Abstract

A new, to the author’s knowledge, method for time-averaged TV holography measurements of small-amplitude vibrations is presented. In time-averaged TV holography with sinusoidal phase modulation of the reference arm of the interferometer, two phasors describe the object vibration and the modulation of the reference arm. By inversion of the squared zero-order Bessel function of the first kind, it is possible to measure the distance between these two phasors. The distances from an object-vibration phasor to a number of known reference phasors are measured to determine the amplitude and the phase of the object vibration. The method is demonstrated by the measurement of a vibration mode of a circular metal disk. The results are compared with theoretical data and with data obtained by a commonly used method in phase-modulated TV holography.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. O. J. Løkberg, “ESPI—the ultimate holographic tool for vibration analysis?” J. Acoust. Soc. Am. 75, 1783–1791 (1984).
    [Crossref]
  2. O. J. Løkberg, M. Espeland, H. M. Pedersen, “Tomographic reconstruction of sound fields using TV holography,” Appl. Opt. 34, 1640–1645 (1995).
    [Crossref] [PubMed]
  3. S. Ellingsrud, O. J. Løkberg, “Analysis of high frequency vibrations using TV-holography and digital image processing,” in Laser Interferometry: Quantitative Analysis of Interferograms, R. J. Pryputniewicz, ed., Proc. SPIE1162, 402–410 (1989).
    [Crossref]
  4. S. Ellingsrud, G. O. Rosvold, “Analysis of a data-based TV-holography system used to measure small vibration amplitudes,” J. Opt. Soc. Am. 9, 237–251 (1992).
    [Crossref]
  5. R. J. Pryputniewicz, “Time average holography in vibration analysis,” Opt. Eng. 24, 843–848 (1985).
    [Crossref]
  6. K. A. Stetson, W. R. Brohinsky, “Fringe-shifting technique for numerical analysis of time-average holograms of vibrating objects,” J. Opt. Soc. Am. A 5, 1472–1476 (1988).
    [Crossref]
  7. K. Høgmoen, H. M. Pedersen, “Measurement of small vibrations using electronic speckle pattern interferometry: theory,” J. Opt. Soc. Am. 67, 1578–1583 (1977).
    [Crossref]
  8. C. C. Aleksoff, “Time averaged holography extended,” Appl. Phys. Lett. 14, 23 (1969).
    [Crossref]
  9. R. L. Powell, K. A. Stetson, “Interferometric vibration analysis by wavefront reconstruction,” J. Opt. Soc. Am. 55, 1593–1598 (1965).
    [Crossref]
  10. K. Høgmoen, O. J. Løkberg, “Detection and measurement of small vibrations using electronic speckle pattern interferometry,” Appl. Opt. 16, 1869–1875 (1977).
    [Crossref] [PubMed]
  11. D. B. Neumann, C. F. Jacobson, G. M. Brown, “Holographic technique for determining the phase of vibrating objects,” Appl. Opt. 9, 1357–1362 (1970).
    [Crossref] [PubMed]
  12. L. Gold, “Inverse Bessel functions: solutions for the zeros,” J. Math. Phys. 36, 167–171 (1957).
  13. R. Courant, D. Hilbert, Metoden der mathematischen Physic I (Springer-Verlag, Berlin, 1968), Chap. 5.
    [Crossref]

1995 (1)

1992 (1)

1988 (1)

1985 (1)

R. J. Pryputniewicz, “Time average holography in vibration analysis,” Opt. Eng. 24, 843–848 (1985).
[Crossref]

1984 (1)

O. J. Løkberg, “ESPI—the ultimate holographic tool for vibration analysis?” J. Acoust. Soc. Am. 75, 1783–1791 (1984).
[Crossref]

1977 (2)

1970 (1)

1969 (1)

C. C. Aleksoff, “Time averaged holography extended,” Appl. Phys. Lett. 14, 23 (1969).
[Crossref]

1965 (1)

1957 (1)

L. Gold, “Inverse Bessel functions: solutions for the zeros,” J. Math. Phys. 36, 167–171 (1957).

Aleksoff, C. C.

C. C. Aleksoff, “Time averaged holography extended,” Appl. Phys. Lett. 14, 23 (1969).
[Crossref]

Brohinsky, W. R.

Brown, G. M.

Courant, R.

R. Courant, D. Hilbert, Metoden der mathematischen Physic I (Springer-Verlag, Berlin, 1968), Chap. 5.
[Crossref]

Ellingsrud, S.

S. Ellingsrud, G. O. Rosvold, “Analysis of a data-based TV-holography system used to measure small vibration amplitudes,” J. Opt. Soc. Am. 9, 237–251 (1992).
[Crossref]

S. Ellingsrud, O. J. Løkberg, “Analysis of high frequency vibrations using TV-holography and digital image processing,” in Laser Interferometry: Quantitative Analysis of Interferograms, R. J. Pryputniewicz, ed., Proc. SPIE1162, 402–410 (1989).
[Crossref]

Espeland, M.

Gold, L.

L. Gold, “Inverse Bessel functions: solutions for the zeros,” J. Math. Phys. 36, 167–171 (1957).

Hilbert, D.

R. Courant, D. Hilbert, Metoden der mathematischen Physic I (Springer-Verlag, Berlin, 1968), Chap. 5.
[Crossref]

Høgmoen, K.

Jacobson, C. F.

Løkberg, O. J.

O. J. Løkberg, M. Espeland, H. M. Pedersen, “Tomographic reconstruction of sound fields using TV holography,” Appl. Opt. 34, 1640–1645 (1995).
[Crossref] [PubMed]

O. J. Løkberg, “ESPI—the ultimate holographic tool for vibration analysis?” J. Acoust. Soc. Am. 75, 1783–1791 (1984).
[Crossref]

K. Høgmoen, O. J. Løkberg, “Detection and measurement of small vibrations using electronic speckle pattern interferometry,” Appl. Opt. 16, 1869–1875 (1977).
[Crossref] [PubMed]

S. Ellingsrud, O. J. Løkberg, “Analysis of high frequency vibrations using TV-holography and digital image processing,” in Laser Interferometry: Quantitative Analysis of Interferograms, R. J. Pryputniewicz, ed., Proc. SPIE1162, 402–410 (1989).
[Crossref]

Neumann, D. B.

Pedersen, H. M.

Powell, R. L.

Pryputniewicz, R. J.

R. J. Pryputniewicz, “Time average holography in vibration analysis,” Opt. Eng. 24, 843–848 (1985).
[Crossref]

Rosvold, G. O.

Stetson, K. A.

Appl. Opt. (3)

Appl. Phys. Lett. (1)

C. C. Aleksoff, “Time averaged holography extended,” Appl. Phys. Lett. 14, 23 (1969).
[Crossref]

J. Acoust. Soc. Am. (1)

O. J. Løkberg, “ESPI—the ultimate holographic tool for vibration analysis?” J. Acoust. Soc. Am. 75, 1783–1791 (1984).
[Crossref]

J. Math. Phys. (1)

L. Gold, “Inverse Bessel functions: solutions for the zeros,” J. Math. Phys. 36, 167–171 (1957).

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

R. J. Pryputniewicz, “Time average holography in vibration analysis,” Opt. Eng. 24, 843–848 (1985).
[Crossref]

Other (2)

S. Ellingsrud, O. J. Løkberg, “Analysis of high frequency vibrations using TV-holography and digital image processing,” in Laser Interferometry: Quantitative Analysis of Interferograms, R. J. Pryputniewicz, ed., Proc. SPIE1162, 402–410 (1989).
[Crossref]

R. Courant, D. Hilbert, Metoden der mathematischen Physic I (Springer-Verlag, Berlin, 1968), Chap. 5.
[Crossref]

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 (14)

Fig. 1
Fig. 1

Vector difference a r - a o between the object-vibration phasor a o of a vibration with amplitude a o and phase ϕ and a reference phasor a r of magnitude a r and phase 0.

Fig. 2
Fig. 2

The squared zero-order Bessel function of the first kind. A working point a r is shown, and the part of the function that is assumed to be a straight line is emphasized.

Fig. 3
Fig. 3

Error in measurements of the vibration amplitude when the measurements are performed as described in Ref. 4. The solid curve is the error when the object-vibration phase is 0°, the dotted curve represents an object-vibration phase of 45°.

Fig. 4
Fig. 4

Circle of radius |a r - a o | centered at a r . The object-vibration phasor a o is a vector from the origin to a point on this circle.

Fig. 5
Fig. 5

Squared zero-order Bessel function J 0 2(x). The invertible area is marked. The magnitude of the reference phasors used in this implementation is a r .

Fig. 6
Fig. 6

The inverse function g(x) in the interval [0.2–0.95].

Fig. 7
Fig. 7

Area of valid measurements for a reference modulation a r . For object vibrations described by phasors in the diagonally striped area, the vector difference |a r - a o | can be computed by use of the inverse function g(x). The corresponding part of the function J 0 2(x) is shown to the left.

Fig. 8
Fig. 8

Four reference phasors and the corresponding areas of valid measurements. The reference phasors all have a length of 0.5, and the circles have radii of 0.32 and 1.6. For object vibrations with phasors in the area with cross-hatching, four valid measurements will be available. For object vibrations with phasors in the areas with diagonal striping, three valid measurements will be available. For object vibrations with phasors in the areas with horizontal striping, only two valid measurements will be available.

Fig. 9
Fig. 9

Object-vibration phasor a o = ê Re a o cos(ϕ) + ê Im a o sin(ϕ) measured by measurement of the vector differences d 0, d 90, d 180, and d 270 between this phasor and four reference phasors of magnitude a r and phases of 0°, 90°, 180°, and 270°.

Fig. 10
Fig. 10

Two valid measurements of an object vibration with the object-vibration phasor in the first quadrant are available. The distances to two reference phasors, d 0 and d 90, are known. Circles of radii d 0 and d 90 are drawn about the corresponding reference phasors and intersect at two points, A and B. The angle α is computed with the aid of the cosine law.

Fig. 11
Fig. 11

Four normalized measurement frames I 0, I 90, I 180, and I 270. Pixels with values higher than 0.95 are white; pixels with values below 0.2 are black.

Fig. 12
Fig. 12

Amplitude and phase of the vibration pattern of a circular brass disk 2 mm thick with a diameter of 89 nm. The disk is vibrating at 6115 Hz with a maximum amplitude of 87 nm. To the left are the measurements obtained with the method described here; to the right is the vibration pattern computed from Eq. (13). Note that the phase is wrapped so that white corresponds to π and black to -π.

Fig. 13
Fig. 13

Same vibration pattern as in Fig. 12 measured with the method described in Ref. 4. To the left is a real measurement; to the right is a simulation.

Fig. 14
Fig. 14

Plots of the amplitude and the phase of the vibration pattern along a horizontal line running through the upper points of maximal amplitude: The dotted curve represents the actual values. The solid curve was measured with the new method. The dashed curve was measured with the method described in Ref. 4. Note that the phase is wrapped.

Equations (13)

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

I     J 0 2 2 k | a r - a o | ,
J 0 2 2 k | a r - a o | = J 0 2 2 k a r 2 + a o 2 - 2 a r a o × cos ϕ - θ 1 / 2 J 0 2 a r - d d a r J 0 2 a r a o × cos ϕ - θ + ,
I x ,   y = I b x ,   y + c x ,   y J 0 2 2 k | a r - a o x ,   y | ,
I u x ,   y = I b x ,   y + c x ,   y J 0 2 0 = I b x ,   y + c x ,   y .
I x ,   y = I b x ,   y + c x ,   y J 0 2 2 k | a r | = 0 = I b x ,   y . = 0
I N x ,   y = I x ,   y - I b x ,   y I u x ,   y - I b x ,   y = J 0 2 2 k | a r - a o x ,   y | ,     0 I N 1 .
| a r - a o x ,   y | = 1 2 k   g I N x ,   y .
J 0 2 x = y     g y = x , x     0.32 ,   1.6 ,   y     0.2 ,   0.95 .
J 0 2 x - y = 0
d 0 = | a r 0 - a o | = 1 2 k   g I 0 - I b I u - I b , d 90 = | a r 90 - a o | = 1 2 k   g I 90 - I b I u - I b , d 180 = | a r 180 - a o | = 1 2 k   g I 180 - I b I u - I b , d 270 = | a r 270 - a o | = 1 2 k   g I 270 - I b I u - I b .
a 0 = d 180 2 - d 0 2 4 a r , a 0 = d 270 2 - d 90 2 4 a r ,
α = arccos d 0 2 + 2 a r 2 - d 90 2 2 2   a r d 0 , A : a 0 = a r - 1 2   d 0 cos α + sin α , a 0 = 1 2   d 0 cos α - sin α , B : a 0 = a r - 1 2   d 0 cos α - sin α , a 0 = 1 2   d 0 cos α + sin α .
Z r ,   θ = AJ 2 kr exp i 2 θ + α 1 + exp { - i 2 θ + α 2 , r = x 2 + y 2 ,     θ = tan - 1 y x mod   2 π ,

Metrics