Abstract

This paper reports the successful computational determination of structural detail in a simple transparent object through holographic measurement of scattered monochromatic light. The complex disturbance of the scattered light is measured in amplitude and phase, along a line transverse to the illumination in the Fresnel zone of the object. The scattering potential of the object is then calculated along a parallel line using the field data and a new inverse scattering theory. The results agree well with the known parameters of the two test objects, a high-quality and a low-quality right parallelepiped aligned with two faces normal to the illumination. This experiment is believed to be the first which includes the quantitative reconstruction of structure in a physical object from measurement of scattered light. The technique is somewhat similar to that employed in connection with reconstruction of crystal structures from x-ray diffraction experiments.

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  1. E. Wolf, Optics Commun. 1, 153 (1969).
  2. E. Wolf, J. Opt. Soc. Am. 60, 18 (1970).
  3. J. S. Harris and M. P. Givens, J. Opt. Soc. Am. 56, 862 (1966).
  4. B. R. Brown and A. W. Lohmann, Appl. Opt. 5, 967 (1966).
  5. W. H. Carter and A. A. Dougal, J. Opt. Soc. Am. 56, 1754 (1966).
  6. J. P. Waters, Appl. Phys. Letters 9, 405 (1966).
  7. J. J. Burch, Proc. IEEE 55, 599 (1967).
  8. J. W. Goodman and R. W. Lawrence, Appl. Phys. Letters 11, 77 (1967).
  9. A. W. Lohmann and D. P. Paris, Appl. Opt. 6, 1739 (1967).
  10. S. C. Keeton, Proc. IEEE 56, 325 (1968).
  11. A. W. Lohmann and D. P. Paris, Appl. Opt. 7, 651 (1968).
  12. L. B. Lesem, P. N. Hirsch, and J. A. Jordan, Jr., Commun. Assoc. Computing Machinery, Inc. 11, 661 (1968).
  13. L. B. Lesem, P. N. Hirsch, and J. A. Jordan, Jr., IBM J. Res. Develop. 13, 150 (1969).
  14. B. R. Brown and A. W. Lohmann, IBM J. Res. Develop. 13, 160 (1969).
  15. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill Book Co., New York, 1968), p. 48.
  16. The range of the parameters p, q satisfying Eq. (3b) describes evanescent waves which decay very rapidly outside region II of Fig. 1, and is not readily observed experimentally. These waves carry information of object structures with dimensions smaller than about λ. Since they are hard to detect, they have little effect on the measured field and are neglected here. One of the anonymous reviewers pointed out that evanescent waves have been detected with holograms by K. A. Stetson [Appl. Phys. Letters 12, 362 (1968)], H. Nassenstein [Phys. Letters 29A, 175 (1969)], and Olof Bryngdahl [J. Opt. Soc. Am. 59, 1645 (1969)].
  17. H. L. Van Trees, Detection, Estimation and Modulation Theory (John Wiley & Sons, Inc., New York, 1968), Pt. I, p. 169.
  18. The variation of phase at the discontinuties in Figs. 11 and 13 was faster than 2π rad/sample interval. Thus it is impossible to determine the phases in different discontinuous segments of the data, to the same phase reference. The true phase difference between points on different segments of the phase data is unknown by an interval equal to an integral multiple of 2π. To plot all segments together in Figs. 11 and 13, the 2π intervals have been discarded; the segments are shown on the same scale in a physically reasonable manner.

Brown, B. R.

B. R. Brown and A. W. Lohmann, Appl. Opt. 5, 967 (1966).

B. R. Brown and A. W. Lohmann, IBM J. Res. Develop. 13, 160 (1969).

Burch, J. J.

J. J. Burch, Proc. IEEE 55, 599 (1967).

Carter, W. H.

W. H. Carter and A. A. Dougal, J. Opt. Soc. Am. 56, 1754 (1966).

Dougal, A. A.

W. H. Carter and A. A. Dougal, J. Opt. Soc. Am. 56, 1754 (1966).

Givens, M. P.

J. S. Harris and M. P. Givens, J. Opt. Soc. Am. 56, 862 (1966).

Goodman, J. W.

J. W. Goodman and R. W. Lawrence, Appl. Phys. Letters 11, 77 (1967).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill Book Co., New York, 1968), p. 48.

Harris, J. S.

J. S. Harris and M. P. Givens, J. Opt. Soc. Am. 56, 862 (1966).

Hirsch, P. N.

L. B. Lesem, P. N. Hirsch, and J. A. Jordan, Jr., Commun. Assoc. Computing Machinery, Inc. 11, 661 (1968).

L. B. Lesem, P. N. Hirsch, and J. A. Jordan, Jr., IBM J. Res. Develop. 13, 150 (1969).

Jordan, Jr., J. A.

L. B. Lesem, P. N. Hirsch, and J. A. Jordan, Jr., IBM J. Res. Develop. 13, 150 (1969).

L. B. Lesem, P. N. Hirsch, and J. A. Jordan, Jr., Commun. Assoc. Computing Machinery, Inc. 11, 661 (1968).

Keeton, S. C.

S. C. Keeton, Proc. IEEE 56, 325 (1968).

Lawrence, R. W.

J. W. Goodman and R. W. Lawrence, Appl. Phys. Letters 11, 77 (1967).

Lesem, L. B.

L. B. Lesem, P. N. Hirsch, and J. A. Jordan, Jr., IBM J. Res. Develop. 13, 150 (1969).

L. B. Lesem, P. N. Hirsch, and J. A. Jordan, Jr., Commun. Assoc. Computing Machinery, Inc. 11, 661 (1968).

Lohmann, A. W.

B. R. Brown and A. W. Lohmann, IBM J. Res. Develop. 13, 160 (1969).

A. W. Lohmann and D. P. Paris, Appl. Opt. 7, 651 (1968).

A. W. Lohmann and D. P. Paris, Appl. Opt. 6, 1739 (1967).

B. R. Brown and A. W. Lohmann, Appl. Opt. 5, 967 (1966).

Paris, D. P.

A. W. Lohmann and D. P. Paris, Appl. Opt. 6, 1739 (1967).

A. W. Lohmann and D. P. Paris, Appl. Opt. 7, 651 (1968).

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation and Modulation Theory (John Wiley & Sons, Inc., New York, 1968), Pt. I, p. 169.

Waters, J. P.

J. P. Waters, Appl. Phys. Letters 9, 405 (1966).

Wolf, E.

E. Wolf, Optics Commun. 1, 153 (1969).

E. Wolf, J. Opt. Soc. Am. 60, 18 (1970).

Other

E. Wolf, Optics Commun. 1, 153 (1969).

E. Wolf, J. Opt. Soc. Am. 60, 18 (1970).

J. S. Harris and M. P. Givens, J. Opt. Soc. Am. 56, 862 (1966).

B. R. Brown and A. W. Lohmann, Appl. Opt. 5, 967 (1966).

W. H. Carter and A. A. Dougal, J. Opt. Soc. Am. 56, 1754 (1966).

J. P. Waters, Appl. Phys. Letters 9, 405 (1966).

J. J. Burch, Proc. IEEE 55, 599 (1967).

J. W. Goodman and R. W. Lawrence, Appl. Phys. Letters 11, 77 (1967).

A. W. Lohmann and D. P. Paris, Appl. Opt. 6, 1739 (1967).

S. C. Keeton, Proc. IEEE 56, 325 (1968).

A. W. Lohmann and D. P. Paris, Appl. Opt. 7, 651 (1968).

L. B. Lesem, P. N. Hirsch, and J. A. Jordan, Jr., Commun. Assoc. Computing Machinery, Inc. 11, 661 (1968).

L. B. Lesem, P. N. Hirsch, and J. A. Jordan, Jr., IBM J. Res. Develop. 13, 150 (1969).

B. R. Brown and A. W. Lohmann, IBM J. Res. Develop. 13, 160 (1969).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill Book Co., New York, 1968), p. 48.

The range of the parameters p, q satisfying Eq. (3b) describes evanescent waves which decay very rapidly outside region II of Fig. 1, and is not readily observed experimentally. These waves carry information of object structures with dimensions smaller than about λ. Since they are hard to detect, they have little effect on the measured field and are neglected here. One of the anonymous reviewers pointed out that evanescent waves have been detected with holograms by K. A. Stetson [Appl. Phys. Letters 12, 362 (1968)], H. Nassenstein [Phys. Letters 29A, 175 (1969)], and Olof Bryngdahl [J. Opt. Soc. Am. 59, 1645 (1969)].

H. L. Van Trees, Detection, Estimation and Modulation Theory (John Wiley & Sons, Inc., New York, 1968), Pt. I, p. 169.

The variation of phase at the discontinuties in Figs. 11 and 13 was faster than 2π rad/sample interval. Thus it is impossible to determine the phases in different discontinuous segments of the data, to the same phase reference. The true phase difference between points on different segments of the phase data is unknown by an interval equal to an integral multiple of 2π. To plot all segments together in Figs. 11 and 13, the 2π intervals have been discarded; the segments are shown on the same scale in a physically reasonable manner.

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