u - τ ,   v × exp i   π d λ x - u 2 + y - v 2 d u d v .
g u - τ ,   v = g u ,   v exp 2 τ x W 2 exp - τ 2 W 2 .
U 1 x ,   y = exp [ i   π d λ x 2 + y 2 σ   f u ,   v g u ,   v × exp i   π d λ u 2 + v 2 × exp - i   2 π d λ xu + yv d u d v ,
U 2 x ,   y = exp i   π d λ x 2 + y 2 × exp - τ 2 W 2 σ   f u ,   v g u ,   v × exp i   π d λ u 2 + v 2 × exp - i   2 π d λ x + ic u + yv d u d v ,
c = d λ τ π W 2 .
U 2 x ,   y = exp - τ 2 W 2 U 1 x + ic ,   y .
U 1 x ,   y = M x ,   y exp i ϕ x ,   y .
| U 1 x + ic ,   y | = | M x + ic ,   y | exp - Im ϕ x + ic ,   y ,
ln | U 2 x ,   y | | M x + ic ,   y | + τ 2 W 2 = - Im ϕ x + ic ,   y .
ϕ x ,   y n = 1 N a n y cos n π l   x + b n y sin n π l   x ,
ln | U 2 x ,   y | | M x + ic ,   y | + τ 2 W 2 n = 1 N a n y sin n π l   x - b n y cos n π l   x sinh n π c l .
U 3 x ,   y = σ   f u ,   v g u ,   v - ν × exp i   π d λ x - u 2 + y - v 2 d u d v ,
U 3 x ,   y = exp - ν 2 W 2 U 1 x ,   y + ic ,
c = d λ ν π W 2 .
b u ,   v ,   z = A 0 W 0 W z exp - u 2 + v 2 W 2 z × exp i   2 π λ   z + i   π u 2 + v 2 λ R z - i ζ z ,
W z = W 0 1 + z z 0 2 1 / 2
R z = z 1 + z 0 z 2
ζ z = tan - 1 z z 0
z 0 = π W 0 2 λ
b u ,   v = A 1 W 0 W p exp - u 2 + v 2 W p 2 exp i   π u 2 + v 2 λ R p ,
U 1 x ,   y = σ   f u ,   v b u ,   v × exp i   π d λ x - u 2 + y - v 2 d u d v ,
U 2 x ,   y = σ   f u ,   v b u - τ ,   v × exp i   π d λ x - u 2 + y - v 2 d u d v ,
U 1 x ,   y = exp i   π d λ x 2 + y 2 σ   f u ,   v b u ,   v × exp i   π d λ u 2 + v 2 × exp - i 2 π d λ xu + yv d u d v ,
U 2 x ,   y = exp i   π d λ x 2 + y 2 × exp - τ 2 W p 2 + i   π τ 2 λ R p × σ   f u ,   v b u ,   v × exp i   π d λ u 2 + v 2 × exp - i   2 π d λ x + ic + s u + yv d u d v ,
c = d λ τ π W p 2 ,
s = d τ R p .
U 2 x ,   y = exp - τ 2 W p 2 + i   π τ 2 λ R p U 1 x + ic + s ,   y .

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