Abstract

We present a system that measures the full-field amplitude and phase distributions of objects that vibrate with small amplitudes. The system is based on TV–holography combined with sinusoidal phase modulation, discrete vibration-phase shifts, and digital image processing. Different new algorithms, based on a linear approximation of the fringe function, are discussed. Averaging techniques, used to reduce the effects of noise sources and to increase the resolution of the system, are also introduced. For one of the algorithms, tested in combination with the averaging techniques, the amplitude threshold was approximately 1/3000 of the wavelength of the applied laser light. The amplitude resolution was of the same magnitude. The phase accuracy is amplitude dependent and was about 3° for amplitudes greater than 5 nm.

© 1992 Optical Society of America

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  1. J. N. Butters, J. A. Leendertz, “Holographic and video-techniques applied to engineering measurements,”J. Meas. Cont. 4, 349–354 (1971).
  2. A. Macovski, S. D. Ramsey, L. F. Shaefer, “Time-lapse interferometry and contouring using televisions systems,” Appl. Opt. 10, 2722–2727 (1971).
    [Crossref] [PubMed]
  3. O. Schwomma, Osterreichisches Patent298830, (May25, 1972); see also “Forschung un Technik,” Neu. Zur. Beilage 257, (1975).
  4. O. J. Løkberg, K. Høgmoen, O. M. Holje, “Vibration measurement on the human ear drum in vivo,” Appl. Opt. 18, 763–765 (1979).
    [Crossref] [PubMed]
  5. O. J. Løkberg, “ESPI—the ultimate holographic tool for vibration analysis?”J. Acoust. Soc. Am. 75, 1783–1791 (1984).
    [Crossref]
  6. L. Ek, N. E. Molin, K. Biederman, “Real-time study of vibrations by means of an instrument recording time-average holograms on a TV-vidicon,” in Applications of Holography, L. Huff, ed., Proc. Soc. Photo-Opt. Instrum. Eng.523, 155–159 (1985).
    [Crossref]
  7. M. C. Shellabear, J. R. Tyrer, “Three-dimensional vibration analysis using electronic speckle pattern interferometry (ESPI),” in Laser Technologies in Industry, S. P. Almeida, L. M. Bernardo, O. D. Soares, eds., Proc. Soc. Photo-Opt. Instrum. Eng.952, 251–259 (1988).
  8. J. Davies, C. H. Buckberry, J. D. C. Jones, C. N. Pannell, “Developments in electronic speckle pattern interferometry for automotive vibration analysis,” in Laser Technologies in Industry, S. P. Almeida, L. M. Bernardo, O. D. Soares, eds., Proc. Soc. Photo-Opt. Instrum. Eng.952, 260–274 (1988).
  9. J. T. Malmo, E. Vikhagen, “Vibration analysis of a car body by means of TV–holography,” Exp. Tech. 12, 28–30 (1988).
    [Crossref]
  10. E. Vikhagen, “Vibration measurement using phase shifting TV–holography and digital image processing,” Opt. Commun. 69, 214–218 (1989).
    [Crossref]
  11. S. Ellingsrud, O. J. Løkberg, “Recording and analysis of (high frequency) sinusoidal vibrations using computerized TV–holography,” in Optical Systems in Adverse Environments, Proc. Soc. Photo-Opt. Instrum. Eng.1399, 30–41 (1991).
    [Crossref]
  12. K. Høgmoen, O. J. Løkberg, “Detection and measurement of small vibrations using electronic speckle pattern interferometry,” Appl. Opt. 16, 1869–1875 (1977).
    [Crossref] [PubMed]
  13. C. L. Koliopoulos, “Interferometric optical phase measurement techniques,” Ph.D. dissertation (Optical Sciences Center, University of Arizona, Tucson, Ariz., 1981).
  14. P. Hariharan, B. F. Oreb, N. Brown, “A digital phase-measurement system for real-time holographic interferometry,” Opt. Commun. 22, 393–396 (1982).
    [Crossref]
  15. K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. 24, 3053–3058 (1985).
    [Crossref] [PubMed]
  16. S. Nakadate, H. Saito, “Fringe scanning speckle-pattern interferometry,” Appl. Opt. 24, 2172–2180 (1985).
    [Crossref] [PubMed]
  17. R. J. Pryputniewicz, K. A. Stetson, “Measurement of vibration patterns using electro-optic holography,” in Laser Interferometry: Quantitative Analysis of Interferograms: Third in a Series, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1162, 456–467 (1990).
    [Crossref]
  18. R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, London, 1983).
  19. O. J. Løkberg, G. Å. Slettemoen, “Basic electronic speckle pattern interferometry,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1987), pp. 455–504.
  20. H. Ostberg, “An interferometer method of studying the vibrations of an oscillating quartz plate,”J. Opt. Soc. Am. 22, 19–35 (1932).
    [Crossref]
  21. K. Høgmoen, H. M. Pedersen, “Measurement of small vibrations using electronic speckle pattern interferometry: theory,”J. Opt. Soc. Am. 67, 1578–1583 (1977).
    [Crossref]
  22. H. M. Pedersen, Division of Physics, The Norwegian Institute of Technology, 7034-Trondheim, Norway (personal communication).
  23. G. Å. Slettemoen, “Electronic speckle pattern interferometric system based on a speckle reference beam,” Appl. Opt. 19, 616–623 (1980).
    [Crossref] [PubMed]
  24. R. C. Gonzalez, P. Wintz, Digital Image Processing, 2nd ed. (Addison-Wesley, Reading, Mass.1987).
  25. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–75.
    [Crossref]
  26. H. P. Stahl, “Testing large optics: high-speed phase-measuring interferometry,” Phot. Spec. 23, 105–112 (1989).
  27. S. Ellingsrud, J. T. Malmo, A. Mikkelsen, K. D. Knudsen, A. Elgsæter, “Torsional dynamics of the Birnboim-Schrag multiple lump resonator studied using TV–holography,” submitted to Rheol. Acta.

1989 (2)

E. Vikhagen, “Vibration measurement using phase shifting TV–holography and digital image processing,” Opt. Commun. 69, 214–218 (1989).
[Crossref]

H. P. Stahl, “Testing large optics: high-speed phase-measuring interferometry,” Phot. Spec. 23, 105–112 (1989).

1988 (1)

J. T. Malmo, E. Vikhagen, “Vibration analysis of a car body by means of TV–holography,” Exp. Tech. 12, 28–30 (1988).
[Crossref]

1985 (2)

1984 (1)

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

1982 (1)

P. Hariharan, B. F. Oreb, N. Brown, “A digital phase-measurement system for real-time holographic interferometry,” Opt. Commun. 22, 393–396 (1982).
[Crossref]

1980 (1)

1979 (1)

1977 (2)

1971 (2)

A. Macovski, S. D. Ramsey, L. F. Shaefer, “Time-lapse interferometry and contouring using televisions systems,” Appl. Opt. 10, 2722–2727 (1971).
[Crossref] [PubMed]

J. N. Butters, J. A. Leendertz, “Holographic and video-techniques applied to engineering measurements,”J. Meas. Cont. 4, 349–354 (1971).

1932 (1)

Biederman, K.

L. Ek, N. E. Molin, K. Biederman, “Real-time study of vibrations by means of an instrument recording time-average holograms on a TV-vidicon,” in Applications of Holography, L. Huff, ed., Proc. Soc. Photo-Opt. Instrum. Eng.523, 155–159 (1985).
[Crossref]

Brown, N.

P. Hariharan, B. F. Oreb, N. Brown, “A digital phase-measurement system for real-time holographic interferometry,” Opt. Commun. 22, 393–396 (1982).
[Crossref]

Buckberry, C. H.

J. Davies, C. H. Buckberry, J. D. C. Jones, C. N. Pannell, “Developments in electronic speckle pattern interferometry for automotive vibration analysis,” in Laser Technologies in Industry, S. P. Almeida, L. M. Bernardo, O. D. Soares, eds., Proc. Soc. Photo-Opt. Instrum. Eng.952, 260–274 (1988).

Butters, J. N.

J. N. Butters, J. A. Leendertz, “Holographic and video-techniques applied to engineering measurements,”J. Meas. Cont. 4, 349–354 (1971).

Creath, K.

Davies, J.

J. Davies, C. H. Buckberry, J. D. C. Jones, C. N. Pannell, “Developments in electronic speckle pattern interferometry for automotive vibration analysis,” in Laser Technologies in Industry, S. P. Almeida, L. M. Bernardo, O. D. Soares, eds., Proc. Soc. Photo-Opt. Instrum. Eng.952, 260–274 (1988).

Ek, L.

L. Ek, N. E. Molin, K. Biederman, “Real-time study of vibrations by means of an instrument recording time-average holograms on a TV-vidicon,” in Applications of Holography, L. Huff, ed., Proc. Soc. Photo-Opt. Instrum. Eng.523, 155–159 (1985).
[Crossref]

Elgsæter, A.

S. Ellingsrud, J. T. Malmo, A. Mikkelsen, K. D. Knudsen, A. Elgsæter, “Torsional dynamics of the Birnboim-Schrag multiple lump resonator studied using TV–holography,” submitted to Rheol. Acta.

Ellingsrud, S.

S. Ellingsrud, O. J. Løkberg, “Recording and analysis of (high frequency) sinusoidal vibrations using computerized TV–holography,” in Optical Systems in Adverse Environments, Proc. Soc. Photo-Opt. Instrum. Eng.1399, 30–41 (1991).
[Crossref]

S. Ellingsrud, J. T. Malmo, A. Mikkelsen, K. D. Knudsen, A. Elgsæter, “Torsional dynamics of the Birnboim-Schrag multiple lump resonator studied using TV–holography,” submitted to Rheol. Acta.

Gonzalez, R. C.

R. C. Gonzalez, P. Wintz, Digital Image Processing, 2nd ed. (Addison-Wesley, Reading, Mass.1987).

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–75.
[Crossref]

Hariharan, P.

P. Hariharan, B. F. Oreb, N. Brown, “A digital phase-measurement system for real-time holographic interferometry,” Opt. Commun. 22, 393–396 (1982).
[Crossref]

Høgmoen, K.

Holje, O. M.

Jones, J. D. C.

J. Davies, C. H. Buckberry, J. D. C. Jones, C. N. Pannell, “Developments in electronic speckle pattern interferometry for automotive vibration analysis,” in Laser Technologies in Industry, S. P. Almeida, L. M. Bernardo, O. D. Soares, eds., Proc. Soc. Photo-Opt. Instrum. Eng.952, 260–274 (1988).

Jones, R.

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, London, 1983).

Knudsen, K. D.

S. Ellingsrud, J. T. Malmo, A. Mikkelsen, K. D. Knudsen, A. Elgsæter, “Torsional dynamics of the Birnboim-Schrag multiple lump resonator studied using TV–holography,” submitted to Rheol. Acta.

Koliopoulos, C. L.

C. L. Koliopoulos, “Interferometric optical phase measurement techniques,” Ph.D. dissertation (Optical Sciences Center, University of Arizona, Tucson, Ariz., 1981).

Leendertz, J. A.

J. N. Butters, J. A. Leendertz, “Holographic and video-techniques applied to engineering measurements,”J. Meas. Cont. 4, 349–354 (1971).

Løkberg, O. J.

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

O. J. Løkberg, K. Høgmoen, O. M. Holje, “Vibration measurement on the human ear drum in vivo,” Appl. Opt. 18, 763–765 (1979).
[Crossref] [PubMed]

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, “Recording and analysis of (high frequency) sinusoidal vibrations using computerized TV–holography,” in Optical Systems in Adverse Environments, Proc. Soc. Photo-Opt. Instrum. Eng.1399, 30–41 (1991).
[Crossref]

O. J. Løkberg, G. Å. Slettemoen, “Basic electronic speckle pattern interferometry,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1987), pp. 455–504.

Macovski, A.

Malmo, J. T.

J. T. Malmo, E. Vikhagen, “Vibration analysis of a car body by means of TV–holography,” Exp. Tech. 12, 28–30 (1988).
[Crossref]

S. Ellingsrud, J. T. Malmo, A. Mikkelsen, K. D. Knudsen, A. Elgsæter, “Torsional dynamics of the Birnboim-Schrag multiple lump resonator studied using TV–holography,” submitted to Rheol. Acta.

Mikkelsen, A.

S. Ellingsrud, J. T. Malmo, A. Mikkelsen, K. D. Knudsen, A. Elgsæter, “Torsional dynamics of the Birnboim-Schrag multiple lump resonator studied using TV–holography,” submitted to Rheol. Acta.

Molin, N. E.

L. Ek, N. E. Molin, K. Biederman, “Real-time study of vibrations by means of an instrument recording time-average holograms on a TV-vidicon,” in Applications of Holography, L. Huff, ed., Proc. Soc. Photo-Opt. Instrum. Eng.523, 155–159 (1985).
[Crossref]

Nakadate, S.

Oreb, B. F.

P. Hariharan, B. F. Oreb, N. Brown, “A digital phase-measurement system for real-time holographic interferometry,” Opt. Commun. 22, 393–396 (1982).
[Crossref]

Ostberg, H.

Pannell, C. N.

J. Davies, C. H. Buckberry, J. D. C. Jones, C. N. Pannell, “Developments in electronic speckle pattern interferometry for automotive vibration analysis,” in Laser Technologies in Industry, S. P. Almeida, L. M. Bernardo, O. D. Soares, eds., Proc. Soc. Photo-Opt. Instrum. Eng.952, 260–274 (1988).

Pedersen, H. M.

K. Høgmoen, H. M. Pedersen, “Measurement of small vibrations using electronic speckle pattern interferometry: theory,”J. Opt. Soc. Am. 67, 1578–1583 (1977).
[Crossref]

H. M. Pedersen, Division of Physics, The Norwegian Institute of Technology, 7034-Trondheim, Norway (personal communication).

Pryputniewicz, R. J.

R. J. Pryputniewicz, K. A. Stetson, “Measurement of vibration patterns using electro-optic holography,” in Laser Interferometry: Quantitative Analysis of Interferograms: Third in a Series, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1162, 456–467 (1990).
[Crossref]

Ramsey, S. D.

Saito, H.

Schwomma, O.

O. Schwomma, Osterreichisches Patent298830, (May25, 1972); see also “Forschung un Technik,” Neu. Zur. Beilage 257, (1975).

Shaefer, L. F.

Shellabear, M. C.

M. C. Shellabear, J. R. Tyrer, “Three-dimensional vibration analysis using electronic speckle pattern interferometry (ESPI),” in Laser Technologies in Industry, S. P. Almeida, L. M. Bernardo, O. D. Soares, eds., Proc. Soc. Photo-Opt. Instrum. Eng.952, 251–259 (1988).

Slettemoen, G. Å.

G. Å. Slettemoen, “Electronic speckle pattern interferometric system based on a speckle reference beam,” Appl. Opt. 19, 616–623 (1980).
[Crossref] [PubMed]

O. J. Løkberg, G. Å. Slettemoen, “Basic electronic speckle pattern interferometry,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1987), pp. 455–504.

Stahl, H. P.

H. P. Stahl, “Testing large optics: high-speed phase-measuring interferometry,” Phot. Spec. 23, 105–112 (1989).

Stetson, K. A.

R. J. Pryputniewicz, K. A. Stetson, “Measurement of vibration patterns using electro-optic holography,” in Laser Interferometry: Quantitative Analysis of Interferograms: Third in a Series, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1162, 456–467 (1990).
[Crossref]

Tyrer, J. R.

M. C. Shellabear, J. R. Tyrer, “Three-dimensional vibration analysis using electronic speckle pattern interferometry (ESPI),” in Laser Technologies in Industry, S. P. Almeida, L. M. Bernardo, O. D. Soares, eds., Proc. Soc. Photo-Opt. Instrum. Eng.952, 251–259 (1988).

Vikhagen, E.

E. Vikhagen, “Vibration measurement using phase shifting TV–holography and digital image processing,” Opt. Commun. 69, 214–218 (1989).
[Crossref]

J. T. Malmo, E. Vikhagen, “Vibration analysis of a car body by means of TV–holography,” Exp. Tech. 12, 28–30 (1988).
[Crossref]

Wintz, P.

R. C. Gonzalez, P. Wintz, Digital Image Processing, 2nd ed. (Addison-Wesley, Reading, Mass.1987).

Wykes, C.

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, London, 1983).

Appl. Opt. (6)

Exp. Tech. (1)

J. T. Malmo, E. Vikhagen, “Vibration analysis of a car body by means of TV–holography,” Exp. Tech. 12, 28–30 (1988).
[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. Meas. Cont. (1)

J. N. Butters, J. A. Leendertz, “Holographic and video-techniques applied to engineering measurements,”J. Meas. Cont. 4, 349–354 (1971).

J. Opt. Soc. Am. (2)

Opt. Commun. (2)

E. Vikhagen, “Vibration measurement using phase shifting TV–holography and digital image processing,” Opt. Commun. 69, 214–218 (1989).
[Crossref]

P. Hariharan, B. F. Oreb, N. Brown, “A digital phase-measurement system for real-time holographic interferometry,” Opt. Commun. 22, 393–396 (1982).
[Crossref]

Phot. Spec. (1)

H. P. Stahl, “Testing large optics: high-speed phase-measuring interferometry,” Phot. Spec. 23, 105–112 (1989).

Other (13)

S. Ellingsrud, J. T. Malmo, A. Mikkelsen, K. D. Knudsen, A. Elgsæter, “Torsional dynamics of the Birnboim-Schrag multiple lump resonator studied using TV–holography,” submitted to Rheol. Acta.

S. Ellingsrud, O. J. Løkberg, “Recording and analysis of (high frequency) sinusoidal vibrations using computerized TV–holography,” in Optical Systems in Adverse Environments, Proc. Soc. Photo-Opt. Instrum. Eng.1399, 30–41 (1991).
[Crossref]

H. M. Pedersen, Division of Physics, The Norwegian Institute of Technology, 7034-Trondheim, Norway (personal communication).

R. C. Gonzalez, P. Wintz, Digital Image Processing, 2nd ed. (Addison-Wesley, Reading, Mass.1987).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–75.
[Crossref]

O. Schwomma, Osterreichisches Patent298830, (May25, 1972); see also “Forschung un Technik,” Neu. Zur. Beilage 257, (1975).

R. J. Pryputniewicz, K. A. Stetson, “Measurement of vibration patterns using electro-optic holography,” in Laser Interferometry: Quantitative Analysis of Interferograms: Third in a Series, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1162, 456–467 (1990).
[Crossref]

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, London, 1983).

O. J. Løkberg, G. Å. Slettemoen, “Basic electronic speckle pattern interferometry,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1987), pp. 455–504.

C. L. Koliopoulos, “Interferometric optical phase measurement techniques,” Ph.D. dissertation (Optical Sciences Center, University of Arizona, Tucson, Ariz., 1981).

L. Ek, N. E. Molin, K. Biederman, “Real-time study of vibrations by means of an instrument recording time-average holograms on a TV-vidicon,” in Applications of Holography, L. Huff, ed., Proc. Soc. Photo-Opt. Instrum. Eng.523, 155–159 (1985).
[Crossref]

M. C. Shellabear, J. R. Tyrer, “Three-dimensional vibration analysis using electronic speckle pattern interferometry (ESPI),” in Laser Technologies in Industry, S. P. Almeida, L. M. Bernardo, O. D. Soares, eds., Proc. Soc. Photo-Opt. Instrum. Eng.952, 251–259 (1988).

J. Davies, C. H. Buckberry, J. D. C. Jones, C. N. Pannell, “Developments in electronic speckle pattern interferometry for automotive vibration analysis,” in Laser Technologies in Industry, S. P. Almeida, L. M. Bernardo, O. D. Soares, eds., Proc. Soc. Photo-Opt. Instrum. Eng.952, 260–274 (1988).

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Figures (14)

Fig. 1
Fig. 1

Linear part of the J02 function. The abscissa is valid for wavelength 632.8 nm of a He–Ne laser.

Fig. 2
Fig. 2

Noise reduction effect of a 3 × 3 digital averaging filter displayed as a function of the number of successive times the filter is used (here from 1 to 10 times). The dotted curve was calculated by the analytical method, and the solid curve was computed by standard matrix calculations.

Fig. 3
Fig. 3

Spatial resolution when a 3 × 3 digital averaging filter is used successively several times (sinc2nu function). The three graphs are for n = 1, n = 5, and n = 100, where n is the number of times the filter was used.

Fig. 4
Fig. 4

Schematic drawing of the experimental setup. Optical components: the laser, a beam splitter (BS), an electro-optic modulator (EOM), the speckle-averaging mechanism (SAM), a beam combiner (BC), and a video camera are positioned inside the holographic sensor unit (HSU). An analog rectifier and a high-pass filter are contained in the electronic processing unit (EPU).

Fig. 5
Fig. 5

Fringe function, drawn with a solid curve, was measured with a static object and the voltage of the reference excitation signal increased by 100 small, discrete steps. The measured function was normalized, and the background intensity was subtracted. The function is compared with the Bessel function (dotted curve) and its square (dashed curve).

Fig. 6
Fig. 6

Time-average recording of a steel plate vibrating with a 2.2 mode at 3720 Hz. The recording was speckle averaged 256 times.

Fig. 7
Fig. 7

Mesh plots of the phase and amplitude distributions of the 2.2 mode at excitation levels corresponding to maximum amplitude values of (a) 30 nm, (b) 5 nm, (c) 2 nm, and (d) 1 nm.

Fig. 8
Fig. 8

Amplitude contour plots of the 2.2 mode with excitation levels corresponding to maximum amplitude values of (a) 30 nm, (b) 2 nm, and (c) 1 nm. The contour spacings are (a) 4.3 nm, (b) 0.34 nm, and (c) 0.17 nm. The hatched areas have amplitude values lower than those of the first contour.

Fig. 9
Fig. 9

Graphs from horizontal cross sections starting at the 2-cm vertical label scale of the amplitude contour plots for (a) 1 nm and (b) 2 nm.

Fig. 10
Fig. 10

(a) Display of the amplitude distribution of the 2.2 mode with an amplitude value of ~0.2 nm, represented in gray levels. Display of (c) the phase distribution and (b) and (d) the quadrature components.

Fig. 11
Fig. 11

Cross sections of the phase plots corresponding to the amplitude plots with maximum values of (a) 30 nm, (b) 5 nm, (c) 2 nm, and (d) 1 nm.

Fig. 12
Fig. 12

(a) Amplitude mesh plot of a rigid cylinder vibrating with a torsional motion. (b) Cross section of the amplitude plot averaged over all cross sections. (c) Cross section of the phase distribution.

Fig. 13
Fig. 13

(a) Time-average recording of a steel plate vibrating at 16,940 Hz. (b) A similar amplitude plot recorded with the phase-shifting system at a lower excitation level. The recordings were speckle averaged 128 times with preaveragings. The contour spacing in (b) is 3 nm, and the hatched areas have amplitude values lower than 3 nm.

Fig. 14
Fig. 14

(a) Recording with 48 preaveragings. (b) Recording with 48 postaveragings.

Equations (59)

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

I ( x , y ) J 0 2 [ 2 π λ g φ a O ( x , y ) ] ,
I ( x , y ) J 0 2 { 4 π λ [ a O 2 ( x , y ) + a R 2 ( x , y ) - 2 a O ( x , y ) a R ( x , y ) × cos θ ( x , y ) ] 1 / 2 } ,
E n ( x , y ) = I R ( x , y ) + I O ( x , y ) + 2 R [ u R * ( x , y ) u O ( x , y ) x ( t n ) ] + N ,
R [ u R * ( x , y ) u O ( x , y ) ] = [ I R ( x , y ) I O ( x , y ) ] 1 / 2 γ cos [ α ( x , y ) ] ,
I ( x , y ) = Δ I R 2 ( x , y ) + Δ I O 2 ( x , y ) + 2 Δ I R ( x , y ) Δ I O ( x , y ) + 2 N [ Δ I R ( x , y ) + Δ I O ( x , y ) ] + 4 [ Δ I R ( x , y ) + Δ I O ( x , y ) + N ] R [ u R * ( x , y ) u O ( x , y ) x ( t n ) ] + 4 I R ( x , y ) I O ( x , y ) x 2 ( t n ) + N 2 .
x ( t n ) = J 0 [ ( a O 2 + a R 2 - 2 a O a R cos θ ) 1 / 2 ] J 0 ( a R ) - d d a R [ J 0 ( a R ) ] a O cos θ + ,
x 2 ( t n ) = J 0 2 [ ( a O 2 + a R 2 - 2 a O a R cos θ ) 1 / 2 ] J 0 2 ( a R ) - d d a R [ J 0 2 ( a R ) ] a O cos θ + .
I ( x , y ) = I b ( x , y ) - k ( x , y ) a O ( x , y ) cos θ ( x , y ) ,
I b ( x , y ) = 4 I R ( x , y ) I O ( x , y ) J 0 2 ( a R ) + Δ I R 2 ( x , y ) + Δ I O 2 ( x , y ) + 2 Δ I R ( x , y ) Δ I 0 ( x , y ) + 2 N ( Δ I R ( x , y ) + Δ I O ( x , y ) ] + N 2 + 4 [ Δ I R ( x , y ) + Δ I O ( x , y ) + N ] R [ u R * ( x , y ) u O ( x , y ) J 0 ( a R ) ] , k ( x , y ) = - 4 ( I R ( x , y ) I O ( x , y ) d d a R [ J 0 2 ( a R ) ] + [ Δ I R ( x , y ) + Δ I O ( x , y ) + N ] × R { u R * ( x , y ) u O ( x , y ) d d a R [ J 0 ( a R ) ] } ) .
I = I b + Δ I b - ( k + Δ k ) a O cos θ .
I ¯ = I b + M - 1 / 2 Δ I b - ( k + M - 1 / 2 Δ k ) a O cos θ O .
I 000 ° = I b - k a O cos θ O ,
I 090 ° = I b + k a O sin θ O ,
I 180 ° = I b + k a O cos θ O ,
I 270 ° = I b - k a O sin θ O .
θ O = arctan I 090 ° - I 270 ° I 180 ° - I 000 ° .
I δ R 1 = I b - k ( a O + δ R ) sin θ O .
a O = δ R 2 2 k a O sin θ O k δ R sin θ O = δ R 2 I 090 ° - I 270 ° I 270 ° - I δ R 1 .
I δ R 2 = I b - k ( a O - δ R ) cos θ O .
a 0 = δ R 2 [ ( I 180 ° - I 000 ° ) 2 + ( I 090 ° - I 270 ° ) 2 ( I 000 ° - I δ R 2 ) 2 + ( I 270 ° - I δ R 1 ) 2 ] 1 / 2 .
k a O cos θ O = I 180 ° - I 000 ° 2 ,
k a O sin θ 0 = I 090 ° - I 270 ° 2 ,
k δ R cos θ O = I δ R 2 - I 000 ° ,
k δ R sin θ O = I 270 ° - I δ R 1 .
I + δ R = I b - k δ R ,
I - δ R = I b + k δ R .
2 k a O cos θ O = I 180 ° - I 000 ° ,
2 k a O sin θ O = I 090 ° - I 270 ° ,
2 k δ R = I 180 ° + I 000 ° - 2 I + δ R ,
2 k δ R = I - δ R - I + δ R ,
a O = [ ( 2 k a O cos θ O ) 2 + ( 2 k a O sin θ O ) 2 ] 1 / 2 2 k = δ R [ ( I 180 ° - I 000 ° ) 2 + ( I 090 ° - I 270 ° ) 2 ] 1 / 2 I - δ R - I + δ R ,
θ O = arctan 2 k a O sin θ O 2 k a O cos θ O = arctan I 090 ° - I 270 ° I 180 ° - I 000 ° .
I n ¯ = 1 M m = 1 M I n m             or             I n ¯ = m = 1 M I n m ,             n = 1 , 2 , 3 , 4 , 5 , 6.
2 k a O cos θ O ¯ = I 180 ° ¯ - I 000 ° ¯ or             2 k a O cos θ O ¯ = I 180 ° ¯ - I 000 ° ¯ ,
2 k a O sin θ O ¯ = I 090 ° ¯ - I 270 ° ¯ or             2 k a O sin θ O ¯ = I 090 ° ¯ - I 270 ° ¯ ,
2 k δ R ¯ = I - δ R ¯ - I + δ R ¯ or             2 k δ R ¯ = I - δ R ¯ - I + δ R ¯ .
2 k a O cos θ O ¯ = 1 M k = 1 M ( I 180 ° k - I 000 ° k ) or             2 k a O sin θ O ¯ = k = 1 M ( I 180 ° k - I 000 ° k ) ,
2 k a O sin θ O ¯ = 1 M k = 1 M ( I 090 ° k - I 270 ° k ) or             2 k a O sin θ O ¯ = k = 1 M ( I 090 ° k - I 270 ° k ) ,
2 k δ R ¯ = 1 M k = 1 M ( I - δ R k - I + δ R k ) or             2 k δ R ¯ = k = 1 M ( I - δ R k - I + δ R k ) .
I n ¯ = 1 P m = 1 P I n m             or             I n ¯ = m = 1 P I n m ,             n = 1 , 2 , 3 , 4 , 5 , 6 ,
2 k a O cos θ O ¯ = 1 L k = 1 L ( I 180 ° k ¯ - I 000 ° k ¯ ) or             2 k a O cos θ O ¯ = k = 1 L ( I 180 ° k ¯ - I 000 ° k ¯ ) ,
2 k a O sin θ O ¯ = 1 L k = 1 L ( I 090 ° k ¯ - I 270 ° k ¯ ) or             2 k a O sin θ O ¯ = k = 1 L ( I 090 ° k ¯ - I 270 ° k ¯ ) ,
2 k δ R ¯ = 1 L k = 1 L ( I - δ R k ¯ - I + δ R k ¯ ) or             2 k δ R ¯ = k = 1 L ( I - δ R k ¯ - I + δ R k ¯ ) .
σ n 2 σ 0 2 = 1 π 0 π ( 1 3 + 2 3 cos Ψ ) 2 n d Ψ
( 1 3 + 2 3 cos Ψ ) 2 n [ 1 3 + 2 3 ( 1 - 1 2 Ψ 2 ) ] 2 n = ( 1 - 1 6 n 2 n Ψ 2 ) 2 n exp ( - 1 3 2 n Ψ 2 ) = exp ( - 4 n 6 Ψ 2 ) .
σ n 2 σ 0 2 n 1 1 π π exp ( - 4 n 6 Ψ 2 ) d Ψ = [ 3 / ( 8 π n ) ] 1 / 2 .
σ a a = σ I 2 k a O δ R ( δ R 2 + 3 a O 2 ) 1 / 2 ,
σ a a = σ I 2 k a O δ R ( δ R 2 + a O 2 ) 1 / 2 .
σ θ = σ a 180 a π = σ I 180 2 k a O δ R π ( δ R 2 + 3 a O 2 ) 1 / 2 ,
σ θ = σ a 180 a π = σ I 180 2 k a O δ R π ( δ R 2 + a O 2 ) 1 / 2 .
I 000 ° = I b - k a O cos ( θ O + β 1 ) ,
I 090 ° = I b + k a O sin ( θ O + β 2 ) ,
I 180 ° = I b + k a O cos ( θ O + β 3 ) ,
I 270 ° = I b - k a O sin ( θ O + β 4 ) .
( θ O + Δ θ O ) = arctan I 090 ° - I 270 ° I 180 ° - I 000 ° = arctan ( β 2 + β 4 ) / 2 + tan θ O 1 - ( β 3 + β 1 ) / 2 tan θ O .
( a O + Δ a O ) = δ R [ ( I 180 ° - I 000 ° ) 2 + ( I 090 ° - I 270 ° ) 2 ] 1 / 2 ( I - δ R - I + δ R ) = a O [ 1 + 1 / 2 ( β 2 + β 4 - β 3 - β 1 ) sin 2 θ O ] 1 / 2 .
k = I - δ R - I + δ R 2 δ R .
σ k k = [ σ I - δ R 2 ( I - δ R - I + δ R ) 2 + σ I + δ R 2 ( I - δ R - I + δ R ) 2 + σ δ R 2 ] 1 / 2 .
I 000 ° = I b - ( k ± σ k ) a O cos θ O .

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