Abstract

We propose a sinusoidal phase modulating (SPM) interferometer for displacement measurement. The displacement is obtained by processing the detected signal with a microcomputer. The optical configuration of the SPM interferometer is simple because an optical fiber is effectively used to produce sinusoidal phase modulation. The movements of a piezoelectric transducer and the surface profile of a diamond-turned aluminum disk are measured with high accuracy (a few nanometers).

© 1988 Optical Society of America

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References

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  1. S. Ueha, K. Shiota, T. Okada, J. Tsujiuchi, “Optical Heterodyne Measurement of In-Plane Vibrations,” Opt. Commun. 10, 88 (1974).
    [Crossref]
  2. W. Puschert, “Optical Detection of Amplitude and Phase of Mechanical Displacements in the Angstrom Range,” Opt. Commun. 10, 357 (1974).
    [Crossref]
  3. Y. Ohtsuka, I. Sasaki, “Laser Heterodyne Measurement of Small Arbitrary Displacements,” Opt. Commun. 10, 362 (1974).
    [Crossref]
  4. O. Sasaki, H. Okazaki, “Sinusoidal Phase Modulating Interferometry for Surface Profile Measurement,” Appl. Opt. 25, 3137 (1986).
    [Crossref] [PubMed]
  5. O. Sasaki, H. Okazaki, “Analysis of Measurement Accuracy in Sinusoidal Phase Modulating Interferometry,” Appl. Opt. 25, 3152 (1986).
    [Crossref] [PubMed]
  6. O. Sasaki, H. Okazaki, M. Sakai, “Sinusoidal Phase Modulating Interferometer Using the Integrating-Bucket Method,” Appl. Opt. 26, 1089 (1987).
    [Crossref] [PubMed]
  7. G. E. Sommargren, “Optical Heterodyne Profilometry,” Appl. Opt. 20, 610 (1981).
    [Crossref] [PubMed]
  8. D. Pantzer, J. Politch, L. Ek, “Heterodyne Profiling Instrument for the Angstrom Region,” Appl. Opt. 25, 4168 (1986).
    [Crossref] [PubMed]
  9. T. Kohno, N. Ozawa, K. Miyamoto, T. Musha, “High Precision Optical Surface Sensor,” Appl. Opt. 27, 103 (1988).
    [Crossref] [PubMed]

1988 (1)

1987 (1)

1986 (3)

1981 (1)

1974 (3)

S. Ueha, K. Shiota, T. Okada, J. Tsujiuchi, “Optical Heterodyne Measurement of In-Plane Vibrations,” Opt. Commun. 10, 88 (1974).
[Crossref]

W. Puschert, “Optical Detection of Amplitude and Phase of Mechanical Displacements in the Angstrom Range,” Opt. Commun. 10, 357 (1974).
[Crossref]

Y. Ohtsuka, I. Sasaki, “Laser Heterodyne Measurement of Small Arbitrary Displacements,” Opt. Commun. 10, 362 (1974).
[Crossref]

Ek, L.

Kohno, T.

Miyamoto, K.

Musha, T.

Ohtsuka, Y.

Y. Ohtsuka, I. Sasaki, “Laser Heterodyne Measurement of Small Arbitrary Displacements,” Opt. Commun. 10, 362 (1974).
[Crossref]

Okada, T.

S. Ueha, K. Shiota, T. Okada, J. Tsujiuchi, “Optical Heterodyne Measurement of In-Plane Vibrations,” Opt. Commun. 10, 88 (1974).
[Crossref]

Okazaki, H.

Ozawa, N.

Pantzer, D.

Politch, J.

Puschert, W.

W. Puschert, “Optical Detection of Amplitude and Phase of Mechanical Displacements in the Angstrom Range,” Opt. Commun. 10, 357 (1974).
[Crossref]

Sakai, M.

Sasaki, I.

Y. Ohtsuka, I. Sasaki, “Laser Heterodyne Measurement of Small Arbitrary Displacements,” Opt. Commun. 10, 362 (1974).
[Crossref]

Sasaki, O.

Shiota, K.

S. Ueha, K. Shiota, T. Okada, J. Tsujiuchi, “Optical Heterodyne Measurement of In-Plane Vibrations,” Opt. Commun. 10, 88 (1974).
[Crossref]

Sommargren, G. E.

Tsujiuchi, J.

S. Ueha, K. Shiota, T. Okada, J. Tsujiuchi, “Optical Heterodyne Measurement of In-Plane Vibrations,” Opt. Commun. 10, 88 (1974).
[Crossref]

Ueha, S.

S. Ueha, K. Shiota, T. Okada, J. Tsujiuchi, “Optical Heterodyne Measurement of In-Plane Vibrations,” Opt. Commun. 10, 88 (1974).
[Crossref]

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Figures (10)

Fig. 1
Fig. 1

Sinusoidal phase modulating interferometer using optical fibers for displacement measurement.

Fig. 2
Fig. 2

Amplitude distribution of a discrete Fourier transform of the detected signal s(t) when the object was sinusoidally vibrated.

Fig. 3
Fig. 3

Measured movement of the piezoelectric transducer when a sinusoidal wave voltage was applied to it.

Fig. 4
Fig. 4

Measured movement of the piezoelectric transducer when no voltage was applied to it. This movement indicates the vibrations of the experimental setup.

Fig. 5
Fig. 5

Measured movement of the piezoelectric transducer when a rectangular pulse voltage was applied to it.

Fig. 6
Fig. 6

Experimental setup for surface profile measurement, where two sinusoidal phase modulating interferometers are used.

Fig. 7
Fig. 7

Difference r1r2, where r1 and r2 are the measured displacements of the object and the mirror, respectively, when they were translated.

Fig. 8
Fig. 8

Surface profile obtained by eliminating the tilt component from Fig. 7.

Fig. 9
Fig. 9

Surface profile obtained from the r1 by eliminating its tilt component.

Fig. 10
Fig. 10

Surface profile measured with a Talystep instrument.

Equations (9)

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A ( t ) = a cos ( ω c t + θ ) .
l ( t ) = A ( t ) + r ( t ) + b ,
s ( t ) = cos [ z cos ( ω c t + θ ) + α ( t ) ] ,
F ( ω ) = F { cos [ α ( t ) ] } [ m = - ( - 1 ) m A 2 m δ ( ω - 2 m ω c ) ] + F { sin [ α ( t ) ] } { m = - ( - 1 ) m A 2 m - 1 δ [ ω - ( 2 m - 1 ) ω c ] } ,
F { cos [ α ( t ) ] } = 0 , F { sin [ α ( t ) ] } = 0 ,             ω > ω c / 2 ,
F 1 ( ω + ω c ) = - J 1 ( z ) exp ( j θ ) F { sin [ α ( t ) ] } , F 2 ( ω + 2 ω c ) = - J 2 ( z ) exp ( j 2 θ ) F { cos [ α ( t ) ] } ,             ω < ω c / 2.
- F 1 ( ω + ω c ) / J 1 ( z ) exp ( j θ ) , - F 2 ( ω + 2 ω c ) / J 2 ( z ) exp ( j 2 θ ) ,
r 1 ( t ) = ϕ ( x ) + a 1 x + n ( t ) , r 2 ( t ) = a 2 x + n ( t ) ,
r 1 ( t ) - r 2 ( t ) = ϕ ( x ) + ( a 1 - a 2 ) x ,

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