Abstract

Generally, both the phase and amplitude of vibration of an object's surface vary from point to point. This paper details two methods for extracting vibrational phase information from the higher order fringes of holographic interferograms recorded by phase modulating either the object or reference illumination. Both methods require the superposition of only a few transparencies of interferograms in order to generate useful vibration-phase contour maps.

© 1976 Optical Society of America

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References

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  1. K. A. Stetson, P. A. Taylor, J. Phys. E: 4, 1009 (1971).
    [CrossRef]
  2. W. C. Hurty, M. F. Rubinstein, Dynamics of Structures (Prentice-Hall, Englewood Cliffs, N.J., 1964), Chap. 9, p. 313.
  3. C. C. Aleksoff, Appl. Phys. Lett. 14, 23 (1969).
    [CrossRef]
  4. D. B. Neumann, C. F. Jacobson, G. M. Brown, Appl. Opt. 9, 1357 (1970).
    [CrossRef] [PubMed]

1971 (1)

K. A. Stetson, P. A. Taylor, J. Phys. E: 4, 1009 (1971).
[CrossRef]

1970 (1)

1969 (1)

C. C. Aleksoff, Appl. Phys. Lett. 14, 23 (1969).
[CrossRef]

Aleksoff, C. C.

C. C. Aleksoff, Appl. Phys. Lett. 14, 23 (1969).
[CrossRef]

Brown, G. M.

Hurty, W. C.

W. C. Hurty, M. F. Rubinstein, Dynamics of Structures (Prentice-Hall, Englewood Cliffs, N.J., 1964), Chap. 9, p. 313.

Jacobson, C. F.

Neumann, D. B.

Rubinstein, M. F.

W. C. Hurty, M. F. Rubinstein, Dynamics of Structures (Prentice-Hall, Englewood Cliffs, N.J., 1964), Chap. 9, p. 313.

Stetson, K. A.

K. A. Stetson, P. A. Taylor, J. Phys. E: 4, 1009 (1971).
[CrossRef]

Taylor, P. A.

K. A. Stetson, P. A. Taylor, J. Phys. E: 4, 1009 (1971).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

C. C. Aleksoff, Appl. Phys. Lett. 14, 23 (1969).
[CrossRef]

J. Phys. E: (1)

K. A. Stetson, P. A. Taylor, J. Phys. E: 4, 1009 (1971).
[CrossRef]

Other (1)

W. C. Hurty, M. F. Rubinstein, Dynamics of Structures (Prentice-Hall, Englewood Cliffs, N.J., 1964), Chap. 9, p. 313.

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Figures (11)

Fig. 1
Fig. 1

Hologram camera with a vibrating mirror used to phase modulate the object illumination.

Fig. 2
Fig. 2

The electronics used to drive the mirror (mounted on a high fidelity tweeter diaphragm) and the object (via an eddy current transducer) at the same frequency—while their relative phase of motion was monitored on an oscilloscope.

Fig. 3
Fig. 3

The experimental object: a pair of coupled plates designed to conserve momentum without the motion of the support.

Fig. 4
Fig. 4

Time-average holographic interferograms of the two lowest frequency modes of the object: frequencies of maximum response (a) 77 Hz and (b) 77.4 Hz, bandwidths at half-power approximately 0.7 Hz.

Fig. 5
Fig. 5

An interferogram of the object with both of its lowest order modes excited.

Fig. 6
Fig. 6

Interferograms with phase modulation. The object motion is the same as in Fig. 5, Ωm is the same for both (a) and (b), and ϕm = 150° (a) and 330° (b).

Fig. 7
Fig. 7

Figures 6(a) and 6(b) superposed. The intersection of fringes of equal order (dotted) define a nearly radial line of constant phase.

Fig. 8
Fig. 8

A phase map prepared by superposing modulated interferograms.

Fig. 9
Fig. 9

The superposition of modulated and unmodulated interferograms: (a) Figs. 5 and 6(a) superposed; (b) Figs. 5 and 6(b) superposed.

Fig. 10
Fig. 10

A phase map prepared by superposing modulated and unmodulated interferograms.

Fig. 11
Fig. 11

The phase of vibration of the object vs azimuthal position. Dots indicate measurements made superposing modulated and unmodulated interferograms. Open circles indicate measurements made superposing modulated interferograms.

Equations (11)

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I = I s J 0 2 { [ Ω 0 2 + Ω m 2 2 Ω 0 Ω m cos ( ϕ 0 ϕ m ) ] 1 / 2 } ,
Ω 0 2 + Ω m 2 2 Ω 0 Ω m cos ( ϕ 0 ϕ 1 ) = j 0 , n 2
Ω 0 2 + Ω m 2 2 Ω 0 Ω m cos ( ϕ 0 ϕ 2 ) = j 0 , n 2 ,
0 = cos ( ϕ 0 ϕ 2 ) cos ( ϕ 0 ϕ 1 ) ,
0 = 2 sin ( ϕ 0 ϕ 2 + ϕ 1 2 ) sin ( ϕ 1 ϕ 2 2 ) ,
ϕ 0 = ϕ 1 + ϕ 2 2 ± n π , ( n = 0 , 1 ) .
Ω 0 2 = j 0 , q 2
Ω 0 2 + Ω m 2 2 Ω 0 Ω m cos ( ϕ 0 ϕ m ) = j 0 , P 2 ,
cos ( ϕ 0 ϕ m ) = ( j 0 , q 2 j 0 , P 2 ) + Ω m 2 j 0 , q Ω m .
D = A x cos ( ω t + ϕ 1 ) + B y cos ( ω t + ϕ 2 ) = [ A 2 x 2 + B 2 y 2 + 2 ABxy cos ( ϕ 1 ϕ 2 ) ] 1 / 2 cos ( ω t + ϕ )
ϕ = tan 1 ( A x sin ϕ 1 + B y sin ϕ 2 A x cos ϕ 1 + B y cos ϕ 2 ) .

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