Abstract

A method for determining the internal structure of a localized scattering potential from field measurements performed outside the scattering volume is developed by using the Rytov approximation. The theory is compared with the inverse-scattering method within the Born and eikonal approximations and found to reduce to these methods in the weak-scattering (Born) and very-short-wavelength (eikonal) limits.

© 1981 Optical Society of America

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  1. For a survey of these techniques, see A. C. Kak, Proc. IEEE 67, 1245 (1979).
    [CrossRef]
  2. More specifically, they employ the eikonal approximation (see Ref. 6) to obtain a relationship between the complex phase of the field and so-called projections (see Ref. 1) of the object’s internal structure. This relationship is derived as a limiting case of the inverse-scattering method later in this Letter.
  3. See, for example, E. Wolf, Opt. Commun. 1, 153 (1969); W. H. Carter, J. Opt. Soc. Am. 60, 306 (1970); W. H. Carter, P.-C. Ho, Appl. Opt. 13, 162 (1974); A. F. Fercher et al., Appl Opt. 18, 2427 (1979); A. J. Devaney, J. Math. Phys. 19, 1526 (1978). An overall review of the method of inverse scattering is given by A. J. Devaney in Optics in Four Dimensions, M. A. Machado, L. M. Narducci, eds., Conference Proceedings #65 (American Institute of Physics, New York, 1981). See also the collection of articles in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Heidelberg, 1980).
    [CrossRef]
  4. See, for example, V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), Chap. 7; L. A. Chernov, Wave Propagation in a Random Medium (Dover, New York, 1967). Chap. 5.
  5. The relative merits of the Rytov approximation over the Born approximation are discussed in Ref. 4 and by J. B. Keller, J. Opt. Soc. Am. 59, 1003 (1969). It should be emphasized that the Rytov approximation may lose its advantage over the Born approximation in cases in which the wavelength is not significantly shorter than the finest object detail. (In this connection, see Ref. 8).
  6. For a discussion of the eikonal approximation, see L. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York, 1968), p. 339. The Rytov approximation reduces to the eikonal approximation in the limit of very short wavelengths (see discussions in Ref. 4).
  7. K. Iwata, R. Nagata, Jpn. J. Appl. Phys. 14, Suppl. 14-1, 379 (1975).
    [CrossRef]
  8. R. K. Mueller, M. Kaveh, G. Wade, Proc. IEEE 67, 567 (1979).
    [CrossRef]
  9. See Ref. 4. In deriving Eq. (2) the quantity [δn(r)]2 has been assumed to be negligible in comparison to δn(r).
  10. E. Wolf, Opt. Commun. 1, 153 (1969).
    [CrossRef]
  11. R. Hosemann, S. H. Baghi, Direct Analysis of Diffraction of Matter (North-Holland, Amsterdam, 1962), Chap. I, Sec. 6.
  12. See Ref. 5 and also the discussion in K. Gottfried, Quantum Mechanics (Benjamin, New York, 1966), Sec. 13.

1979 (2)

For a survey of these techniques, see A. C. Kak, Proc. IEEE 67, 1245 (1979).
[CrossRef]

R. K. Mueller, M. Kaveh, G. Wade, Proc. IEEE 67, 567 (1979).
[CrossRef]

1975 (1)

K. Iwata, R. Nagata, Jpn. J. Appl. Phys. 14, Suppl. 14-1, 379 (1975).
[CrossRef]

1969 (3)

E. Wolf, Opt. Commun. 1, 153 (1969).
[CrossRef]

The relative merits of the Rytov approximation over the Born approximation are discussed in Ref. 4 and by J. B. Keller, J. Opt. Soc. Am. 59, 1003 (1969). It should be emphasized that the Rytov approximation may lose its advantage over the Born approximation in cases in which the wavelength is not significantly shorter than the finest object detail. (In this connection, see Ref. 8).

See, for example, E. Wolf, Opt. Commun. 1, 153 (1969); W. H. Carter, J. Opt. Soc. Am. 60, 306 (1970); W. H. Carter, P.-C. Ho, Appl. Opt. 13, 162 (1974); A. F. Fercher et al., Appl Opt. 18, 2427 (1979); A. J. Devaney, J. Math. Phys. 19, 1526 (1978). An overall review of the method of inverse scattering is given by A. J. Devaney in Optics in Four Dimensions, M. A. Machado, L. M. Narducci, eds., Conference Proceedings #65 (American Institute of Physics, New York, 1981). See also the collection of articles in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Heidelberg, 1980).
[CrossRef]

Baghi, S. H.

R. Hosemann, S. H. Baghi, Direct Analysis of Diffraction of Matter (North-Holland, Amsterdam, 1962), Chap. I, Sec. 6.

Gottfried, K.

See Ref. 5 and also the discussion in K. Gottfried, Quantum Mechanics (Benjamin, New York, 1966), Sec. 13.

Hosemann, R.

R. Hosemann, S. H. Baghi, Direct Analysis of Diffraction of Matter (North-Holland, Amsterdam, 1962), Chap. I, Sec. 6.

Iwata, K.

K. Iwata, R. Nagata, Jpn. J. Appl. Phys. 14, Suppl. 14-1, 379 (1975).
[CrossRef]

Kak, A. C.

For a survey of these techniques, see A. C. Kak, Proc. IEEE 67, 1245 (1979).
[CrossRef]

Kaveh, M.

R. K. Mueller, M. Kaveh, G. Wade, Proc. IEEE 67, 567 (1979).
[CrossRef]

Keller, J. B.

Mueller, R. K.

R. K. Mueller, M. Kaveh, G. Wade, Proc. IEEE 67, 567 (1979).
[CrossRef]

Nagata, R.

K. Iwata, R. Nagata, Jpn. J. Appl. Phys. 14, Suppl. 14-1, 379 (1975).
[CrossRef]

Schiff, L.

For a discussion of the eikonal approximation, see L. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York, 1968), p. 339. The Rytov approximation reduces to the eikonal approximation in the limit of very short wavelengths (see discussions in Ref. 4).

Tatarski, V. I.

See, for example, V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), Chap. 7; L. A. Chernov, Wave Propagation in a Random Medium (Dover, New York, 1967). Chap. 5.

Wade, G.

R. K. Mueller, M. Kaveh, G. Wade, Proc. IEEE 67, 567 (1979).
[CrossRef]

Wolf, E.

E. Wolf, Opt. Commun. 1, 153 (1969).
[CrossRef]

See, for example, E. Wolf, Opt. Commun. 1, 153 (1969); W. H. Carter, J. Opt. Soc. Am. 60, 306 (1970); W. H. Carter, P.-C. Ho, Appl. Opt. 13, 162 (1974); A. F. Fercher et al., Appl Opt. 18, 2427 (1979); A. J. Devaney, J. Math. Phys. 19, 1526 (1978). An overall review of the method of inverse scattering is given by A. J. Devaney in Optics in Four Dimensions, M. A. Machado, L. M. Narducci, eds., Conference Proceedings #65 (American Institute of Physics, New York, 1981). See also the collection of articles in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Heidelberg, 1980).
[CrossRef]

J. Opt. Soc. Am. (1)

Jpn. J. Appl. Phys. (1)

K. Iwata, R. Nagata, Jpn. J. Appl. Phys. 14, Suppl. 14-1, 379 (1975).
[CrossRef]

Opt. Commun. (2)

E. Wolf, Opt. Commun. 1, 153 (1969).
[CrossRef]

See, for example, E. Wolf, Opt. Commun. 1, 153 (1969); W. H. Carter, J. Opt. Soc. Am. 60, 306 (1970); W. H. Carter, P.-C. Ho, Appl. Opt. 13, 162 (1974); A. F. Fercher et al., Appl Opt. 18, 2427 (1979); A. J. Devaney, J. Math. Phys. 19, 1526 (1978). An overall review of the method of inverse scattering is given by A. J. Devaney in Optics in Four Dimensions, M. A. Machado, L. M. Narducci, eds., Conference Proceedings #65 (American Institute of Physics, New York, 1981). See also the collection of articles in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Heidelberg, 1980).
[CrossRef]

Proc. IEEE (2)

For a survey of these techniques, see A. C. Kak, Proc. IEEE 67, 1245 (1979).
[CrossRef]

R. K. Mueller, M. Kaveh, G. Wade, Proc. IEEE 67, 567 (1979).
[CrossRef]

Other (6)

See Ref. 4. In deriving Eq. (2) the quantity [δn(r)]2 has been assumed to be negligible in comparison to δn(r).

R. Hosemann, S. H. Baghi, Direct Analysis of Diffraction of Matter (North-Holland, Amsterdam, 1962), Chap. I, Sec. 6.

See Ref. 5 and also the discussion in K. Gottfried, Quantum Mechanics (Benjamin, New York, 1966), Sec. 13.

More specifically, they employ the eikonal approximation (see Ref. 6) to obtain a relationship between the complex phase of the field and so-called projections (see Ref. 1) of the object’s internal structure. This relationship is derived as a limiting case of the inverse-scattering method later in this Letter.

See, for example, V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), Chap. 7; L. A. Chernov, Wave Propagation in a Random Medium (Dover, New York, 1967). Chap. 5.

For a discussion of the eikonal approximation, see L. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York, 1968), p. 339. The Rytov approximation reduces to the eikonal approximation in the limit of very short wavelengths (see discussions in Ref. 4).

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