Abstract

Three-dimensional position and velocity information can be extracted by directly analyzing the interference patterns of in-line, Fraunhofer holograms of small spherical particles. Spherically diverging light is shown to enhance resolution accuracy of the components of displacement perpendicular to the film plane. Data are obtained by measuring the radii of the interference patterns. The theory, experimental verification, and error analysis of this basic problem are discussed.

© 1970 Optical Society of America

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References

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  1. R. Menzel, T. G. Russell, F. M. Shofner, Soc. Phot. Eng. 8, 275 (1968).
  2. F. M. Shofner, R. Menzel, T. H. Gee, R. O. Webb, “Fundamentals of Holographic Velocimetry,” Intern. Congr. Instrumentation Aerospace Test Facilities Record, Farmingdale, N. Y., May1969 (IEEE, New York, 1969).
  3. D. Gabor, Nature 161, 777 (1948). Also Proc. Roy. Soc. A197, 454 (1949).
    [CrossRef] [PubMed]
  4. M. Born, B. Wolf, Principles of Optics (Pergamon, New York, 1965).
  5. G. W. Stroke, An Introduction to Coherent Optics and Holography (Academic, New York, 1966).
  6. J. B. DeVelis, G. O. Reynolds, Theory and Applications of Holography (Addison-Wesley, Reading, Mass., 1967).
  7. G. B. Parrent, B. J. Thompson, Opt. Acta 11, 183 (1964).
    [CrossRef]
  8. B. J. Thompson, Japan. J. Appl. Phys. Suppl. 1 4, 302 (1965).
  9. B. J. Thompson, J. H. Ward, W. R. Zinky, Appl. Opt. 6, 519 (1967).
    [CrossRef] [PubMed]
  10. R. A. Belz, M.S. thesis, University of Tennessee Space Institute (June1968).
  11. F. M. Shofner, R. Menzel, T. H. Gee, R. O. Webb, “Processing Holographic Velocimetry Data,” Intern. Congr. Instrumentation Aerospace Test Facilities Record, Farmingdale, New York, 1969 (IEEE, New York, 1969).
  12. F. M. Shofner, R. O. Webb, R. L. Fradenburg, Laser J. 2, 13–17 (1970).

1970 (1)

F. M. Shofner, R. O. Webb, R. L. Fradenburg, Laser J. 2, 13–17 (1970).

1968 (1)

R. Menzel, T. G. Russell, F. M. Shofner, Soc. Phot. Eng. 8, 275 (1968).

1967 (1)

1965 (1)

B. J. Thompson, Japan. J. Appl. Phys. Suppl. 1 4, 302 (1965).

1964 (1)

G. B. Parrent, B. J. Thompson, Opt. Acta 11, 183 (1964).
[CrossRef]

1948 (1)

D. Gabor, Nature 161, 777 (1948). Also Proc. Roy. Soc. A197, 454 (1949).
[CrossRef] [PubMed]

Belz, R. A.

R. A. Belz, M.S. thesis, University of Tennessee Space Institute (June1968).

Born, M.

M. Born, B. Wolf, Principles of Optics (Pergamon, New York, 1965).

DeVelis, J. B.

J. B. DeVelis, G. O. Reynolds, Theory and Applications of Holography (Addison-Wesley, Reading, Mass., 1967).

Fradenburg, R. L.

F. M. Shofner, R. O. Webb, R. L. Fradenburg, Laser J. 2, 13–17 (1970).

Gabor, D.

D. Gabor, Nature 161, 777 (1948). Also Proc. Roy. Soc. A197, 454 (1949).
[CrossRef] [PubMed]

Gee, T. H.

F. M. Shofner, R. Menzel, T. H. Gee, R. O. Webb, “Fundamentals of Holographic Velocimetry,” Intern. Congr. Instrumentation Aerospace Test Facilities Record, Farmingdale, N. Y., May1969 (IEEE, New York, 1969).

F. M. Shofner, R. Menzel, T. H. Gee, R. O. Webb, “Processing Holographic Velocimetry Data,” Intern. Congr. Instrumentation Aerospace Test Facilities Record, Farmingdale, New York, 1969 (IEEE, New York, 1969).

Menzel, R.

R. Menzel, T. G. Russell, F. M. Shofner, Soc. Phot. Eng. 8, 275 (1968).

F. M. Shofner, R. Menzel, T. H. Gee, R. O. Webb, “Fundamentals of Holographic Velocimetry,” Intern. Congr. Instrumentation Aerospace Test Facilities Record, Farmingdale, N. Y., May1969 (IEEE, New York, 1969).

F. M. Shofner, R. Menzel, T. H. Gee, R. O. Webb, “Processing Holographic Velocimetry Data,” Intern. Congr. Instrumentation Aerospace Test Facilities Record, Farmingdale, New York, 1969 (IEEE, New York, 1969).

Parrent, G. B.

G. B. Parrent, B. J. Thompson, Opt. Acta 11, 183 (1964).
[CrossRef]

Reynolds, G. O.

J. B. DeVelis, G. O. Reynolds, Theory and Applications of Holography (Addison-Wesley, Reading, Mass., 1967).

Russell, T. G.

R. Menzel, T. G. Russell, F. M. Shofner, Soc. Phot. Eng. 8, 275 (1968).

Shofner, F. M.

F. M. Shofner, R. O. Webb, R. L. Fradenburg, Laser J. 2, 13–17 (1970).

R. Menzel, T. G. Russell, F. M. Shofner, Soc. Phot. Eng. 8, 275 (1968).

F. M. Shofner, R. Menzel, T. H. Gee, R. O. Webb, “Fundamentals of Holographic Velocimetry,” Intern. Congr. Instrumentation Aerospace Test Facilities Record, Farmingdale, N. Y., May1969 (IEEE, New York, 1969).

F. M. Shofner, R. Menzel, T. H. Gee, R. O. Webb, “Processing Holographic Velocimetry Data,” Intern. Congr. Instrumentation Aerospace Test Facilities Record, Farmingdale, New York, 1969 (IEEE, New York, 1969).

Stroke, G. W.

G. W. Stroke, An Introduction to Coherent Optics and Holography (Academic, New York, 1966).

Thompson, B. J.

B. J. Thompson, J. H. Ward, W. R. Zinky, Appl. Opt. 6, 519 (1967).
[CrossRef] [PubMed]

B. J. Thompson, Japan. J. Appl. Phys. Suppl. 1 4, 302 (1965).

G. B. Parrent, B. J. Thompson, Opt. Acta 11, 183 (1964).
[CrossRef]

Ward, J. H.

Webb, R. O.

F. M. Shofner, R. O. Webb, R. L. Fradenburg, Laser J. 2, 13–17 (1970).

F. M. Shofner, R. Menzel, T. H. Gee, R. O. Webb, “Processing Holographic Velocimetry Data,” Intern. Congr. Instrumentation Aerospace Test Facilities Record, Farmingdale, New York, 1969 (IEEE, New York, 1969).

F. M. Shofner, R. Menzel, T. H. Gee, R. O. Webb, “Fundamentals of Holographic Velocimetry,” Intern. Congr. Instrumentation Aerospace Test Facilities Record, Farmingdale, N. Y., May1969 (IEEE, New York, 1969).

Wolf, B.

M. Born, B. Wolf, Principles of Optics (Pergamon, New York, 1965).

Zinky, W. R.

Appl. Opt. (1)

Japan. J. Appl. Phys. Suppl. 1 (1)

B. J. Thompson, Japan. J. Appl. Phys. Suppl. 1 4, 302 (1965).

Laser J. (1)

F. M. Shofner, R. O. Webb, R. L. Fradenburg, Laser J. 2, 13–17 (1970).

Nature (1)

D. Gabor, Nature 161, 777 (1948). Also Proc. Roy. Soc. A197, 454 (1949).
[CrossRef] [PubMed]

Opt. Acta (1)

G. B. Parrent, B. J. Thompson, Opt. Acta 11, 183 (1964).
[CrossRef]

Soc. Phot. Eng. (1)

R. Menzel, T. G. Russell, F. M. Shofner, Soc. Phot. Eng. 8, 275 (1968).

Other (6)

F. M. Shofner, R. Menzel, T. H. Gee, R. O. Webb, “Fundamentals of Holographic Velocimetry,” Intern. Congr. Instrumentation Aerospace Test Facilities Record, Farmingdale, N. Y., May1969 (IEEE, New York, 1969).

M. Born, B. Wolf, Principles of Optics (Pergamon, New York, 1965).

G. W. Stroke, An Introduction to Coherent Optics and Holography (Academic, New York, 1966).

J. B. DeVelis, G. O. Reynolds, Theory and Applications of Holography (Addison-Wesley, Reading, Mass., 1967).

R. A. Belz, M.S. thesis, University of Tennessee Space Institute (June1968).

F. M. Shofner, R. Menzel, T. H. Gee, R. O. Webb, “Processing Holographic Velocimetry Data,” Intern. Congr. Instrumentation Aerospace Test Facilities Record, Farmingdale, New York, 1969 (IEEE, New York, 1969).

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Figures (8)

Fig. 1
Fig. 1

Experimental configuration.

Fig. 2
Fig. 2

Holographic doublet: a = 110 ± 7 μm; λ = 0.6328 μm; z1 = 40 cm; zo = 20 cm; Δz1 = 1 cm. Film: Kodak SO 243.

Fig. 3
Fig. 3

The effect of the Bessel function and the sine function on the intensity versus r: ——two-term theory; –-–-–-three-term theory; –··· typical experimental result.

Fig. 4
Fig. 4

Microdensitometer trace of a singlet.

Fig. 5
Fig. 5

Constant (zo + z1) = 175.5 cm.

Fig. 6
Fig. 6

Constant (zo + z1) = 60.0 cm.

Fig. 7
Fig. 7

Spread in eleven microdensitometer readings of singlets for different n and different z1.

Fig. 8
Fig. 8

Histograms of multiple microdensitometer traces across a singlet; the abscissa represents z1 in centimeters.

Equations (22)

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U ( P ) = i A z o ( a r ) exp [ i k ( z o + z 1 ) ] exp ( i k r 2 2 z 1 ) J 1 ( k a r z 1 ) ,
U 1 + U 2 = U i .
U 2 = U i U 1 .
U i = ( A / s ) exp ( i k s ) ,
s = ( Z 2 + r 2 ) 1 2 Z + r 2 / 2 Z = z o + z 1 z o r 2 / 2 z 1 ( z o + z 1 ) + r 2 / 2 z 1 .
s Z = z o + z 1
U i A ( z o + z 1 ) exp [ i k ( z o + z 1 ) ] exp ( i k r 2 2 z 1 ) × exp [ i k z o r 2 2 z 1 ( z o + z 1 ) ] ,
I = ( U i U 1 ) ( U i U 1 ) * = A 2 [ 1 ( z o + z 1 ) 2 + a 2 J 1 2 ( k a r z 1 ) r 2 z o 2 2 ( a r ) J 1 ( k a r z 1 ) z o ( z o + z 1 ) sin k z o r 2 2 z 1 ( z o + z 1 ) ] .
r n = { [ ( 2 n + 1 ) λ z 1 / 2 ] ( 1 + z 1 / z o ) } 1 2 ,
z 1 = r n 2 ( n + 1 2 ) λ + r n 2 / Z .
I plane wave = lim z o ( z o 2 I spherical wave ) = A o 2 × [ 1 + a 2 J 1 2 ( k a r / z 1 ) r 2 2 ( a r ) J 1 ( k a r z 1 ) sin ( k r 2 2 z 1 ) ] .
δ z 1 = λ ( 2 n + 1 ) [ r n 2 / Z + λ ( n + 1 2 ) ] 2 r n δ r n .
δ z 1 1.52 × 10 2 δ r n .
δ z 1 0.87 × 10 2 δ r n .
I = 1 g ( r ) f ( r ) ,
d I d r = 0 = g r f ( r ) g ( r ) f r ,
f / r g / r = f ( r ) g ( r ) .
tan β = k z o z 1 Z / [ 2 r n 2 k a z 1 r n J o ( α ) J 1 ( α ) ] .
z 1 = [ Z / ( 2 Z β k r n 2 + 1 ) ] ,
z 1 = 39.84 cm for n = 1 , z 1 = 39.87 cm for n = 2 , z 1 = 39.88 cm for n = 3 , z 1 = 39.89 cm for n = 4 ;
z 1 = 40.03 cm for n = 1 , z 1 = 39.99 cm for n = 2 , z 1 = 39.98 cm for n = 3 , z 1 = 39.98 cm for n = 4.
Δ z = u ( λ / 2 π ) ( f / a ) 2 ,

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