Abstract

There is increasing demand for in situ shape measurements performed on ultraprecision processing machines. One major source of error during interferometric measurements performed on machines is fringe displacement due to external disturbances. We have developed an interferometer equipped with an electro-optic phase modulator that measures the phase of interference fringes before they are displaced by air turbulence. The frequency characteristics of air turbulence induced by a heat source are derived from successive measurements of a test surface. Experimental results show that the phase of the interference fringes can be accurately measured in the presence of air turbulence when the intensity of the fringes is sampled at a speed of several hundred hertz.

© 2003 Optical Society of America

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References

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  1. A. Marquez, M. Yamauchi, J. A. Davis, D. J. Franich, “Phase measurement of a twisted nematic liquid crystal spatial light modulator with a common-path interferometer,” Opt. Commun. 190, 129–133 (2001).
    [CrossRef]
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    [CrossRef] [PubMed]
  3. O. Sasaki, K. Takahashi, T. Suzuki, “Sinusoidal phase modulating laser diode interferometer with a feedback control system to eliminate external disturbance,” Opt. Eng. 29, 1511–1515 (1990).
    [CrossRef]
  4. I. Yamaguchi, J. Liu, J. Kato, “Active phase-shifting interferometers for shape and deformation measurements,” Opt. Eng. 35, 2930–2937 (1996).
    [CrossRef]
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    [CrossRef]
  6. M. Melozzi, L. Pezzati, A. Mazzoni, “Vibration-insensitive interferometer for on-line measurements,” Appl. Opt. 34, 5595–5601 (1995).
    [CrossRef] [PubMed]
  7. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics XIX, E. Wolf, ed. (North-Holland, Amsterdam, 1981), pp. 281–376.
    [CrossRef]
  8. M. Yamauchi, A. Marquez, J. A. Davis, D. J. Franich, “Interferometric phase measurements for polarization eigenvectors in twisted nematic liquid crystal spatial light modulators,” Opt. Commun. 181, 1–6 (2000).
    [CrossRef]
  9. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics XXVI, E. Wolf, ed. (North-Holland, Amsterdam, 1988), pp. 349–393.
    [CrossRef]
  10. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef] [PubMed]
  11. K. Hibino, “Susceptibility of systematic error-compensating algorithms to random noise in phase-shifting interferometry,” Appl. Opt. 36, 2084–2093 (1997).
    [CrossRef] [PubMed]

2001

A. Marquez, M. Yamauchi, J. A. Davis, D. J. Franich, “Phase measurement of a twisted nematic liquid crystal spatial light modulator with a common-path interferometer,” Opt. Commun. 190, 129–133 (2001).
[CrossRef]

2000

M. Yamauchi, A. Marquez, J. A. Davis, D. J. Franich, “Interferometric phase measurements for polarization eigenvectors in twisted nematic liquid crystal spatial light modulators,” Opt. Commun. 181, 1–6 (2000).
[CrossRef]

1997

1996

I. Yamaguchi, J. Liu, J. Kato, “Active phase-shifting interferometers for shape and deformation measurements,” Opt. Eng. 35, 2930–2937 (1996).
[CrossRef]

1995

1990

O. Sasaki, K. Takahashi, T. Suzuki, “Sinusoidal phase modulating laser diode interferometer with a feedback control system to eliminate external disturbance,” Opt. Eng. 29, 1511–1515 (1990).
[CrossRef]

1987

1981

1974

Brangaccio, D. J.

Bruning, J. H.

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics XXVI, E. Wolf, ed. (North-Holland, Amsterdam, 1988), pp. 349–393.
[CrossRef]

Davis, J. A.

A. Marquez, M. Yamauchi, J. A. Davis, D. J. Franich, “Phase measurement of a twisted nematic liquid crystal spatial light modulator with a common-path interferometer,” Opt. Commun. 190, 129–133 (2001).
[CrossRef]

M. Yamauchi, A. Marquez, J. A. Davis, D. J. Franich, “Interferometric phase measurements for polarization eigenvectors in twisted nematic liquid crystal spatial light modulators,” Opt. Commun. 181, 1–6 (2000).
[CrossRef]

Franich, D. J.

A. Marquez, M. Yamauchi, J. A. Davis, D. J. Franich, “Phase measurement of a twisted nematic liquid crystal spatial light modulator with a common-path interferometer,” Opt. Commun. 190, 129–133 (2001).
[CrossRef]

M. Yamauchi, A. Marquez, J. A. Davis, D. J. Franich, “Interferometric phase measurements for polarization eigenvectors in twisted nematic liquid crystal spatial light modulators,” Opt. Commun. 181, 1–6 (2000).
[CrossRef]

Gallagher, J. E.

Herriott, D. R.

Hibino, K.

Ina, H.

Kato, J.

I. Yamaguchi, J. Liu, J. Kato, “Active phase-shifting interferometers for shape and deformation measurements,” Opt. Eng. 35, 2930–2937 (1996).
[CrossRef]

Kobayashi, S.

Lee, B. S.

Liu, J.

I. Yamaguchi, J. Liu, J. Kato, “Active phase-shifting interferometers for shape and deformation measurements,” Opt. Eng. 35, 2930–2937 (1996).
[CrossRef]

Marquez, A.

A. Marquez, M. Yamauchi, J. A. Davis, D. J. Franich, “Phase measurement of a twisted nematic liquid crystal spatial light modulator with a common-path interferometer,” Opt. Commun. 190, 129–133 (2001).
[CrossRef]

M. Yamauchi, A. Marquez, J. A. Davis, D. J. Franich, “Interferometric phase measurements for polarization eigenvectors in twisted nematic liquid crystal spatial light modulators,” Opt. Commun. 181, 1–6 (2000).
[CrossRef]

Mazzoni, A.

Melozzi, M.

Mnatzakanian, S.

Nara, M.

Pezzati, L.

Roddier, F.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics XIX, E. Wolf, ed. (North-Holland, Amsterdam, 1981), pp. 281–376.
[CrossRef]

Rosenfeld, D. P.

Sasaki, O.

O. Sasaki, K. Takahashi, T. Suzuki, “Sinusoidal phase modulating laser diode interferometer with a feedback control system to eliminate external disturbance,” Opt. Eng. 29, 1511–1515 (1990).
[CrossRef]

Strand, T. C.

Suzuki, T.

O. Sasaki, K. Takahashi, T. Suzuki, “Sinusoidal phase modulating laser diode interferometer with a feedback control system to eliminate external disturbance,” Opt. Eng. 29, 1511–1515 (1990).
[CrossRef]

Takahashi, K.

O. Sasaki, K. Takahashi, T. Suzuki, “Sinusoidal phase modulating laser diode interferometer with a feedback control system to eliminate external disturbance,” Opt. Eng. 29, 1511–1515 (1990).
[CrossRef]

Takeda, M.

White, A. D.

Yamaguchi, I.

I. Yamaguchi, J. Liu, J. Kato, “Active phase-shifting interferometers for shape and deformation measurements,” Opt. Eng. 35, 2930–2937 (1996).
[CrossRef]

Yamauchi, M.

A. Marquez, M. Yamauchi, J. A. Davis, D. J. Franich, “Phase measurement of a twisted nematic liquid crystal spatial light modulator with a common-path interferometer,” Opt. Commun. 190, 129–133 (2001).
[CrossRef]

M. Yamauchi, A. Marquez, J. A. Davis, D. J. Franich, “Interferometric phase measurements for polarization eigenvectors in twisted nematic liquid crystal spatial light modulators,” Opt. Commun. 181, 1–6 (2000).
[CrossRef]

Yoshino, T.

Appl. Opt.

J. Opt. Soc. Am.

Opt. Commun.

A. Marquez, M. Yamauchi, J. A. Davis, D. J. Franich, “Phase measurement of a twisted nematic liquid crystal spatial light modulator with a common-path interferometer,” Opt. Commun. 190, 129–133 (2001).
[CrossRef]

M. Yamauchi, A. Marquez, J. A. Davis, D. J. Franich, “Interferometric phase measurements for polarization eigenvectors in twisted nematic liquid crystal spatial light modulators,” Opt. Commun. 181, 1–6 (2000).
[CrossRef]

Opt. Eng.

O. Sasaki, K. Takahashi, T. Suzuki, “Sinusoidal phase modulating laser diode interferometer with a feedback control system to eliminate external disturbance,” Opt. Eng. 29, 1511–1515 (1990).
[CrossRef]

I. Yamaguchi, J. Liu, J. Kato, “Active phase-shifting interferometers for shape and deformation measurements,” Opt. Eng. 35, 2930–2937 (1996).
[CrossRef]

Other

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics XIX, E. Wolf, ed. (North-Holland, Amsterdam, 1981), pp. 281–376.
[CrossRef]

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics XXVI, E. Wolf, ed. (North-Holland, Amsterdam, 1988), pp. 349–393.
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Data sampling of phase ϕ(t).

Fig. 2
Fig. 2

On-machine interferometry in the presence of air turbulence (side view).

Fig. 3
Fig. 3

Comprehensive experimental setup for measuring the phase change induced by air turbulence in the object beam path. The reference signal for the lock-in amplifier is supplied by the function generator when the frequency response of the change is measured. It is supplied by PD1 via the filter when the spatial distribution of the change is measured. In this case an aperture in front of PD2 is scanned by the linear stage.

Fig. 4
Fig. 4

Examples of signals transmitted to the lock-in amplifier. Top, function generator; middle, PD1 via a filter; bottom, PD2.

Fig. 5
Fig. 5

Detected phase as measured in the absence of induced air turbulence: circles, phase ϕ2; triangles, phase difference ϕ2 - ϕ1. The tilt component is removed.

Fig. 6
Fig. 6

Raw phase data measured with our system in Fig. 3. The modulation frequency equals 1 kHz, the reference signal for the lock-in amplifier is provided by the function generator, and air turbulence is induced by a portable body warmer.

Fig. 7
Fig. 7

Measured phase error as defined by the standard deviation of raw data in Fig. 6. A comparison of the circles and triangles shows the effect of air turbulence induced by a portable body warmer as a function of the sampling frequency.

Fig. 8
Fig. 8

Phase error measured when the reference signal transmitted to the lock-in amplifier is supplied by PD1. A comparison of the solid and open circles shows the spatial distribution of the air turbulence.

Fig. 9
Fig. 9

Measured phase error as a function of total measurement period needed to sample all ten data. Equation (6) is used to obtain the solid curves.

Fig. 10
Fig. 10

Time chart of image sampling for a four-sample Fourier phase-shifting algorithm.

Fig. 11
Fig. 11

Measurement accuracy of the phase-shifting technique compared with the single-interferogram technique.

Equations (13)

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

I=I0+I1 cos ϕ,
ϕt=ϕ+j=1n/2 Aj sin2π jτ t+εj,
σϕ=1τ0τϕt-ϕ2dt1/2=1τ0τj=1n/2 Aj sin2π jτ t+εj2dt1/2=12j=1n/2 Aj21/2.
Aj=Afja,
σϕτ=Aτa2j=1n/21j2a1/2.
σϕτ=A2τ2a2j=1n/21j2a+σ021/2.
Rz, t =R0 expiωt-kz+2πfmt,
Mz, t =M0 expiωt-kz-ϕ,
I1,2=R02+M02+2R0M0 cos2πfmt+ϕ1,2,
ϕ=arctanIπ/2-I3π/2Iπ-I0,
τp=1720fm.
σp=σϕτpm.
σs=σϕτe.

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