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.

© 1970 Optical Society of America

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References

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  1. E. Wolf, Optics Commun. 1, 153 (1969).
    [CrossRef]
  2. E. Wolf, J. Opt. Soc. Am. 60, 18 (1970).
    [CrossRef]
  3. J. S. Harris and M. P. Givens, J. Opt. Soc. Am. 56, 862 (1966).
    [CrossRef]
  4. B. R. Brown and A. W. Lohmann, Appl. Opt. 5, 967 (1966).
    [CrossRef] [PubMed]
  5. W. H. Carter and A. A. Dougal, J. Opt. Soc. Am. 56, 1754 (1966).
    [CrossRef]
  6. J. P. Waters, Appl. Phys. Letters 9, 405 (1966).
    [CrossRef]
  7. J. J. Burch, Proc. IEEE 55, 599 (1967).
    [CrossRef]
  8. J. W. Goodman and R. W. Lawrence, Appl. Phys. Letters 11, 77 (1967).
    [CrossRef]
  9. A. W. Lohmann and D. P. Paris, Appl. Opt. 6, 1739 (1967).
    [CrossRef] [PubMed]
  10. S. C. Keeton, Proc. IEEE 56, 325 (1968).
    [CrossRef]
  11. A. W. Lohmann and D. P. Paris, Appl. Opt. 7, 651 (1968).
    [CrossRef] [PubMed]
  12. L. B. Lesem, P. N. Hirsch, and J. A. Jordan, Commun. Assoc. Computing Machinery, Inc. 11, 661 (1968).
  13. L. B. Lesem, P. N. Hirsch, and J. A. Jordan, IBM J. Res. Develop. 13, 150 (1969).
    [CrossRef]
  14. B. R. Brown and A. W. Lohmann, IBM J. Res. Develop. 13, 160 (1969).
    [CrossRef]
  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)].
    [CrossRef]
  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.

1970 (1)

1969 (3)

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

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

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

1968 (4)

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)].
[CrossRef]

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

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

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

1967 (3)

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

J. J. Burch, Proc. IEEE 55, 599 (1967).
[CrossRef]

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

1966 (4)

Brown, B. R.

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

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

Burch, J. J.

J. J. Burch, Proc. IEEE 55, 599 (1967).
[CrossRef]

Carter, W. H.

Dougal, A. A.

Givens, M. P.

Goodman, J. W.

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

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

Harris, J. S.

Hirsch, P. N.

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

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

Jordan, J. A.

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

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

Keeton, S. C.

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

Lawrence, R. W.

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

Lesem, L. B.

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

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

Lohmann, A. W.

Paris, D. P.

Stetson, K. A.

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)].
[CrossRef]

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).
[CrossRef]

Wolf, E.

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

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

Appl. Opt. (3)

Appl. Phys. Letters (3)

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

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

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)].
[CrossRef]

Commun. Assoc. Computing Machinery, Inc. (1)

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

IBM J. Res. Develop. (2)

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

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

J. Opt. Soc. Am. (3)

Optics Commun. (1)

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

Proc. IEEE (2)

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

J. J. Burch, Proc. IEEE 55, 599 (1967).
[CrossRef]

Other (3)

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.

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

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Figures (13)

Fig. 1
Fig. 1

Scattering experiment. The illumination is incident from the left side of the figure as shown.

Fig. 2
Fig. 2

Optical system. The experiment was set up on a Tech/Ops research optical bench utilizing good-quality commercial components. The numbers in this figure identify (1) He–Ne laser (λ = 6328 Å), (2) spatial filter (9.5-μ pinhole), (3) diffraction grating (600 lines/mm), (4) stop, (5) lens (+5.0 cm fl), (6) lens (+60.9 cm fl), (7) scatterer, (8) hologram plate (649F).

Fig. 3
Fig. 3

Sequence of computation. The dotted lines indicate computation done by hand and the solid lines those done by computer.

Fig. 4
Fig. 4

The diffraction pattern from a rectangular fused-silica bar shown in the hologram plane without reference beam. The actual width of this pattern is the order of 2 mm.

Fig. 5
Fig. 5

Same as Fig. 4 with reference beam added. Note that the fringe period is much smaller than details of the diffraction pattern.

Fig. 6
Fig. 6

Photomicrograph of a hologram that was made of the pattern in Fig. 5. Note the clean carrier fringes that are necessary for quantitative data. The amplitdue modulation is clearly visible; however, the phase modulation is too slight to be detected visually.

Fig. 7
Fig. 7

Effective exposure vs density for the hologram in Fig. 6. This curve shows the known exposure values used to lay down a step wedge, plotted as a function of the measured density produced in a Kodak 649F spectroscopic plate after development.

Fig. 8
Fig. 8

Amplitude of the complex field U(x,z0) along a line perpendicular to the carrier fringes. These data were found from measurements made on the hologram shown in Fig. 6. Note that the field is the same as that which produced the diffraction pattern in Fig. 4.

Fig. 9
Fig. 9

Phase of U(x,z0) corresponding to the amplitude in Fig. 8.

Fig. 10
Fig. 10

Amplitude of the scattering potential Fx(x) biased by the illumination U(i)(x) as discussed in the Appendix. These data were computed from those in Figs. 8 and 9. The edges of the bar and the shape of the illuminating wavefront can be clearly seen.

Fig. 11
Fig. 11

Phase of the scattering potential Fx(x) corresponding to the amplitude in Fig. 10. The phase differences between disconnected segments of this curve are unknown (see Ref. 18).

Fig. 12
Fig. 12

Amplitude of the scattering potential and illumination for an inhomogeneous bar made of casting resin. Note that the pattern is similar to that in Fig. 10.

Fig. 13
Fig. 13

Phase of the scattering potential Fx(x) corresponding to the amplitude in Fig. 12. Note that if compared with Fig. 11, the phase suggests cylindrical error of the surface flatness, in excess of 5λ. The actual height of the phase peak is unknown since the different segments of this curve do not have a common phase reference (see Ref. 18).

Equations (35)

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U ( s ) ( x , y , ± z ) = - A ( ± ) ( p , q ) × exp [ i k ( p x + q y ± m z ) ] d p d q ,
A ( ± ) ( p , q ) = - i k 8 π 2 m - F ( x , y , z ) U ( i ) ( x , y , z ) × exp [ - i k ( p x + q y ± m z ) ] d x d y d z ,
m = ( 1 - p 2 - q 2 ) 1 2             if             p 2 + q 2 1 ,
m = i ( p 2 + q 2 - 1 ) 1 2             if             p 2 + q 2 > 1.
A ( ± ) ( p , q ) = 1 λ 2 e i k m z - U ( s ) ( x , y , ± z ) × e - i k ( p x + q y ) d x d y .
B ˆ ( ± ) ( u , v , w ) = - F ( x , y , z ) U ( i ) ( x , y , z ) × exp [ - i ( u x + v y ± w z ) ] d x d y d z .
A ( ± ) ( p , q ) = ( - i k / 8 π 2 m ) B ˆ ( ± ) ( k p , k q , k m ) ,
k m = k ( 1 - p 2 + q 2 ) 1 2 .
U ( i ) ( x , y , z ) = exp [ i k ( p 0 x + q 0 y + m 0 z ) ]
F ˆ ( U , V , W ± ) = ( i w / π ) e i k m z ± U ˆ ( s ) ( u , v ; w ± ) ,
U = u - k p 0 ,             V = v - k q 0 ,             W ± = ± w - k m 0
w = ( k 0 2 - u 2 - v 2 ) 1 2 .
F ˆ ( U , V , W ) = 1 ( 2 π ) 3 - F ( x , y , z ) × exp [ - i ( U x + V y + W z ) ] d x d y d z ,
U ˆ ( s ) ( u , v ; w ± ) = 1 ( 2 π ) 2 - U ( s ) ( x , y , z ± ) × exp [ - i ( u x + v y ) ] d x d y .
F ( x , y , z ) U ( i ) ( x , y , z ) = F x ( x ) F y ( y ) F z ( z ) e i k z .
A ( p , q ) = - i k 2 m F ˆ x ( p ) F ˆ y ( q ) - F z ( z ) e - i k ( m - 1 ) z d z ,
F ˆ x ( p ) = 1 2 π - F x ( x ) e - i k p x d x ,
F ˆ y ( q ) = 1 2 π - F y ( y ) e - i k q y d y .
m = ( 1 - p 2 - q 2 ) 1 2 1 - p 2 / 2 - q 2 / 2             to second order
1             to zero order .
- F z ( z ) exp [ - i k ( m - 1 ) z ] d z = C 1 ,
( π / λ ) ( p 2 + q 2 ) Δ z 0.1 ,
= ( 10 π λ Δ z ) 1 2
U ( s ) ( x , y , z 0 ) = - i k C 1 e i k z 0 2 - F ˆ x ( p ) exp ( - i k p 2 z 0 2 ) × e i k p x d p - F ˆ y ( q ) exp ( - i k q 2 z 0 2 ) e - i k q y d q ,
U ( s ) ( x , y , z 0 ) = ( k C 1 e i k z 0 / 2 ) V ( s ) ( x , z 0 ) W ( s ) ( y , z 0 ) ,
F ˆ x ( p ) = i λ exp ( i k p 2 z 0 2 ) - V ( s ) ( x , z 0 ) e - i k p x d x ,
F ˆ y ( q ) = 1 λ exp ( i k q 2 z 0 2 ) - W ( s ) ( y , z 0 ) e - i k q y d y .
F ˆ x ( p ) = i λ exp ( i k p 2 z 0 2 ) - U ( s ) ( x , z 0 ) e - i k p x d x ,
U ( x , z 0 ) = x - 3 d / 2 x + 3 d / 2 I ( x , y 0 ) exp ( 2 π i x d ) × sin [ 2 π ( x - x ) / 3 d ] [ 2 π ( x - x ) / 3 d ] d x ,
Δ x = 3 d / 2 ,
p λ / 2 Δ x ,
Δ θ = ( δ / d ) 360 ° ,
F ( x ) = F ( x ) + i λ - d p exp ( i k p 2 z 0 2 ) × - d x U ( ι ) ( x , y , z 0 ) e - i k p x e i k p x
U ( i ) = e i k z 0
F ( x ) = F ( x ) + i e i k z 0 .