Abstract

A generalized tomographic reconstruction procedure is described for determining the complex-valuedindex-of-refraction distribution of a semitransparent, three-dimensional inhomogeneous object from observations of the far-field intensity patterns generated by the object in a sequence of scattering experiments. The inversion procedure is based on the wave equation governing the scattered optical field and fully accounts for the diffraction and propagation effects associated with the interaction of the incident wave with the object and the subsequent free-space propagation of the scattered wave to the wave zone (far field). The reconstruction of the object’s index-of-refraction distribution is performed digitally directly from the far-field intensity of the scattered wave and does not require direct measurement or retrieval of the phase of the scattered field. An optical scattering experiment is reported in which the cross-sectional profile of the index-of-refraction distribution of an optical fiber is reconstructed from the measured intensity of the diffraction pattern of the fiber by using the described inversion procedure.

© 1992 Optical Society of America

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References

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  1. H. Lipson, W. Cochran, The Determination of Crystal Structures (Cornell U. Press, Ithaca, N.Y, 1966).
  2. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
    [CrossRef]
  3. E. Wolf, “Determination of the amplitude and the phase of the scattered field by holography,” J. Opt. Soc. Am. 60, 18–20 (1970).
    [CrossRef]
  4. See, for example, H. A. Hauptman, “The phase problem of X-ray crystallography,” Phys. Today 42(11), 24–29 (1990); R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990).
    [CrossRef]
  5. W. H. Carter, P.-C. Ho, “Reconstruction of inhomogeneous scattering objects from holograms,” Appl. Opt. 13, 162–172 (1974); A. Gretzula, W. H. Carter, “Structural measurements by inverse scattering in the Rytov approximation,” J. Opt. Soc. Am. A 2, 1958–1960 (1985).
    [CrossRef] [PubMed]
  6. A. F. Fercher, H. Bartelt, H. Becker, E. Wiltschko, “Image formation by inversion of scattered field data: experiments and computational simulation,” Appl. Opt. 18, 2427–2439 (1979).
    [CrossRef] [PubMed]
  7. K. Iwata, R. Nagata, “Calculation of refractive index distribution from interferograms using the Born and Rytov approximation,” Jpn J. Appl. Phys. 14(Suppl. 14-1), 379–383 (1975).
  8. D. W. Sweeney, C. M. Vest, “Reconstruction of three-dimensional refractive index fields from multidimensional interferometric data,” Appl. Opt. 12, 2649–2664 (1973).
    [CrossRef] [PubMed]
  9. R. Snyder, L. Hesselink, “High speed optical tomography for flow visualization,” Appl. Opt. 24, 4046–4051 (1985).
    [CrossRef] [PubMed]
  10. S. Kawata, O. Nakamura, T. Noda, H. Ooki, K. Ogino, Y. Kuroiwa, S. Minami, “Laser computed-tomography microscope,” Appl. Opt. 29, 3805–3809 (1990).
    [CrossRef] [PubMed]
  11. J. F. Greenleaf, S. A. Johnson, S. L. Lee, G. T. Herman, E. H. Wood, “Algebraic reconstruction of spatial distributions of acoustic absorption in tissues from their two-dimensional acoustic projections,” in Acoustical Holography, P. S. Green, ed. (Plenum, New York, 1974), Vol. 5, 591–603.
    [CrossRef]
  12. A. C. Kak, “Computerized tomography with x-ray emission and ultrasound sources,” Proc. IEEE 67, 1245–1272 (1979).
    [CrossRef]
  13. J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1990), Vol. 28.
    [CrossRef]
  14. G. W. Faris, H. M. Hertz, “Tunable differential interferometer for optical tomography,” Appl. Opt. 28, 4662–4667 (1989).
    [CrossRef] [PubMed]
  15. I. H. Lira, C. M. Vest, “Refraction correction in holographic interferometry and tomography of transparent objects,” Appl. Opt. 26, 3919–3928 (1987).
    [CrossRef] [PubMed]
  16. R. K. Mueller, M. Kaveh, G. Wade, “Reconstructive tomography and applications to ultrasonics,” Proc. Inst. Electr. Eng. 67, 567–587 (1979).
    [CrossRef]
  17. J. F. Greenleaf, R. C. Bah, “Clinical imaging with transmissive ultrasonic computerized tomography,” IEEE Trans. Biomed. Eng. BME-28, 496–505 (1981).
    [CrossRef]
  18. S. X. Pan, A. C. Kak, “A computational study of reconstruction algorithms for diffraction tomography: interpolation versus filtered backpropagation,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1262–1275 (1983).
    [CrossRef]
  19. A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrasonic Imag. 4, 336–350 (1982).
  20. M. A. Fiddy, “Inversion of optical scattered field data,” J. Phys. D 19, 301–317 (1986).
    [CrossRef]
  21. 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).
    [CrossRef]
  22. L. S. Taylor, “The phase retrieval problem,” IEEE Trans. Antennas Propag. AP-29, 386–391 (1981).
    [CrossRef]
  23. 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).
  24. J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
    [CrossRef]
  25. J. S. Jaffe, R. Fricke, “Constrained reconstruction of complex waveforms,” J. Opt. Soc. Am. A 4, 216–220 (1987).
    [CrossRef]
  26. J. M. Tribolet, “A new phase unwrapping algorithm,” IEEE Trans. Acoust. Speech Signal Process. ASSP-26, 170–177 (1977).
    [CrossRef]
  27. A. J. Devaney, “Structure determination from intensity measurements in scattering experiments,” Phys. Rev. Lett. 62, 2385–2388 (1989).
    [CrossRef] [PubMed]
  28. A. J. Devaney, “Diffraction tomographic reconstruction from intensity data,” IEEE Trans. Image Process. 1, 221–228 (1992).
    [CrossRef] [PubMed]
  29. A. J. Devaney, “Reconstructive tomography with diffracting wavefields,” Inverse Probl. 2, 161–183 (1986).
    [CrossRef]
  30. V. T. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
  31. J. M. Cowley, Diffraction Physics, 2nd rev. ed. (North-Holland, New York, 1981).
  32. The radius of the Ewald limiting circle differs from its value of 2kin x-ray crystallography because only forward-scattered radiation is measured in DT.
  33. See J. R. Shewell, E. Wolf, “Inverse diffraction and a new reciprocity theorem,” J. Opt. Soc. Am. 58, 1596–1603 (1968); G. C. Sherman, “Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” J. Opt. Soc. Am. 59, 697–711 (1969).
    [CrossRef]
  34. A. Schatzberg, A. J. Devaney, “An optical microscope for imaging three dimensional semi-transparent structures,” Final Project Rep. Small Business Innovation Research Phase I National Science Foundation grant ISI-8960413 (National Science Foundation, Washington, D.C., 1990).
  35. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  36. N. Sponheim, I. Johansen, A. J. Devaney, “Initial testing of a clinical ultrasound mammograph,” in Acoustical Imaging, H. Lee, G. Wade, eds. (Plenum, New York, 1990), Vol. 18.
  37. D. Colton, P. Monk, “The inverse scattering problem for acoustic waves in an inhomogeneous medium,” in Inverse Problems in Partial Differential Equations, D. Colton, R. Ewing, W. Rundell, eds. (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1990).
  38. The intensity profile for this fiber was also computed with an exact eigenfunction-based method and was found to coincide almost exactly with that computed with the hybrid method.

1992 (1)

A. J. Devaney, “Diffraction tomographic reconstruction from intensity data,” IEEE Trans. Image Process. 1, 221–228 (1992).
[CrossRef] [PubMed]

1990 (2)

See, for example, H. A. Hauptman, “The phase problem of X-ray crystallography,” Phys. Today 42(11), 24–29 (1990); R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990).
[CrossRef]

S. Kawata, O. Nakamura, T. Noda, H. Ooki, K. Ogino, Y. Kuroiwa, S. Minami, “Laser computed-tomography microscope,” Appl. Opt. 29, 3805–3809 (1990).
[CrossRef] [PubMed]

1989 (2)

G. W. Faris, H. M. Hertz, “Tunable differential interferometer for optical tomography,” Appl. Opt. 28, 4662–4667 (1989).
[CrossRef] [PubMed]

A. J. Devaney, “Structure determination from intensity measurements in scattering experiments,” Phys. Rev. Lett. 62, 2385–2388 (1989).
[CrossRef] [PubMed]

1987 (3)

1986 (2)

M. A. Fiddy, “Inversion of optical scattered field data,” J. Phys. D 19, 301–317 (1986).
[CrossRef]

A. J. Devaney, “Reconstructive tomography with diffracting wavefields,” Inverse Probl. 2, 161–183 (1986).
[CrossRef]

1985 (1)

1983 (1)

S. X. Pan, A. C. Kak, “A computational study of reconstruction algorithms for diffraction tomography: interpolation versus filtered backpropagation,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1262–1275 (1983).
[CrossRef]

1982 (1)

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrasonic Imag. 4, 336–350 (1982).

1981 (2)

L. S. Taylor, “The phase retrieval problem,” IEEE Trans. Antennas Propag. AP-29, 386–391 (1981).
[CrossRef]

J. F. Greenleaf, R. C. Bah, “Clinical imaging with transmissive ultrasonic computerized tomography,” IEEE Trans. Biomed. Eng. BME-28, 496–505 (1981).
[CrossRef]

1979 (3)

R. K. Mueller, M. Kaveh, G. Wade, “Reconstructive tomography and applications to ultrasonics,” Proc. Inst. Electr. Eng. 67, 567–587 (1979).
[CrossRef]

A. C. Kak, “Computerized tomography with x-ray emission and ultrasound sources,” Proc. IEEE 67, 1245–1272 (1979).
[CrossRef]

A. F. Fercher, H. Bartelt, H. Becker, E. Wiltschko, “Image formation by inversion of scattered field data: experiments and computational simulation,” Appl. Opt. 18, 2427–2439 (1979).
[CrossRef] [PubMed]

1977 (1)

J. M. Tribolet, “A new phase unwrapping algorithm,” IEEE Trans. Acoust. Speech Signal Process. ASSP-26, 170–177 (1977).
[CrossRef]

1975 (1)

K. Iwata, R. Nagata, “Calculation of refractive index distribution from interferograms using the Born and Rytov approximation,” Jpn J. Appl. Phys. 14(Suppl. 14-1), 379–383 (1975).

1974 (1)

1973 (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).

1970 (1)

1969 (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

1968 (1)

Bah, R. C.

J. F. Greenleaf, R. C. Bah, “Clinical imaging with transmissive ultrasonic computerized tomography,” IEEE Trans. Biomed. Eng. BME-28, 496–505 (1981).
[CrossRef]

Bartelt, H.

Becker, H.

Carter, W. H.

Cochran, W.

H. Lipson, W. Cochran, The Determination of Crystal Structures (Cornell U. Press, Ithaca, N.Y, 1966).

Colton, D.

D. Colton, P. Monk, “The inverse scattering problem for acoustic waves in an inhomogeneous medium,” in Inverse Problems in Partial Differential Equations, D. Colton, R. Ewing, W. Rundell, eds. (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1990).

Cowley, J. M.

J. M. Cowley, Diffraction Physics, 2nd rev. ed. (North-Holland, New York, 1981).

Devaney, A. J.

A. J. Devaney, “Diffraction tomographic reconstruction from intensity data,” IEEE Trans. Image Process. 1, 221–228 (1992).
[CrossRef] [PubMed]

A. J. Devaney, “Structure determination from intensity measurements in scattering experiments,” Phys. Rev. Lett. 62, 2385–2388 (1989).
[CrossRef] [PubMed]

A. J. Devaney, “Reconstructive tomography with diffracting wavefields,” Inverse Probl. 2, 161–183 (1986).
[CrossRef]

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrasonic Imag. 4, 336–350 (1982).

A. Schatzberg, A. J. Devaney, “An optical microscope for imaging three dimensional semi-transparent structures,” Final Project Rep. Small Business Innovation Research Phase I National Science Foundation grant ISI-8960413 (National Science Foundation, Washington, D.C., 1990).

N. Sponheim, I. Johansen, A. J. Devaney, “Initial testing of a clinical ultrasound mammograph,” in Acoustical Imaging, H. Lee, G. Wade, eds. (Plenum, New York, 1990), Vol. 18.

Faris, G. W.

Fercher, A. F.

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).
[CrossRef]

Fiddy, M. A.

M. A. Fiddy, “Inversion of optical scattered field data,” J. Phys. D 19, 301–317 (1986).
[CrossRef]

Fienup, J. R.

Fricke, 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).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Greenleaf, J. F.

J. F. Greenleaf, R. C. Bah, “Clinical imaging with transmissive ultrasonic computerized tomography,” IEEE Trans. Biomed. Eng. BME-28, 496–505 (1981).
[CrossRef]

J. F. Greenleaf, S. A. Johnson, S. L. Lee, G. T. Herman, E. H. Wood, “Algebraic reconstruction of spatial distributions of acoustic absorption in tissues from their two-dimensional acoustic projections,” in Acoustical Holography, P. S. Green, ed. (Plenum, New York, 1974), Vol. 5, 591–603.
[CrossRef]

Hauptman, H. A.

See, for example, H. A. Hauptman, “The phase problem of X-ray crystallography,” Phys. Today 42(11), 24–29 (1990); R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990).
[CrossRef]

Herman, G. T.

J. F. Greenleaf, S. A. Johnson, S. L. Lee, G. T. Herman, E. H. Wood, “Algebraic reconstruction of spatial distributions of acoustic absorption in tissues from their two-dimensional acoustic projections,” in Acoustical Holography, P. S. Green, ed. (Plenum, New York, 1974), Vol. 5, 591–603.
[CrossRef]

Hertz, H. M.

Hesselink, L.

Ho, P.-C.

Iwata, K.

K. Iwata, R. Nagata, “Calculation of refractive index distribution from interferograms using the Born and Rytov approximation,” Jpn J. Appl. Phys. 14(Suppl. 14-1), 379–383 (1975).

Jaffe, J. S.

Johansen, I.

N. Sponheim, I. Johansen, A. J. Devaney, “Initial testing of a clinical ultrasound mammograph,” in Acoustical Imaging, H. Lee, G. Wade, eds. (Plenum, New York, 1990), Vol. 18.

Johnson, S. A.

J. F. Greenleaf, S. A. Johnson, S. L. Lee, G. T. Herman, E. H. Wood, “Algebraic reconstruction of spatial distributions of acoustic absorption in tissues from their two-dimensional acoustic projections,” in Acoustical Holography, P. S. Green, ed. (Plenum, New York, 1974), Vol. 5, 591–603.
[CrossRef]

Kak, A. C.

S. X. Pan, A. C. Kak, “A computational study of reconstruction algorithms for diffraction tomography: interpolation versus filtered backpropagation,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1262–1275 (1983).
[CrossRef]

A. C. Kak, “Computerized tomography with x-ray emission and ultrasound sources,” Proc. IEEE 67, 1245–1272 (1979).
[CrossRef]

Kaveh, M.

R. K. Mueller, M. Kaveh, G. Wade, “Reconstructive tomography and applications to ultrasonics,” Proc. Inst. Electr. Eng. 67, 567–587 (1979).
[CrossRef]

Kawata, S.

Kuroiwa, Y.

Lee, S. L.

J. F. Greenleaf, S. A. Johnson, S. L. Lee, G. T. Herman, E. H. Wood, “Algebraic reconstruction of spatial distributions of acoustic absorption in tissues from their two-dimensional acoustic projections,” in Acoustical Holography, P. S. Green, ed. (Plenum, New York, 1974), Vol. 5, 591–603.
[CrossRef]

Lipson, H.

H. Lipson, W. Cochran, The Determination of Crystal Structures (Cornell U. Press, Ithaca, N.Y, 1966).

Lira, I. H.

Minami, S.

Monk, P.

D. Colton, P. Monk, “The inverse scattering problem for acoustic waves in an inhomogeneous medium,” in Inverse Problems in Partial Differential Equations, D. Colton, R. Ewing, W. Rundell, eds. (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1990).

Mueller, R. K.

R. K. Mueller, M. Kaveh, G. Wade, “Reconstructive tomography and applications to ultrasonics,” Proc. Inst. Electr. Eng. 67, 567–587 (1979).
[CrossRef]

Nagata, R.

K. Iwata, R. Nagata, “Calculation of refractive index distribution from interferograms using the Born and Rytov approximation,” Jpn J. Appl. Phys. 14(Suppl. 14-1), 379–383 (1975).

Nakamura, O.

Noda, T.

Ogino, K.

Ooki, H.

Pan, S. X.

S. X. Pan, A. C. Kak, “A computational study of reconstruction algorithms for diffraction tomography: interpolation versus filtered backpropagation,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1262–1275 (1983).
[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).

Schatzberg, A.

A. Schatzberg, A. J. Devaney, “An optical microscope for imaging three dimensional semi-transparent structures,” Final Project Rep. Small Business Innovation Research Phase I National Science Foundation grant ISI-8960413 (National Science Foundation, Washington, D.C., 1990).

Schwider, J.

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1990), Vol. 28.
[CrossRef]

Shewell, J. R.

Snyder, R.

Sponheim, N.

N. Sponheim, I. Johansen, A. J. Devaney, “Initial testing of a clinical ultrasound mammograph,” in Acoustical Imaging, H. Lee, G. Wade, eds. (Plenum, New York, 1990), Vol. 18.

Sweeney, D. W.

Tatarski, V. T.

V. T. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

Taylor, L. S.

L. S. Taylor, “The phase retrieval problem,” IEEE Trans. Antennas Propag. AP-29, 386–391 (1981).
[CrossRef]

Tribolet, J. M.

J. M. Tribolet, “A new phase unwrapping algorithm,” IEEE Trans. Acoust. Speech Signal Process. ASSP-26, 170–177 (1977).
[CrossRef]

Vest, C. M.

Wade, G.

R. K. Mueller, M. Kaveh, G. Wade, “Reconstructive tomography and applications to ultrasonics,” Proc. Inst. Electr. Eng. 67, 567–587 (1979).
[CrossRef]

Wiltschko, E.

Wolf, E.

Wood, E. H.

J. F. Greenleaf, S. A. Johnson, S. L. Lee, G. T. Herman, E. H. Wood, “Algebraic reconstruction of spatial distributions of acoustic absorption in tissues from their two-dimensional acoustic projections,” in Acoustical Holography, P. S. Green, ed. (Plenum, New York, 1974), Vol. 5, 591–603.
[CrossRef]

Appl. Opt. (7)

IEEE Trans. Acoust. Speech Signal Process. (2)

J. M. Tribolet, “A new phase unwrapping algorithm,” IEEE Trans. Acoust. Speech Signal Process. ASSP-26, 170–177 (1977).
[CrossRef]

S. X. Pan, A. C. Kak, “A computational study of reconstruction algorithms for diffraction tomography: interpolation versus filtered backpropagation,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1262–1275 (1983).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

L. S. Taylor, “The phase retrieval problem,” IEEE Trans. Antennas Propag. AP-29, 386–391 (1981).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

J. F. Greenleaf, R. C. Bah, “Clinical imaging with transmissive ultrasonic computerized tomography,” IEEE Trans. Biomed. Eng. BME-28, 496–505 (1981).
[CrossRef]

IEEE Trans. Image Process. (1)

A. J. Devaney, “Diffraction tomographic reconstruction from intensity data,” IEEE Trans. Image Process. 1, 221–228 (1992).
[CrossRef] [PubMed]

Inverse Probl. (1)

A. J. Devaney, “Reconstructive tomography with diffracting wavefields,” Inverse Probl. 2, 161–183 (1986).
[CrossRef]

J. Opt. Soc. Am. (2)

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

J. Phys. D (1)

M. A. Fiddy, “Inversion of optical scattered field data,” J. Phys. D 19, 301–317 (1986).
[CrossRef]

Jpn J. Appl. Phys. (1)

K. Iwata, R. Nagata, “Calculation of refractive index distribution from interferograms using the Born and Rytov approximation,” Jpn J. Appl. Phys. 14(Suppl. 14-1), 379–383 (1975).

Opt. Commun. (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[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).

Phys. Rev. Lett. (1)

A. J. Devaney, “Structure determination from intensity measurements in scattering experiments,” Phys. Rev. Lett. 62, 2385–2388 (1989).
[CrossRef] [PubMed]

Phys. Today (1)

See, for example, H. A. Hauptman, “The phase problem of X-ray crystallography,” Phys. Today 42(11), 24–29 (1990); R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990).
[CrossRef]

Proc. IEEE (1)

A. C. Kak, “Computerized tomography with x-ray emission and ultrasound sources,” Proc. IEEE 67, 1245–1272 (1979).
[CrossRef]

Proc. Inst. Electr. Eng. (1)

R. K. Mueller, M. Kaveh, G. Wade, “Reconstructive tomography and applications to ultrasonics,” Proc. Inst. Electr. Eng. 67, 567–587 (1979).
[CrossRef]

Ultrasonic Imag. (1)

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrasonic Imag. 4, 336–350 (1982).

Other (12)

H. Lipson, W. Cochran, The Determination of Crystal Structures (Cornell U. Press, Ithaca, N.Y, 1966).

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).
[CrossRef]

V. T. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

J. M. Cowley, Diffraction Physics, 2nd rev. ed. (North-Holland, New York, 1981).

The radius of the Ewald limiting circle differs from its value of 2kin x-ray crystallography because only forward-scattered radiation is measured in DT.

J. F. Greenleaf, S. A. Johnson, S. L. Lee, G. T. Herman, E. H. Wood, “Algebraic reconstruction of spatial distributions of acoustic absorption in tissues from their two-dimensional acoustic projections,” in Acoustical Holography, P. S. Green, ed. (Plenum, New York, 1974), Vol. 5, 591–603.
[CrossRef]

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1990), Vol. 28.
[CrossRef]

A. Schatzberg, A. J. Devaney, “An optical microscope for imaging three dimensional semi-transparent structures,” Final Project Rep. Small Business Innovation Research Phase I National Science Foundation grant ISI-8960413 (National Science Foundation, Washington, D.C., 1990).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

N. Sponheim, I. Johansen, A. J. Devaney, “Initial testing of a clinical ultrasound mammograph,” in Acoustical Imaging, H. Lee, G. Wade, eds. (Plenum, New York, 1990), Vol. 18.

D. Colton, P. Monk, “The inverse scattering problem for acoustic waves in an inhomogeneous medium,” in Inverse Problems in Partial Differential Equations, D. Colton, R. Ewing, W. Rundell, eds. (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1990).

The intensity profile for this fiber was also computed with an exact eigenfunction-based method and was found to coincide almost exactly with that computed with the hybrid method.

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Figures (7)

Fig. 1
Fig. 1

Conventional scan configuration of DT. The scatterer is illuminated by a monochromatic plane wave, and the sum of the incident and the scattered fields is measured over a measurement line z = l0 that is perpendicular to the direction of propagation ŝ0 of the incident plane wave.

Fig. 2
Fig. 2

Fourier-transform domain of the object showing the semicircular arc (bold) of the Ewald circle K = k(ŝŝ0) over which the data are collected. The set of all such semicircular arcs for different ŝ0 maps out the interior of the Ewald limiting circle.

Fig. 3
Fig. 3

Experimental setup. The fiber is embedded in an index-matching fluid contained in the cell, and the intensity distribution on the diffraction plane is imaged onto the photodetector (PMT, photomultiplier tube).

Fig. 4
Fig. 4

Normalized intensity distribution on the diffraction plane located at l0 = 7.6 mm from the fiber: (a) experimental data, (b) simulated data, using the hybrid method, and (c) simulated data, using the usual Rytov approximation at l0.

Fig. 5
Fig. 5

Reconstruction from experimental and hybrid scattering data. The dotted curve shows the ideal fiber profile, and the solid curve gives the reconstruction obtained from the full (amplitude and phase) simulated data. The dashed curve is the reconstruction obtained from the hybrid intensity data, and the reconstruction from the experimental intensity data is displayed with pluses (+).

Fig. 6
Fig. 6

Attainable resolution in the reconstruction as a function of the half-angle subtending the measurement line.

Fig. 7
Fig. 7

Reconstruction from simulated full and intensity-only data for a 4-mm span of the measurement line. The dotted curve shows the actual fiber profile, and the solid and the dashed curves correspond, respectively, to the reconstructions obtained from simulated full and intensity data.

Equations (20)

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

U ( r ) = exp ( i k s ^ 0 · r ) + U ( s ) ( r ) ,
[ 2 + k 2 ] U ( r ) = - 2 k 2 O ( r ) U ( r ) ,
O ( r ) = 1 2 [ n 2 ( r ) n 0 2 - 1 ]
O ( r ) δ n ( r ) ,             δ n ( r ) n ( r ) n 0 - 1
U ( r ) = exp { i k [ s ^ 0 · r + δ W ( r ) ] } ,
δ W ( r ) = 2 i k exp ( - i k s ^ 0 · r ) d r δ n ( r ) × exp ( i k s ^ 0 · r ) G ( r - r ) ,
G ( r - r ) = - ( i / 4 ) H 0 ( k r - r ) ,
δ n ^ ( r ) = 1 2 π 0 k d K K δ W ˜ ( K ) × exp [ i ( k - γ ) l 0 ] J 0 [ ( 2 k ) 1 / 2 ( k - γ ) 1 / 2 r ] ,
δ W ˜ ( K ) = - + d x δ W ( x ) exp ( - i K x )
D ( r ) 1 i k log e I ( r ) = δ W ( r ) - δ W ¯ ( r ) ,
I ( r ) = U ( r ) 2 = exp { i k [ δ W ( r ) - δ W ¯ ( r ) ] }
I ( r 0 ) = 1 + exp ( - i k s ^ 0 · r 0 ) U ( s ) ( r 0 ) + exp ( i k s ^ 0 · r 0 ) U ( s ) ¯ ( r 0 ) + U ( s ) ( r 0 ) 2 ,
L ( r 0 ) exp ( i k s ^ 0 · r 0 ) I ( r 0 )
= U ( r 0 ) + exp ( 2 i k s ^ 0 · r 0 ) U ( s ) ¯ ( r 0 ) + exp ( i k s ^ 0 · r 0 ) U ( s ) ( r 0 ) 2
U ( r ) B L ( r ) ,
δ W ( r ) = 1 i k log e [ exp ( - i k s ^ 0 · r ) U ( r ) ] ,
B L ( r ) = B L ( x , z ) = 1 2 π - k k d K L ˜ ( K , l 0 ) × exp { i [ K x + γ ( z - l 0 ) ] } ,
L ˜ ( K , l 0 ) = - + d x L ( x , l 0 ) exp ( - i K x )
K c = k s ^ - s ^ 0 = k [ 2 ( 1 - cos θ c ) ] 1 / 2 ,
δ r = π K c = λ / 2 [ 2 ( 1 - cos θ c ) ] 1 / 2 .

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