Abstract

Fourier-transform profilometry (FTP) and data-dependent system profilometry (DDSP) are the two major phase-extraction methods that use a single interferogram. The difficulty in verifying surface profiles obtained by these methods is that the exact spot on an actual surface cannot be measured with two different instruments. An interferogram regeneration procedure is developed to solve this problem. The surface profile is then extracted from the regenerated interferogram by both FTP and DDSP. Comparisons of the actual surface profile with the extracted surface profiles show that both methods perform equally well in measuring the root mean square and the center line average, but only DDSP is able to reproduce the detailed surface profile of the reference surface.

© 1999 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 interferogram,” J. Opt. Soc. Am. A 72, 156–160 (1982).
    [CrossRef]
  2. S. M. Pandit, N. Jordache, “Interferogram analysis based on the data-dependent systems method for nanometrology applications,” Appl. Opt. 34, 6695–6703 (1995).
    [CrossRef] [PubMed]
  3. S. M. Pandit, N. Jordache, “Data-dependent-systems and Fourier-transform methods for single-interferogram analysis,” Appl. Opt. 34, 5945–5951 (1995).
    [CrossRef] [PubMed]
  4. 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]
  5. M. Takeda, K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22, 3977–3982 (1983).
    [CrossRef] [PubMed]
  6. C. Roddier, F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. 26, 1668–1673 (1987).
    [CrossRef] [PubMed]
  7. M. Takeda, H. Yamamoto, “Fourier-transform speckle profilometry: three-dimensional shape measurements of diffuse objects with large height steps and/or spatially isolated surfaces,” Appl. Opt. 33, 7829–7837 (1994).
    [CrossRef] [PubMed]
  8. H. J. Kim, B. W. James, “Two-dimensional Fourier-transform techniques for the analysis of hook interferograms,” Appl. Opt. 36, 1352–1356 (1997).
    [CrossRef] [PubMed]
  9. M. Takeda, Q. Gu, M. Kinoshita, H. Takai, Y. Takahashi, “Frequency-multiplex Fourier-transform profilometry: a single-shot three-dimensional shape measurement of objects with large height discontinuities and/or surface isolations,” Appl. Opt. 36, 5347–5354 (1997).
    [CrossRef] [PubMed]
  10. S. M. Pandit, S. M. Wu, Time Series and System Analysis with Applications (Wiley, New York, 1983; reprinted by Krieger, Malabar, Fla., 1993).
  11. S. M. Pandit, Modal and Spectrum Analysis: Data-Dependent Systems in State Space (Wiley, New York, 1991).
  12. S. M. Pandit, N. Jordache, G. A. Joshi, “Data-dependent systems methodology for noise-insensitive phase unwrapping in laser interferometric surface characterization,” J. Opt. Soc. Am. A 11, 2584–2592 (1994).
    [CrossRef]
  13. D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis, D. Robinson, G. T. Reid, eds. (Institute of Physics, University of Reading, Berkshire, UK, 1993).
  14. J. Schmit, K. Creath, M. Kujawinska, “Spatial and temporal phase-measurement techniques: a comparison of major error sources in one dimension,” in Interferometry: Techniques and Analysis, G. M. Brown, M. Kujawinska, O. Y. Kwon, G. T. Reid, eds., Proc. SPIE1755, 202–211 (1992).

1997 (2)

1995 (2)

1994 (2)

1987 (1)

1986 (1)

1983 (1)

1982 (1)

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

Bachor, H. A.

Bone, D. J.

Creath, K.

J. Schmit, K. Creath, M. Kujawinska, “Spatial and temporal phase-measurement techniques: a comparison of major error sources in one dimension,” in Interferometry: Techniques and Analysis, G. M. Brown, M. Kujawinska, O. Y. Kwon, G. T. Reid, eds., Proc. SPIE1755, 202–211 (1992).

Gu, Q.

Ina, H.

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

James, B. W.

Jordache, N.

Joshi, G. A.

Kim, H. J.

Kinoshita, M.

Kobayashi, S.

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

Kujawinska, M.

J. Schmit, K. Creath, M. Kujawinska, “Spatial and temporal phase-measurement techniques: a comparison of major error sources in one dimension,” in Interferometry: Techniques and Analysis, G. M. Brown, M. Kujawinska, O. Y. Kwon, G. T. Reid, eds., Proc. SPIE1755, 202–211 (1992).

Mutoh, K.

Pandit, S. M.

Robinson, D. W.

D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis, D. Robinson, G. T. Reid, eds. (Institute of Physics, University of Reading, Berkshire, UK, 1993).

Roddier, C.

Roddier, F.

Sandeman, R. J.

Schmit, J.

J. Schmit, K. Creath, M. Kujawinska, “Spatial and temporal phase-measurement techniques: a comparison of major error sources in one dimension,” in Interferometry: Techniques and Analysis, G. M. Brown, M. Kujawinska, O. Y. Kwon, G. T. Reid, eds., Proc. SPIE1755, 202–211 (1992).

Takahashi, Y.

Takai, H.

Takeda, M.

Wu, S. M.

S. M. Pandit, S. M. Wu, Time Series and System Analysis with Applications (Wiley, New York, 1983; reprinted by Krieger, Malabar, Fla., 1993).

Yamamoto, H.

Appl. Opt. (8)

M. Takeda, K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22, 3977–3982 (1983).
[CrossRef] [PubMed]

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]

M. Takeda, H. Yamamoto, “Fourier-transform speckle profilometry: three-dimensional shape measurements of diffuse objects with large height steps and/or spatially isolated surfaces,” Appl. Opt. 33, 7829–7837 (1994).
[CrossRef] [PubMed]

H. J. Kim, B. W. James, “Two-dimensional Fourier-transform techniques for the analysis of hook interferograms,” Appl. Opt. 36, 1352–1356 (1997).
[CrossRef] [PubMed]

S. M. Pandit, N. Jordache, “Data-dependent-systems and Fourier-transform methods for single-interferogram analysis,” Appl. Opt. 34, 5945–5951 (1995).
[CrossRef] [PubMed]

S. M. Pandit, N. Jordache, “Interferogram analysis based on the data-dependent systems method for nanometrology applications,” Appl. Opt. 34, 6695–6703 (1995).
[CrossRef] [PubMed]

M. Takeda, Q. Gu, M. Kinoshita, H. Takai, Y. Takahashi, “Frequency-multiplex Fourier-transform profilometry: a single-shot three-dimensional shape measurement of objects with large height discontinuities and/or surface isolations,” Appl. Opt. 36, 5347–5354 (1997).
[CrossRef] [PubMed]

C. Roddier, F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. 26, 1668–1673 (1987).
[CrossRef] [PubMed]

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

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

S. M. Pandit, N. Jordache, G. A. Joshi, “Data-dependent systems methodology for noise-insensitive phase unwrapping in laser interferometric surface characterization,” J. Opt. Soc. Am. A 11, 2584–2592 (1994).
[CrossRef]

Other (4)

S. M. Pandit, S. M. Wu, Time Series and System Analysis with Applications (Wiley, New York, 1983; reprinted by Krieger, Malabar, Fla., 1993).

S. M. Pandit, Modal and Spectrum Analysis: Data-Dependent Systems in State Space (Wiley, New York, 1991).

D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis, D. Robinson, G. T. Reid, eds. (Institute of Physics, University of Reading, Berkshire, UK, 1993).

J. Schmit, K. Creath, M. Kujawinska, “Spatial and temporal phase-measurement techniques: a comparison of major error sources in one dimension,” in Interferometry: Techniques and Analysis, G. M. Brown, M. Kujawinska, O. Y. Kwon, G. T. Reid, eds., Proc. SPIE1755, 202–211 (1992).

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Figures (16)

Fig. 1
Fig. 1

Twyman–Green interferometer.

Fig. 2
Fig. 2

Fourier spectrum of an interferogram.

Fig. 3
Fig. 3

Filtered Fourier spectrum.

Fig. 4
Fig. 4

Wrapped phase.

Fig. 5
Fig. 5

Unwrapped phase and υ̂ x .

Fig. 6
Fig. 6

Interferogram (step 1).

Fig. 7
Fig. 7

Intensity plot of column 220 (step 2).

Fig. 8
Fig. 8

Height map extracted by DDSP (step 3).

Fig. 9
Fig. 9

Visibility functions.

Fig. 10
Fig. 10

Regenerated intensity plot with the FTP visibility function (step 4).

Fig. 11
Fig. 11

Regenerated intensity plot with DDSP visibility function (step 4).

Fig. 12
Fig. 12

Regenerated intensity plot with 1 as visibility (step 4).

Fig. 13
Fig. 13

Surface height extracted with the FTP visibility function.

Fig. 14
Fig. 14

Surface height extracted with the DDSP visibility function.

Fig. 15
Fig. 15

Surface height extracted with 1 as the visibility function.

Fig. 16
Fig. 16

Power spectrum of the reference surface.

Tables (1)

Tables Icon

Table 1 Comparisons of FTP and DDSP Based on rms and RA Obtained with Several Visibility Functions

Equations (21)

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

Ix=k22Io,x+Γx+Γx*,
Ix=2k2Io,x1+γˆx cosγx+ϕo+υx,
Ix=2Io,x+γx exp2πifox+γx*exp-2πifox,
wx=1-cos2πx/N.
Ifx=2Iofx+Sfx-fo+S*fx+fo,
Ix=ϕ1Ix-1++ϕnIx-n+ax,
1-ϕ1B-ϕ2B2--ϕnBn=1-λ1B1-λ2B1-λnB.
Gj=g1λ1j+g2λ2j++gn-1λn-1j+gnλnj,
gi=λin-11jinλi-λj,
Γx=j=0x giλijax-j,
Γx=λiΓx-1+giax,
Υ0=d1+d2++dn,
di=gig11-λiλ1+gig21-λiλ2++gign1-λiλnσa2,
argΓx=atanImΓxReΓx.
ωx=xω0+c+ϕx.
cˆ=ωˆ0x¯-ω¯,
ωˆ0=12 x=1Nωx-ω¯x-x¯NN-1N+1,
υˆx=xωˆ0,
ϕˆx=ωx-xωˆ0-cˆ.
h=λϕx/4π,
γˆx=|Γx|max|Γx|.

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