Abstract

We describe a tomographic microscope, for imaging phase objects, that makes use of the transport-of-intensity equation to estimate the phase of the transmitted light through the object. The wave-front data from optical fibers are reconstructed with an algorithm that incorporates correction for the ray bending. The reconstructed refractive-index cross sections of the fibers are found to be in agreement with the available values specified in the catalogs.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. Kawata, “The optical computed tomography microscope,” in Advances in Optical and Electron Microscopy, T. Mulvey, C. R. J. Sheppard, eds. (Academic, San Diego, Calif., 1994), Vol. 14.
  2. A. J. Devaney, A. Schatzberg, “Coherent optical tomographic microscope,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc SPIE1767, 62–71 (1992).
  3. T. C. Wedberg, J. J. Stamnes, “Comparison of phase retrieval methods in optical diffraction tomography,” Pure Appl. Opt. 4, 39–54 (1995).
    [CrossRef]
  4. M. H. Maleki, A. J. Devany, “Phase retrieval and intensity-only reconstruction algorithm for optical diffraction tomography,” J. Opt. Soc. Am. A 10, 1086–1092 (1993).
    [CrossRef]
  5. M. H. Maleki, A. J. Devaney, A. Schatzberg, “Tomographic reconstruction from optical scattered intensities,” J. Opt. Soc. Am. A 9, 1356–1363 (1992).
    [CrossRef]
  6. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
  7. 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]
  8. T. E. Gureyev, A. Roberts, 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]
  9. G. Vdovin, “Reconstruction of an object shape from the near-field intensity of a reflected paraxial beam,” Appl. Opt. 36, 5508–5513 (1997).
    [CrossRef] [PubMed]
  10. I. H. Lira, C. M. Vest, “Refraction correction in holographic interferometry and tomography of transparent objects,” Appl. Opt. 26, 3919–3928 (1987).
    [CrossRef] [PubMed]
  11. K. Ichiwaka, A. W. Lohmann, M. Takeda, “Phase retrieval based on the irradiance transport equation and the Fourier transform method: experiments,” Appl. Opt. 27, 3433–3436 (1988).
    [CrossRef]
  12. D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A 11, 107–117 (1994).
    [CrossRef]
  13. A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988).

1997

1996

T. E. Gureyev, A. Roberts, 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]

1995

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]

T. C. Wedberg, J. J. Stamnes, “Comparison of phase retrieval methods in optical diffraction tomography,” Pure Appl. Opt. 4, 39–54 (1995).
[CrossRef]

1994

1993

1992

1988

1987

1983

Devaney, A. J.

M. H. Maleki, A. J. Devaney, A. Schatzberg, “Tomographic reconstruction from optical scattered intensities,” J. Opt. Soc. Am. A 9, 1356–1363 (1992).
[CrossRef]

A. J. Devaney, A. Schatzberg, “Coherent optical tomographic microscope,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc SPIE1767, 62–71 (1992).

Devany, A. J.

Ghiglia, D. C.

Gureyev, T. E.

T. E. Gureyev, A. Roberts, 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]

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]

Ichiwaka, K.

Kak, A. C.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988).

Kawata, S.

S. Kawata, “The optical computed tomography microscope,” in Advances in Optical and Electron Microscopy, T. Mulvey, C. R. J. Sheppard, eds. (Academic, San Diego, Calif., 1994), Vol. 14.

Lira, I. H.

Lohmann, A. W.

Maleki, M. H.

Nugent, K. A.

T. E. Gureyev, A. Roberts, 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]

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]

Roberts, A.

T. E. Gureyev, A. Roberts, 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]

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]

Romero, L. A.

Schatzberg, A.

M. H. Maleki, A. J. Devaney, A. Schatzberg, “Tomographic reconstruction from optical scattered intensities,” J. Opt. Soc. Am. A 9, 1356–1363 (1992).
[CrossRef]

A. J. Devaney, A. Schatzberg, “Coherent optical tomographic microscope,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc SPIE1767, 62–71 (1992).

Slaney, M.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988).

Stamnes, J. J.

T. C. Wedberg, J. J. Stamnes, “Comparison of phase retrieval methods in optical diffraction tomography,” Pure Appl. Opt. 4, 39–54 (1995).
[CrossRef]

Takeda, M.

Teague, M. R.

Vdovin, G.

Vest, C. M.

Wedberg, T. C.

T. C. Wedberg, J. J. Stamnes, “Comparison of phase retrieval methods in optical diffraction tomography,” Pure Appl. Opt. 4, 39–54 (1995).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. A.

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]

T. E. Gureyev, A. Roberts, 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]

Pure Appl. Opt.

T. C. Wedberg, J. J. Stamnes, “Comparison of phase retrieval methods in optical diffraction tomography,” Pure Appl. Opt. 4, 39–54 (1995).
[CrossRef]

Other

S. Kawata, “The optical computed tomography microscope,” in Advances in Optical and Electron Microscopy, T. Mulvey, C. R. J. Sheppard, eds. (Academic, San Diego, Calif., 1994), Vol. 14.

A. J. Devaney, A. Schatzberg, “Coherent optical tomographic microscope,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc SPIE1767, 62–71 (1992).

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Laboratory model of the tomographic microscope. Passage through SD and RD, the static diffuser and the rotating diffuser, respectively, spoils the spatial coherence of the input illumination. The imaging lens O is a 3× microscopic objective, which images exit plane E of the cuvette C onto the CCD array connected to a computer. L, lens.

Fig. 2
Fig. 2

Phase reconstructed from intensity measurements at the plane E of Fig. 1. The object is the first fiber in toluene (a) when the transverse plane intensity is assumed constant, (b) when the TIE is solved with Fourier harmonic expansion, and (c) when the TIE is reduced to a finite-difference equation and solved iteratively.

Fig. 3
Fig. 3

Phase reconstructions from the second fiber. Details of (a), (b), and (c) are as in Fig. 2.

Fig. 4
Fig. 4

Phase reconstruction from a typical view of the three-fiber object.

Fig. 5
Fig. 5

Refractive-index profiles of the first fiber reconstructed through the straight-path CBP algorithm. (a) and (b) are obtained with data from Figs. 2(a) and 2(b), respectively.

Fig. 6
Fig. 6

Refractive-index profiles of the second fiber reconstructed through the straight-path CBP algorithm. (a) and (b) are obtained with data from Figs. 3(a) and 3(b), respectively.

Fig. 7
Fig. 7

Refractive-index profiles of the first fiber reconstructed through the modified CBP algorithm that incorporates the refraction-correction loop. (a) and (b) are obtained with data from Figs. 2(a) and 2(b), respectively.

Fig. 8
Fig. 8

Refractive-index profiles of the second fiber reconstructed through the modified CBP algorithm that incorporates the refraction-correction loop. (a) and (b) are obtained with data from Figs. 3(a) and 3(b), respectively.

Fig. 9
Fig. 9

Cross-sectional image of the three-fiber object reconstructed from the data of Fig. 4 with straight-path CBP.

Fig. 10
Fig. 10

Cross-sectional image of the two fibers: (a) first, (b) second.

Equations (12)

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

kzI=-·Iϕ,
Wmnx, y=expj2πmx/aexpj2πny/b,
f, gab=1ab0a0b fx, yg*x, ydxdy,
-ab 0a0b ·IϕWmnx, ydxdy=ab 0a0b DWmnx, ydxdy.
ϕx, y=i=1j=1ϕijWijx, y.
1abij ϕij0a0b IWmnx, y·Wijx, ydxdy=Dmn.
Amnij=0a0b Ix, yWmnx, y·Wijx, ydxdy.
i,j ϕi,jAmnij=abDmn.
Amnij=2π2i mbaj mabIˆm-i,n-j,
Φij=abm,nAmnij-1Dm,n.
2Φ=-kI0Iz.
Φi,j=ab22π2i2b2+j2a2I0 Di,j.

Metrics