Abstract

A novel phase-shifts nπ/2 calibration method for phase-stepping interferometry, in which the sum intensities and the squared sum of a series of phase-shifted interferograms are firstly calculated, and then by the minimization of the variances of these sums perform the nπ/2 phase-shifts calibration, is proposed in this paper. The proposed method can overcome effectively the effect of the variations of background and modulation intensities in interferograms with any phase structure, and it is also insensitive to the nonlinearity of phase shifter. Numerical simulation and experiments are implemented to verify the effectiveness of this method.

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References

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  1. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland Elsevier, 1988), 26, pp.350–393.
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2006 (1)

2004 (1)

2001 (1)

2000 (1)

1999 (2)

Y. Zhu, J. B. Chen, R. H. Zhu, L. Chen, and Y. L. Xiao, “Phase calibrating method for phase-shifting adapter of interferometer,” Acta Photon. Sinca 28, 951–954 (1999).

H. van Brug, “Phase-step calibration for phase-stepped interferometry,” Appl. Opt. 38(16), 3549–3555 (1999).
[CrossRef]

1990 (1)

1987 (1)

1985 (1)

1982 (1)

1966 (1)

P. Carré, “Installation et utilisation du comparateur photoélectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2(1), 13–23 (1966).
[CrossRef]

Bokor, J.

Carré, P.

P. Carré, “Installation et utilisation du comparateur photoélectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2(1), 13–23 (1966).
[CrossRef]

Chen, J. B.

Y. Zhu, J. B. Chen, R. H. Zhu, L. Chen, and Y. L. Xiao, “Phase calibrating method for phase-shifting adapter of interferometer,” Acta Photon. Sinca 28, 951–954 (1999).

Chen, L.

Y. Zhu, J. B. Chen, R. H. Zhu, L. Chen, and Y. L. Xiao, “Phase calibrating method for phase-shifting adapter of interferometer,” Acta Photon. Sinca 28, 951–954 (1999).

Chen, X.

Cheng, Y. Y.

Fu, S.

Goldberg, K. A.

Gramaglia, M.

Ina, H.

Kerr, D.

Kobayashi, S.

Liu, X.

Mendoza Santoyo, F.

Roddier, C.

Roddier, F.

Sun, X.

Takeda, M.

Tyrer, J. R.

van Brug, H.

Wyant, J. C.

Xiao, Y. L.

Y. Zhu, J. B. Chen, R. H. Zhu, L. Chen, and Y. L. Xiao, “Phase calibrating method for phase-shifting adapter of interferometer,” Acta Photon. Sinca 28, 951–954 (1999).

Yang, X.

Yeazell, J. A.

Yu, Q.

Zhong, X. H.

Zhu, R. H.

Y. Zhu, J. B. Chen, R. H. Zhu, L. Chen, and Y. L. Xiao, “Phase calibrating method for phase-shifting adapter of interferometer,” Acta Photon. Sinca 28, 951–954 (1999).

Zhu, Y.

Y. Zhu, J. B. Chen, R. H. Zhu, L. Chen, and Y. L. Xiao, “Phase calibrating method for phase-shifting adapter of interferometer,” Acta Photon. Sinca 28, 951–954 (1999).

Acta Photon. Sinca (1)

Y. Zhu, J. B. Chen, R. H. Zhu, L. Chen, and Y. L. Xiao, “Phase calibrating method for phase-shifting adapter of interferometer,” Acta Photon. Sinca 28, 951–954 (1999).

Appl. Opt. (5)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Metrologia (1)

P. Carré, “Installation et utilisation du comparateur photoélectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2(1), 13–23 (1966).
[CrossRef]

Opt. Express (1)

Other (2)

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland Elsevier, 1988), 26, pp.350–393.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker Inc., New York, 1998).

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Figures (11)

Fig. 1
Fig. 1

Interferograms with varying background intensity and modulation amplitude both described by Gaussian function, the variable r equals 0 (a), 0.3 (b) and 0.6 (c).

Fig. 2
Fig. 2

The variances of the sum intensities (a) and squared sums (b) as a function of assumed phase shifts.

Fig. 3
Fig. 3

Simulated interferograms: (a) Case 1, with four full fringes and square aperture, (b) Case 2, with nonintegral fringes and circular aperture.

Fig. 4
Fig. 4

Calibration results for π (a) and for π/2 and 3π/2 (b) with the proposed method for two cases. Sign ‘*’,’ × ’: Computed from data; Curves: 5-order polynomial fits.

Fig. 5
Fig. 5

Wrapped phase-shifts extracted with correlation method for two cases.

Fig. 6
Fig. 6

Relationship between phase-shift and voltage. Black real curve, ideal phase-shift, gray dotted curve, fitted result of unwrapped phase for case 1 in Fig. 5; squares, results of proposed method for two cases.

Fig. 7
Fig. 7

Phase-shifts calibration optical setup: L1-4, lens; BS1-2, beam splitter.

Fig. 8
Fig. 8

Interferograms from Exps.1 and 2.

Fig. 9
Fig. 9

Wrapped phase-shifts extracted by correlation method.

Fig. 10
Fig. 10

Calibration results with correlation method (a) and FTM (b) for Exp.1

Fig. 11
Fig. 11

Calibration results for π (a) and for π/2 and 3π/2 (b) with the proposed method for two experiments. Sign ‘ + ’,’ × ’: Computed from data; curves: 5-order polynomial fits.

Tables (1)

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Table 1 Comparison of calibration results of the proposed method and correlation method.

Equations (11)

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Ik(x,y)=A(x,y)+B(x,y)cos[ϕ(x,y)+δk] ,
S=I1+Ik=2A+2Bcos(ϕ+δk/2)cos(δk/2)  .
E[S]=2A+2Bcos(ϕ+δk/2)cos(δk/2)  .
E[S]=2A ,
Var[S]=(SE[S])2=4B2cos2(δk/2)cos2(ϕ+δk/2)
=2B2cos2(δk/2)  .
Ss=I12+Ik2=B2cos2ϕ+B2cos2(ϕ+δk)  .
E[Ss]=B2,
Var[Ss]=(SsE[Ss])2=1/2B4cos2δk  .
Var[S]=4Var[A]+4cos2(δk/2)Var[Bcos(ϕ+δk/2)] ,
Var[Ss]=Var[B2cos2ϕ+B2cos2(ϕ+δk)] .

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