Abstract

The Fourier-transform method is often used to evaluate fringe patterns. The fundamental limitations of its accuracy are examined. Special filter functions leading to an improved spatial definition and a fringe-extrapolation algorithm that reduces the errors at the border of the pattern are presented. Numerical simulations predict an accuracy of the phase evaluation of less than 6 mrad under certain conditions. We investigated the reproducibility by experiments with a Michelson interferometer. Deviations of approximately 10 mrad were found. In a second test a Ronchi ruling was imaged, and a well-defined phase change was introduced. We deduce an accuracy of less than 5 mrad rms.

© 2001 Optical Society of America

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References

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  1. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am 72, 156–160 (1982).
    [Crossref]
  2. M. Küchel, “The new Zeiss interferometer,” in Optical Testing and Metrology III: Recent Advances in Industrial Inspection, C. P. Grover, ed., Proc. SPIE1332, 665–663 (1990).
  3. L. Mertz, “Real-time fringe-pattern analysis,” Appl. Opt. 22, 3977–3982 (1983).
  4. K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
  5. E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974), pp. 102–105.
  6. D. J. Bone, H.-A. Bachor, R. J. Sandeman, “Fringe-pattern analysis using a 2-D Fourier transform,” Appl. Opt. 25, 1653–1660 (1986).
    [Crossref] [PubMed]
  7. T. Kreis, Holographic Interferometry, W. Jüptner, W. Osten, eds. (Akademie-Verlag, Berlin, 1996), Chap. 4.6, pp. 141–142.
  8. E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974), pp. 115–118.
  9. M. Kujawinska, J. Wójciak, “High accuracy Fourier transform fringe pattern analysis,” Opt. Lasers Eng. 14, 325–339 (1991).
    [Crossref]
  10. C. Roddier, F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. 26, 1668–1673 (1987).
    [Crossref] [PubMed]

1991 (1)

M. Kujawinska, J. Wójciak, “High accuracy Fourier transform fringe pattern analysis,” Opt. Lasers Eng. 14, 325–339 (1991).
[Crossref]

1987 (1)

1986 (1)

1984 (1)

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).

1983 (1)

1982 (1)

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am 72, 156–160 (1982).
[Crossref]

Bachor, H.-A.

Bone, D. J.

Brigham, E. O.

E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974), pp. 102–105.

E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974), pp. 115–118.

Ina, H.

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am 72, 156–160 (1982).
[Crossref]

Kobayashi, S.

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am 72, 156–160 (1982).
[Crossref]

Kreis, T.

T. Kreis, Holographic Interferometry, W. Jüptner, W. Osten, eds. (Akademie-Verlag, Berlin, 1996), Chap. 4.6, pp. 141–142.

Küchel, M.

M. Küchel, “The new Zeiss interferometer,” in Optical Testing and Metrology III: Recent Advances in Industrial Inspection, C. P. Grover, ed., Proc. SPIE1332, 665–663 (1990).

Kujawinska, M.

M. Kujawinska, J. Wójciak, “High accuracy Fourier transform fringe pattern analysis,” Opt. Lasers Eng. 14, 325–339 (1991).
[Crossref]

Mertz, L.

Roddier, C.

Roddier, F.

Sandeman, R. J.

Takeda, M.

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am 72, 156–160 (1982).
[Crossref]

Wójciak, J.

M. Kujawinska, J. Wójciak, “High accuracy Fourier transform fringe pattern analysis,” Opt. Lasers Eng. 14, 325–339 (1991).
[Crossref]

Womack, K. H.

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).

Appl. Opt. (3)

J. Opt. Soc. Am (1)

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am 72, 156–160 (1982).
[Crossref]

Opt. Eng. (1)

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).

Opt. Lasers Eng. (1)

M. Kujawinska, J. Wójciak, “High accuracy Fourier transform fringe pattern analysis,” Opt. Lasers Eng. 14, 325–339 (1991).
[Crossref]

Other (4)

T. Kreis, Holographic Interferometry, W. Jüptner, W. Osten, eds. (Akademie-Verlag, Berlin, 1996), Chap. 4.6, pp. 141–142.

E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974), pp. 115–118.

E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974), pp. 102–105.

M. Küchel, “The new Zeiss interferometer,” in Optical Testing and Metrology III: Recent Advances in Industrial Inspection, C. P. Grover, ed., Proc. SPIE1332, 665–663 (1990).

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Figures (11)

Fig. 1
Fig. 1

(a) Simulated fringe pattern, (b) the magnitude of its discrete transform.

Fig. 2
Fig. 2

(a)–(c) Examples of frequency filters, (e) and (f) the magnitudes of their transforms.

Fig. 3
Fig. 3

Illustration of the fringe extrapolation, (a) complemented data ic, (b) magnitude of the spectrum of ic (solid curve) and the filter function (dashed curve), and (c) data with extrapolated pattern.

Fig. 4
Fig. 4

Sketch of the filter used for the extrapolation with the 2D FTM.

Fig. 5
Fig. 5

Simulation: (a) input phase measured in radians and (b) error of the FTM.

Fig. 6
Fig. 6

Sketch of the Michelson interferometer used in the test.

Fig. 7
Fig. 7

Interferometer experiment: (a) intensities i 1 (solid curve) and i 2 (dashed curve) at row 64 coming from single interferometer arms, (b) difference of evaluated phases (solid curve) and fitting plane (dashed line) at the same row. The arrows indicate imperfections of the mirror surfaces.

Fig. 8
Fig. 8

Sketch of the experiment with the Ronchi ruling.

Fig. 9
Fig. 9

Fringe pattern measured with the cover slip on the Ronchi ruling.

Fig. 10
Fig. 10

Phase difference caused by the insertion of the cover slip displayed as gray levels. The arrows indicate the column displayed in Fig. 11.

Fig. 11
Fig. 11

Plot of the evaluated phase difference (solid curve) and the reference plane (dashed line) for the column indicated in Fig. 10.

Equations (6)

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ix, y=ax, y+bx, ycosΦx, y,
Φx, y=Φcx, y+Φsx, y.
Φcx, y=2πfc,xx+fc,yy+Φ0.
cx, y=½bx, yexpjΦsx, y,
ix, y=ax, y+cx, yexpjΦcx, y+c*x, yexp-jΦcx, y,
Ifx, fy=Afx, fy+Cfx-fc,x, fy-fc,y+C*fx+fc,x, fy+fc,y,

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