Abstract

The problem of determining a localized scattering potential V(r) from its associated scattering amplitude f(s, s0) is addressed within the first Born approximation. The conventional methods, based on Fourier synthesis, for obtaining approximate solutions to the problem are reviewed briefly. A new reconstruction method is proposed that is in the form of an integral transform over the scattering amplitude, assumed to be specified over all observation directions s and over all incident unit wave vectors s0. The proposed method is shown also to be applicable to the problem of determining the interatomic distance function ∫d3rV(r + r′) V*(r′) from the magnitude square of the scattering amplitude.

© 1982 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. For a discussion on the conditions required for the first Born approximation see K. Gottfried, Quantum Mechanics (Benjamin, New York, 1966), Sec. 13.
  2. See Ref. 1 and J. R. Taylor, Scattering Theory (Wiley, New York, 1972), Chap. 9.
  3. In optical applications the phase of the scattered field can be determined using holography. The use of holography in inverse scattering was first proposed by E. Wolf [ Opt. Commun. 1, 153 (1969)]. An interesting new application of holography to inverse scattering is presented by R. P. Porter, Opt. Commun. 39, 362 (1981).
    [CrossRef]
  4. B. K. Vainshtein, Diffraction of X-Rays by Chain Molecules (Elsevier, New York, 1966).
  5. See Ref. 4 and H. P. Klug, L. E. Alexander, X-Ray Diffraction Procedures (Wiley, New York, 1974), Chap. 3.
  6. See, for example, the discussion and references in A. J. Devaney in Optics in Four Dimensions (American Institute of Physics, New York, 1981), Chap. 15.
  7. W. H. Carter, J. Opt. Soc. Am. 60, 306 (1970); W. H. Carter, P. C. Ho, Appl. Opt. 18, 162 (1974).
    [CrossRef]
  8. See Ref. 6 and references therein.
  9. A. C. Kak, Proc. IEEE 67, 1245 (1979).
    [CrossRef]
  10. A. J. Devaney, Opt. Lett. 6, 374 (1981).
    [CrossRef] [PubMed]
  11. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Sec.7.7.

1981

In optical applications the phase of the scattered field can be determined using holography. The use of holography in inverse scattering was first proposed by E. Wolf [ Opt. Commun. 1, 153 (1969)]. An interesting new application of holography to inverse scattering is presented by R. P. Porter, Opt. Commun. 39, 362 (1981).
[CrossRef]

A. J. Devaney, Opt. Lett. 6, 374 (1981).
[CrossRef] [PubMed]

1979

A. C. Kak, Proc. IEEE 67, 1245 (1979).
[CrossRef]

1970

Alexander, L. E.

See Ref. 4 and H. P. Klug, L. E. Alexander, X-Ray Diffraction Procedures (Wiley, New York, 1974), Chap. 3.

Carter, W. H.

Devaney, A. J.

A. J. Devaney, Opt. Lett. 6, 374 (1981).
[CrossRef] [PubMed]

See, for example, the discussion and references in A. J. Devaney in Optics in Four Dimensions (American Institute of Physics, New York, 1981), Chap. 15.

Gottfried, K.

For a discussion on the conditions required for the first Born approximation see K. Gottfried, Quantum Mechanics (Benjamin, New York, 1966), Sec. 13.

Kak, A. C.

A. C. Kak, Proc. IEEE 67, 1245 (1979).
[CrossRef]

Klug, H. P.

See Ref. 4 and H. P. Klug, L. E. Alexander, X-Ray Diffraction Procedures (Wiley, New York, 1974), Chap. 3.

Porter, R. P.

In optical applications the phase of the scattered field can be determined using holography. The use of holography in inverse scattering was first proposed by E. Wolf [ Opt. Commun. 1, 153 (1969)]. An interesting new application of holography to inverse scattering is presented by R. P. Porter, Opt. Commun. 39, 362 (1981).
[CrossRef]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Sec.7.7.

Taylor, J. R.

See Ref. 1 and J. R. Taylor, Scattering Theory (Wiley, New York, 1972), Chap. 9.

Vainshtein, B. K.

B. K. Vainshtein, Diffraction of X-Rays by Chain Molecules (Elsevier, New York, 1966).

J. Opt. Soc. Am.

Opt. Commun.

In optical applications the phase of the scattered field can be determined using holography. The use of holography in inverse scattering was first proposed by E. Wolf [ Opt. Commun. 1, 153 (1969)]. An interesting new application of holography to inverse scattering is presented by R. P. Porter, Opt. Commun. 39, 362 (1981).
[CrossRef]

Opt. Lett.

Proc. IEEE

A. C. Kak, Proc. IEEE 67, 1245 (1979).
[CrossRef]

Other

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Sec.7.7.

For a discussion on the conditions required for the first Born approximation see K. Gottfried, Quantum Mechanics (Benjamin, New York, 1966), Sec. 13.

See Ref. 1 and J. R. Taylor, Scattering Theory (Wiley, New York, 1972), Chap. 9.

B. K. Vainshtein, Diffraction of X-Rays by Chain Molecules (Elsevier, New York, 1966).

See Ref. 4 and H. P. Klug, L. E. Alexander, X-Ray Diffraction Procedures (Wiley, New York, 1974), Chap. 3.

See, for example, the discussion and references in A. J. Devaney in Optics in Four Dimensions (American Institute of Physics, New York, 1981), Chap. 15.

See Ref. 6 and references therein.

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.


Equations (21)

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

f ( s , s 0 ) = ( 1 / 4 π ) d 3 r V ( r ) exp [ i k ( s s 0 ) · r ] .
| f ( s , s 0 ) | 2 = 1 / ( 4 π ) 2 d 3 r d 3 r V ( r ) V * ( r ) × exp [ i k ( s s 0 ) · ( r r ) ] = 1 / ( 4 π ) 2 d 3 r Q ( r ) exp [ i k ( s s 0 ) · r ] ,
Q ( r ) = d 3 r V ( r r ) V * ( r )
F ˜ ( K ) = d 3 r F ( r ) e i K · r
K = k ( s s 0 ) .
F ( r ) F LP ( r ) 1 / ( 2 π ) 3 K 2 k d 3 K F ˜ ( K ) e i K · r .
K = k ( s s 0 ) = l π / R 0 x ˆ + m π / R 0 y ˆ + n π / R 0 z ˆ ,
l 2 + m 2 + n 2 ( 2 k R 0 / π ) 2 .
F LP ( r ) = k 3 / ( 2 π ) 4 d Ω s d Ω s 0 | s s 0 | F ˜ [ k ( s s 0 ) ] × exp [ i k ( s s 0 ) · r ] .
V LP ( r ) = k 3 / 4 π 3 d Ω s d Ω s 0 | s s 0 | f ( s , s 0 ) × exp [ i k ( s s 0 ) · r ] ,
Q LP ( r ) = k 3 / π 2 d Ω s d Ω s 0 | s s 0 | | f ( s , s 0 ) | 2 × exp [ i k ( s s 0 ) · r ] .
F LP ( r ) = 1 / ( 2 π ) 3 d 3 K F ˜ ( K ) A ˜ ( K ) B ˜ ( K ) e i K · r ,
B ˜ ( K ) = { 1 / K if K 2 k 0 if K > 2 k .
F LP ( r ) = d 3 r F ( r ) Γ ( r r ) ,
Γ ( R ) = d 3 r A ( R r ) B ( r ) ,
A ( r ) 1 / ( 2 π ) 3 d 3 K A ˜ ( K ) e i K · r ,
B ( r ) 1 / ( 2 π ) 3 d 3 K B ˜ ( K ) e i K · r = 1 / ( 2 π ) 3 0 2 k K 2 d K d Ω s 1 / K e i K · r = k 2 / π 2 ( sin k r / k r ) 2 .
sin K r / K r = 1 / 4 π d Ω s e i K · r .
B ( r ) = k 2 / ( 2 π ) 4 d Ω s d Ω s 0 exp [ i k ( s s 0 ) · r ] ,
Γ ( R ) = k 2 / ( 2 π ) 4 d Ω s d Ω s 0 × exp [ i k ( s s 0 ) · R ] d 3 r A ( r ) × exp [ i k ( s s 0 ) · r ] = k 3 / ( 2 π ) 4 d Ω s d Ω s 0 | s s 0 | × exp [ i k ( s s 0 ) · R ] .
F LP ( r ) = k 3 / ( 2 π ) 4 d Ω s d Ω s 0 | s s 0 | × exp [ i k ( s s 0 ) · r ] d 3 r F ( r ) × exp [ i k ( s s 0 ) · r ] = k 3 / ( 2 π ) 4 d Ω s d Ω s 0 | s s 0 | F ˜ [ k ( s s 0 ) ] × exp [ i k ( s s 0 ) · r ] ,

Metrics