Abstract

The noniterative phase-retrieval method by use of Gaussian filtering is applied to the reconstruction of phase objects from experimental far-field intensities. In this method, the complex amplitude of transmitted light through an object is reconstructed from three far-field intensities, which are measured with the modulation of the object by laterally shifted and unshifted Gaussian filters. In the experiment, the amplitude of a Gaussian beam illuminating objects is utilized as a Gaussian filter, and, as the phase objects, a converging lens with a small exit pupil and a plastic fiber immersed in optical adhesive are used. The experimental results show that the Gaussian beam of a laser is capable of retrieving the phases of those objects with the accuracy of the range from ∼1/10 to 1/4 of the laser’s wavelength.

© 2002 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. 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.
  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. 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.
  7. 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.
  8. 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]
  9. 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).
  10. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  11. J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. 4, 118–123 (1987).
    [CrossRef]
  12. J. G. Walker, “Computer simulation of a method for object reconstruction from stellar speckle interferometry data,” Appl. Opt. 21, 3132–3137 (1982).
    [CrossRef] [PubMed]
  13. W. Kim, “Two-dimensional phase retrieval using a window function,” Opt. Lett. 26, 134–136 (2001).
    [CrossRef]
  14. R. W. Gonsalves, “Small-phase solution to the phase retrieval problem,” Opt. Lett. 26, 684–685 (2001).
    [CrossRef]
  15. 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]
  16. M. R. Teague, “Irradiance moments: their propagation and use for unique retrieval of phase,” J. Opt. Soc. Am. 72, 1199–1209 (1982).
    [CrossRef]
  17. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
  18. 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]
  19. 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]
  20. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
    [CrossRef] [PubMed]
  21. 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]
  22. 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] [PubMed]
  23. N. Nakajima, “Phase retrieval system using a shifted Gaussian filter,” J. Opt. Soc. Am. A 15, 402–406 (1998).
    [CrossRef]
  24. N. Nakajima, “Phase retrieval from Fresnel zone intensity measurements by use of Gaussian filtering,” Appl. Opt. 37, 6219–6226 (1998).
    [CrossRef]
  25. T. Iwai, H. Masui, “Application of the phase retrieval method to the refractive-index profiling of an optical fiber,” Opt. Commun. 72, 195–201 (1989).
    [CrossRef]

2001

2000

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] [PubMed]

1999

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]

1998

1996

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]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

1995

1989

T. Iwai, H. Masui, “Application of the phase retrieval method to the refractive-index profiling of an optical fiber,” Opt. Commun. 72, 195–201 (1989).
[CrossRef]

1987

J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. 4, 118–123 (1987).
[CrossRef]

1983

1982

1972

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] [PubMed]

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] [PubMed]

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]

Barnea, Z.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[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]

Cookson, D. F.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

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. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. 4, 118–123 (1987).
[CrossRef]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

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).

Gonsalves, R. W.

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.

Iwai, T.

T. Iwai, H. Masui, “Application of the phase retrieval method to the refractive-index profiling of an optical fiber,” Opt. Commun. 72, 195–201 (1989).
[CrossRef]

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] [PubMed]

Kim, W.

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]

Masui, H.

T. Iwai, H. Masui, “Application of the phase retrieval method to the refractive-index profiling of an optical fiber,” Opt. Commun. 72, 195–201 (1989).
[CrossRef]

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] [PubMed]

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.

N. Nakajima, “Phase retrieval system using a shifted Gaussian filter,” J. Opt. Soc. Am. A 15, 402–406 (1998).
[CrossRef]

N. Nakajima, “Phase retrieval from Fresnel zone intensity measurements by use of Gaussian filtering,” Appl. Opt. 37, 6219–6226 (1998).
[CrossRef]

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]

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.

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] [PubMed]

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]

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]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

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]

Paganin, D.

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] [PubMed]

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]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[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] [PubMed]

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nature

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] [PubMed]

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]

Opt. Commun.

T. Iwai, H. Masui, “Application of the phase retrieval method to the refractive-index profiling of an optical fiber,” Opt. Commun. 72, 195–201 (1989).
[CrossRef]

Opt. Lett.

Optik (Stuttgart)

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).

Phys. Rev. Lett.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Ultramicroscopy

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

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 (6)

Fig. 1
Fig. 1

Schematic arrangement of the experiment: f 2 is a focal length of lens L2.

Fig. 2
Fig. 2

Reconstruction of the object consisting of the converging lens with the focal length f 1 = 202 mm and the circular aperture of 1-mm diameter: (a) is the Fourier intensity of the object. (b) and (c) are the Fourier intensities of the object shifted with displacement 0.05 mm in each direction of the u and v axes in Fig. 1, respectively. (d) and (e) are the modulus and the phase, respectively, of the reconstructed object from the data of (a), (b), and (c).

Fig. 3
Fig. 3

Same as in Fig. 2 except that the focal length of the object lens is f 1 = 253 mm.

Fig. 4
Fig. 4

Cross-sectional profiles of the reconstructed object phases: (a) and (b) correspond to the phase distributions in Figs. 2(e) and 3(e), respectively. In each figure, the solid and dashed curves show the cross-sectional profiles at the center of the reconstructed phases along the lines parallel to the u and v axes, respectively, and the dotted curve shows the calculated phase from the focal length of the object lens.

Fig. 5
Fig. 5

Schematic diagram of the fiber object.

Fig. 6
Fig. 6

Reconstruction of the complex amplitude in the object plane of the fiber object: (a) and (b) are the reconstructed modulus and phase, respectively. The reconstructed phase is represented only within the extent of the fiber’s diameter for display purpose. The dotted curve shows the phase distribution calculated from the known refractive indexes of the fiber and the surrounding optical adhesive.

Equations (11)

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

bu, v=A W0Wexp-u2+v2W2expi πu2+v2λR,
F1x, y=σ fu, vbu, v×exp-i 2πdλxu+yvdudv,
bu-τ, v=bu, vexp2τuW2exp-i 2πτuλR×exp-τ2W2+i πτ2λR.
F2x, y=exp-τ2W2+i πτ2λRσ fu, vbu, v×exp-i 2πdλx+ic+su+yvdudv,
c=dλτπW2,
s=dτR.
F2x, y=exp-τ2W2+i πτ2λRF1x+ic+s, y.
F1x, y=Mx, yexpiϕx, y.
ln|F2x-s, y||Mx+ic, y|+τ2W2=-Im ϕx+ic, y,
ϕx, yn=1Nanycosnπl x+bnysinnπl x,
ln|F2x-s, y||Mx+ic, y|+τ2W2n=1Nanysinnπl x-bnycosnπl xsinhnπcl.

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