Abstract

An equation describing the reconstructed holographic images of objects in any state of motion is obtained starting from consideration of the effect of motion on coherence. The equation explicitly shows the dependence of the reconstruction on both the total displacement and the form of the law of motion. Applications to the measurement of motion for the special cases of vibration, constant velocity motion, and their combination are presented together with several experimental results. In particular the theory is shown to be valid for both double exposure and time average exposure methods of displacement, distortion, and vibration analysis.

© 1970 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. O. Heflinger, R. F. Wuerker, R. F. Brooks, J. Appl. Phys. 37, 642 (1966).
    [CrossRef]
  2. J. M. Burch, A. E. Ennos, R. J. Wilton, Nature 209, 1015 (1966).
    [CrossRef]
  3. R. L. Powell, K. A. Stetson, J. Opt. Soc. Amer. 55, 1593 (1965).
    [CrossRef]
  4. M. Lurie, M. Zambuto, Appl. Opt. 7, 2323 (1968).
    [CrossRef] [PubMed]
  5. M. O. Fein, E. L. Green, Appl. Opt. 7, 1864 (1968).
    [CrossRef] [PubMed]
  6. M. Lurie, J. Opt. Soc. Amer. 57, 573A (1967).
  7. J. D. Redmon, J. Sci. Instrum. 44, 1032 Note (1967).
    [CrossRef]
  8. H. M. Smith, Principles of Holography (Wiley, New York, 1969).
  9. D. B. Neumann, J. Opt. Soc. Amer. 58, 447 (1968).
    [CrossRef]
  10. M. Lurie, J. Opt. Soc. Amer. 56, 1369 (1966); J. Opt. Soc. Amer. 58, 614 (1968).
    [CrossRef]
  11. The evaluation of this integral was first pointed out in a private communication from H. H. Hopkins, University of Reading, Reading, Berks., England.
  12. R. I. Collier, E. I. Doherty, K. Pennington, Appl. Phys. Lett. 7, 223 (1965).
    [CrossRef]
  13. K. A. Haines, B. P. Hildebrand, Appl. Opt. 5, 595 (1966).
    [CrossRef] [PubMed]

1968 (3)

1967 (2)

M. Lurie, J. Opt. Soc. Amer. 57, 573A (1967).

J. D. Redmon, J. Sci. Instrum. 44, 1032 Note (1967).
[CrossRef]

1966 (4)

L. O. Heflinger, R. F. Wuerker, R. F. Brooks, J. Appl. Phys. 37, 642 (1966).
[CrossRef]

J. M. Burch, A. E. Ennos, R. J. Wilton, Nature 209, 1015 (1966).
[CrossRef]

M. Lurie, J. Opt. Soc. Amer. 56, 1369 (1966); J. Opt. Soc. Amer. 58, 614 (1968).
[CrossRef]

K. A. Haines, B. P. Hildebrand, Appl. Opt. 5, 595 (1966).
[CrossRef] [PubMed]

1965 (2)

R. I. Collier, E. I. Doherty, K. Pennington, Appl. Phys. Lett. 7, 223 (1965).
[CrossRef]

R. L. Powell, K. A. Stetson, J. Opt. Soc. Amer. 55, 1593 (1965).
[CrossRef]

Brooks, R. F.

L. O. Heflinger, R. F. Wuerker, R. F. Brooks, J. Appl. Phys. 37, 642 (1966).
[CrossRef]

Burch, J. M.

J. M. Burch, A. E. Ennos, R. J. Wilton, Nature 209, 1015 (1966).
[CrossRef]

Collier, R. I.

R. I. Collier, E. I. Doherty, K. Pennington, Appl. Phys. Lett. 7, 223 (1965).
[CrossRef]

Doherty, E. I.

R. I. Collier, E. I. Doherty, K. Pennington, Appl. Phys. Lett. 7, 223 (1965).
[CrossRef]

Ennos, A. E.

J. M. Burch, A. E. Ennos, R. J. Wilton, Nature 209, 1015 (1966).
[CrossRef]

Fein, M. O.

Green, E. L.

Haines, K. A.

Heflinger, L. O.

L. O. Heflinger, R. F. Wuerker, R. F. Brooks, J. Appl. Phys. 37, 642 (1966).
[CrossRef]

Hildebrand, B. P.

Hopkins, H. H.

The evaluation of this integral was first pointed out in a private communication from H. H. Hopkins, University of Reading, Reading, Berks., England.

Lurie, M.

M. Lurie, M. Zambuto, Appl. Opt. 7, 2323 (1968).
[CrossRef] [PubMed]

M. Lurie, J. Opt. Soc. Amer. 57, 573A (1967).

M. Lurie, J. Opt. Soc. Amer. 56, 1369 (1966); J. Opt. Soc. Amer. 58, 614 (1968).
[CrossRef]

Neumann, D. B.

D. B. Neumann, J. Opt. Soc. Amer. 58, 447 (1968).
[CrossRef]

Pennington, K.

R. I. Collier, E. I. Doherty, K. Pennington, Appl. Phys. Lett. 7, 223 (1965).
[CrossRef]

Powell, R. L.

R. L. Powell, K. A. Stetson, J. Opt. Soc. Amer. 55, 1593 (1965).
[CrossRef]

Redmon, J. D.

J. D. Redmon, J. Sci. Instrum. 44, 1032 Note (1967).
[CrossRef]

Smith, H. M.

H. M. Smith, Principles of Holography (Wiley, New York, 1969).

Stetson, K. A.

R. L. Powell, K. A. Stetson, J. Opt. Soc. Amer. 55, 1593 (1965).
[CrossRef]

Wilton, R. J.

J. M. Burch, A. E. Ennos, R. J. Wilton, Nature 209, 1015 (1966).
[CrossRef]

Wuerker, R. F.

L. O. Heflinger, R. F. Wuerker, R. F. Brooks, J. Appl. Phys. 37, 642 (1966).
[CrossRef]

Zambuto, M.

Appl. Opt. (3)

Appl. Phys. Lett. (1)

R. I. Collier, E. I. Doherty, K. Pennington, Appl. Phys. Lett. 7, 223 (1965).
[CrossRef]

J. Appl. Phys. (1)

L. O. Heflinger, R. F. Wuerker, R. F. Brooks, J. Appl. Phys. 37, 642 (1966).
[CrossRef]

J. Opt. Soc. Amer. (4)

M. Lurie, J. Opt. Soc. Amer. 57, 573A (1967).

D. B. Neumann, J. Opt. Soc. Amer. 58, 447 (1968).
[CrossRef]

M. Lurie, J. Opt. Soc. Amer. 56, 1369 (1966); J. Opt. Soc. Amer. 58, 614 (1968).
[CrossRef]

R. L. Powell, K. A. Stetson, J. Opt. Soc. Amer. 55, 1593 (1965).
[CrossRef]

J. Sci. Instrum. (1)

J. D. Redmon, J. Sci. Instrum. 44, 1032 Note (1967).
[CrossRef]

Nature (1)

J. M. Burch, A. E. Ennos, R. J. Wilton, Nature 209, 1015 (1966).
[CrossRef]

Other (2)

H. M. Smith, Principles of Holography (Wiley, New York, 1969).

The evaluation of this integral was first pointed out in a private communication from H. H. Hopkins, University of Reading, Reading, Berks., England.

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.


Figures (9)

Fig. 1
Fig. 1

The geometry for calculating the doppler shift produced by object motion.

Fig. 2
Fig. 2

A schematic of the moving object used to test holographic measurements. A steel frame holding the screws and transducer is omitted for clarity. A signal applied to the transducer causes the object to rotate about the line between the two support screws. The mirror M is in one arm of a Twyman-Green interferometer which provides a nonholographic measurement of the motion.

Fig. 3
Fig. 3

The calculated effect of object vibration with amplitude Am on the radiance of a reconstruction. BI(m)/B0(m) is the ratio of the radiance at a point on the reconstruction to the radiance that would have been obtained if the object had not moved. The upper curve shows the motion of any point on the object as a function of time. The lower curve shows the effect of that motion on the radiance of that point of the reconstruction.

Fig. 4
Fig. 4

Reconstructions of the object shown in Figs. 2 and 9. The object moved during the exposure time T in the following ways: (a) no motion, (b) vibration with amplitude A, (c) uniform velocity υ, and (d) combined vibration with amplitude A and uniform velocity υ. In each example, the motion was a small rotation about the two support screws, so the magnitude of the motion was zero between the screws and increased with distance from them. The numbers beside the dark fringes are the magnitudes of the motions at those fringes, as determined holographically. The numbers in parentheses are the comparable magnitudes determined with the Twyman-Green interferometer.

Fig. 5
Fig. 5

The calculated effect on the radiance of a reconstruction of uniform object motion with constant speed υm and final displacement υmT.

Fig. 6
Fig. 6

The calculated effect on the radiance of a reconstruction of a combined uniform motion and vibration of the object. The maximum displacement of the uniform motion component is υmT. The amplitude of the vibration is Am. The appearance of the reconstruction depends on the ratio υmT/Am. For this figure, as for our experiment, υmT/Am = 1.68. Curve 1 shows the effect of the constant velocity motion alone; curve 2 shows the effect of the vibration alone. The solid curve shows the result of the combined motions and corresponds to the radiance distribution indicated by Figure 4(d).

Fig. 7
Fig. 7

A microdensitometer trace showing the locations of the zeros in the reconstruction of an object moving with combined vibration and uniform velocity. The trace was made from the negative used to print Fig. 4(d) and shows the dark fringes more clearly than the print. The positive numbers indicate the distance in centimeters from the zeros to the rotation axis of the object. The negative number is the distance from the axis to the bottom edge of the object which was the reference for these measurements.

Fig. 8
Fig. 8

The calculated effect on the radiance of the reconstruction of a step displacement of magnitude Dm occurring at the middle of the exposure time. This motion corresponds to the double exposure techniques.

Fig. 9
Fig. 9

The moving object and its supporting frame are in the center of the photograph. (Compare with Fig. 2.) The wires connect to the transducer located behind the object. To the left is the plane mirror which provides a reference beam. The components behind the object are part of the Twyman-Green interferometer used to verify the holographic measurements of object motion.

Equations (27)

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

B I ( m ) = B 0 ( m ) | γ m R ( 0 ) | 2 ,
V R ( t ) = A R exp ( j ω t ) = A R exp ( j 0 t ω d τ ) .
V m ( t ) = A m exp [ j 0 t ω m ( τ ) d τ ] ,
I R = A R 2 , I m = A m 2 .
γ m R ( 0 ) = V m ( t ) V R ( t ) / ( I R I m ) 1 2 ,
g m R ( 0 ) = 1 T 0 T V m ( t ) V R ( t ) d t ( I R I m ) 1 2 ,
g m R ( 0 ) = 1 T 0 T exp { j 0 t [ ω m ( τ ) ω ] d τ } d t .
d φ m = ( 2 π / λ ) d s m = ( 2 π / λ ) d x m ( cos α + cos β ) ,
ω m ( t ) = ω + ( d φ m / d t ) = ω + ( 2 π / λ ) ( d x m / d t ) ( cos α + cos β ) .
g m R ( 0 ) = 1 T 0 T exp [ j 2 π λ ( cos α + cos β ) x m ( t ) ] d t ,
g m R ( 0 ) = 1 T 0 T exp [ j 4 π λ x m ( t ) ] d t .
B I ( m ) = B 0 ( m ) | 1 T 0 T exp [ j 4 π λ x m ( t ) ] d t | 2 .
x m ( t ) = A m sin Ω t ,
g m R ( 0 ) = 1 T 0 T exp ( j 4 π λ A m sin Ω t ) d t = n = + J n ( 4 π A m λ ) exp ( j n Ω T ) 1 j n Ω T ,
g m R ( 0 ) = J 0 ( 4 π A m / λ ) .
B I ( m ) B 0 ( m ) { J 0 [ 4 π ( A m / λ ) ] } 2 .
g m R ( 0 ) = 1 T 0 T exp ( j 4 π υ m λ t ) d t = exp [ j ( 4 π / λ ) υ m T ] 1 j ( 4 π / λ ) υ m T .
B I ( m ) B 0 ( m ) = 1 cos ( 4 π / λ ) υ m T 8 [ π ( υ m / λ ) T ] 2 .
x m ( t ) = υ m t + A m sin Ω t ,
g m R ( 0 ) = 1 T 0 T exp [ j 4 π λ ( υ m t + A m sin Ω t ) ] d t .
g m R ( 0 ) = J 0 ( 4 π A m λ ) exp ( j 4 π υ m T / λ ) 1 j 4 π υ m T / λ + n = 1 J n ( 4 π A m λ ) { exp [ j ( 4 π υ m T / λ + n Ω T ) ] 1 j ( 4 π υ m T / λ + n Ω T ) + ( 1 ) n exp [ j ( 4 π υ m T / λ n Ω T ) ] 1 j ( 4 π υ m T / λ n Ω T ) } ,
g m R ( 0 ) J 0 ( 4 π A m λ ) exp [ j ( 4 π / λ ) υ m T ] 1 j ( 4 π / λ ) υ m T ,
B I ( m ) B 0 ( m ) = [ J 0 4 π A m λ ] 2 1 cos [ ( 4 π / λ ) υ m T ] 8 [ ( π / λ ) υ m T ] 2 ,
x 5.46 ( t ) = ( 0.028 λ ) t + ( 0.63 λ ) sin ( 2 π × 50 t ) ,
x m ( t ) = D m u [ t ( T / 2 ) ] ,
g m R ( 0 ) = 1 T [ 0 T / 2 d t + T / 2 T exp ( j 4 π λ D m ) d t ] = 1 2 [ 1 + exp ( j 4 π λ D m ) ] ,
B I ( m ) B 0 ( m ) = 1 + cos ( 4 π D m / λ ) 2 ,

Metrics