Abstract

A new algorithm for precise determination of the global phase shift between two interferograms is introduced. First we calculate the frame difference between the first and the second interferogram; the difference is multiplied by a properly chosen test phase factor, and then we implement a two-dimensional Fourier transform of the frame difference and calculate the energy of the first positive (or negative) diffraction order. An iterative approach is used for the test phase to ensure that the minimum energy is obtained, and then the correct phase shift value is found. This method is called the energy-minimum Fourier transform method, which is accurate and noise insensitive compared with the single-point Fourier transform method. Both the theoretical analysis and experimental results are given.

© 2003 Optical Society of America

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References

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  1. F. Bucholtz, K. P. Koo, G. H. Sigel, A. Dandridge, “Optimization of the fiber/metallic glass bond in fiber-optic magnetic sensors,” J. Lightwave Technol. 3, 814–817 (1985).
    [CrossRef]
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  12. T. Yamaguchi, K. Hamano, “Inteferometric method of measuring complex piezoelectric constants of crystals in a frequency range up to about 50 kHz,” Jpn. J. Appl. Phys. 18, 927–932 (1979).
    [CrossRef]
  13. A. Nesci, R. Dändliker, H. P. Herzig, “Quantitative amplitude and phase measurement by use of a heterodyne scanning near-field optical microscope,” Opt. Lett. 26, 208–210 (2001).
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2002 (1)

2001 (3)

1996 (1)

1993 (1)

1992 (1)

1987 (2)

1985 (1)

F. Bucholtz, K. P. Koo, G. H. Sigel, A. Dandridge, “Optimization of the fiber/metallic glass bond in fiber-optic magnetic sensors,” J. Lightwave Technol. 3, 814–817 (1985).
[CrossRef]

1983 (1)

1982 (1)

1979 (1)

T. Yamaguchi, K. Hamano, “Inteferometric method of measuring complex piezoelectric constants of crystals in a frequency range up to about 50 kHz,” Jpn. J. Appl. Phys. 18, 927–932 (1979).
[CrossRef]

1975 (1)

1974 (1)

1969 (1)

R. Crane, “Interference phase measurement,” Appl. Opt. 8, 538–542 (1969).

Bokor, J.

Brangaccio, D. J.

Bruning, J. H.

Bucholtz, F.

F. Bucholtz, K. P. Koo, G. H. Sigel, A. Dandridge, “Optimization of the fiber/metallic glass bond in fiber-optic magnetic sensors,” J. Lightwave Technol. 3, 814–817 (1985).
[CrossRef]

Burrow, R.

Crane, R.

R. Crane, “Interference phase measurement,” Appl. Opt. 8, 538–542 (1969).

Dändliker, R.

Dandridge, A.

F. Bucholtz, K. P. Koo, G. H. Sigel, A. Dandridge, “Optimization of the fiber/metallic glass bond in fiber-optic magnetic sensors,” J. Lightwave Technol. 3, 814–817 (1985).
[CrossRef]

Eiju, T.

Elssner, K. E.

Gallagher, J. E.

Goldberg, K. A.

Grzanna, J.

Guo, C. S.

Hamano, K.

T. Yamaguchi, K. Hamano, “Inteferometric method of measuring complex piezoelectric constants of crystals in a frequency range up to about 50 kHz,” Jpn. J. Appl. Phys. 18, 927–932 (1979).
[CrossRef]

Hariharan, P.

Herriott, D. R.

Herzig, H. P.

Ina, H.

Kieran, G.

Kobayashi, S.

Koo, K. P.

F. Bucholtz, K. P. Koo, G. H. Sigel, A. Dandridge, “Optimization of the fiber/metallic glass bond in fiber-optic magnetic sensors,” J. Lightwave Technol. 3, 814–817 (1985).
[CrossRef]

Larkin, A.

Larkin, K. G.

Liao, J.

Merkel, K.

Nesci, A.

Oreb, B. F.

Roddier, C.

Roddier, F.

Rosenfeld, D. P.

Schwider, J.

Sigel, G. H.

F. Bucholtz, K. P. Koo, G. H. Sigel, A. Dandridge, “Optimization of the fiber/metallic glass bond in fiber-optic magnetic sensors,” J. Lightwave Technol. 3, 814–817 (1985).
[CrossRef]

Spolaczyk, R.

Surrel, Y.

Takeda, M.

Wang, H. T.

White, A. D.

Wyant, J. C.

Yamaguchi, T.

T. Yamaguchi, K. Hamano, “Inteferometric method of measuring complex piezoelectric constants of crystals in a frequency range up to about 50 kHz,” Jpn. J. Appl. Phys. 18, 927–932 (1979).
[CrossRef]

Yu, F. T. S.

F. T. S. Yu, Optical Information Processing (Wiley-Interscience, New York, 1982), pp. 415–424.

Zhang, L.

Zhu, Y. Y.

Appl. Opt. (9)

J. Lightwave Technol. (1)

F. Bucholtz, K. P. Koo, G. H. Sigel, A. Dandridge, “Optimization of the fiber/metallic glass bond in fiber-optic magnetic sensors,” J. Lightwave Technol. 3, 814–817 (1985).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Jpn. J. Appl. Phys. (1)

T. Yamaguchi, K. Hamano, “Inteferometric method of measuring complex piezoelectric constants of crystals in a frequency range up to about 50 kHz,” Jpn. J. Appl. Phys. 18, 927–932 (1979).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Other (1)

F. T. S. Yu, Optical Information Processing (Wiley-Interscience, New York, 1982), pp. 415–424.

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Figures (8)

Fig. 1
Fig. 1

Optical setup for precise displacement measurement with a Michelson interferometer. EL, two-lens group; BS, beam splitter; M1, M2, mirrors.

Fig. 2
Fig. 2

Complex-plane representations (a) of the influence of the noise Q n on the measured phase shift (b) of the subtraction described by Eq. (10).

Fig. 3
Fig. 3

Two simulated interferograms with a relative phase shift of 0.58 rad.

Fig. 4
Fig. 4

Section curves (a) of the spatial spectrum of the first interferogram shown in Fig. 2(a) and (b) of the spatial spectrum of the difference between two interferograms in Fig. 2, resulting from Eq. (4) with a test phase value of 0.40 rad.

Fig. 5
Fig. 5

Curve of total energy E of the +1-order diffraction versus the test phase shift δ.

Fig. 6
Fig. 6

Phase shift values measured with a different method. Curve A was calculated by use of our EMFT method, proposed in Section 3, and curve B was measured by the single-point Fourier-transform method. The dashed line gives the theoretical value (0.58 rad).

Fig. 7
Fig. 7

One of the interferograms recorded by the CCD camera and (b) its spatial spectrum. (c) and (d), spectra of the differences [from Eq. (4)] between two recorded interferograms, with test phases δ = 3.50 and δ = 3.83 rad, respectively.

Fig. 8
Fig. 8

Experimental displacement curve of a calibrated PZT nanopositioner.

Equations (12)

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

ξ0=sin θ/λ,
fnx, y=ax, y+bx, ycos2πξ0x+φ0x, y+δn,
Fnξ, η=Aξ, η+Bnξ-ξ0, η, δn+Bn*ξ+ξ0, η, δn,
Bnξ, η, δn=expiδn12 bx, yexpiφ0x, y.
Fnξ0, 0=Aξ0, 0+Bn0, 0, δn+Bn*2ξ0, 0, δnBn0, 0, δn.
δn=tan-1ImFnξ0, 0, δnF1*ξ0, 0, δ1ReFnξ0, 0, δnF1*ξ0, 0, δ1.
fnx, y=m=- gm expim2πξ0x+Δφ+δn,
Fnξ0, 0=Bn0, 0, δn+Qnξ0, 0,
Δδmax|Q1|+|Qn||B1|.
Sξ, η, δ=F1ξ, η, δ1-Fnξ, η, δnexp-iδ.
Eδ=+1 order |Sξ, η, δ|2dξdη.
expiδ-δn=1+iδ-δn, |δ-δn|=ε  1,

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