Abstract

Combining phase and coherence information for improved precision in white-light interference microscopy requires a robust strategy for dealing with the inconsistencies between these two types of information. We correct for these inconsistencies on every measurement by direct analysis of the difference map between the coherence and the phase profiles. The algorithm adapts to surface texture and noise level and dynamically compensates for optical aberrations, distortions, diffraction, and dispersion that would otherwise lead to incorrect fringe order. The same analysis also provides the absolute height data that are essential to relational measurements between disconnected surfaces.

© 2002 Optical Society of America

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References

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  1. M. Davidson, K. Kaufman, I. Mazor, “The coherence probe microscope,” Solid State Technol. 30, 57–59 (1987).
  2. G. S. Kino, S. S. C. Chim, “Mirau correlation microscope,” Appl. Opt. 29, 3775–3783 (1990).
    [CrossRef] [PubMed]
  3. T. Dresel, G. Häusler, H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. 31, 919–925 (1992).
    [CrossRef] [PubMed]
  4. J. C. Wyant, “How to extend interferometry for rough-surface tests,” Laser Focus World (Sept.1993), 131–135 (1993).
  5. P. de Groot, L. Deck, “Three-dimensional imaging by sub-Nyquist sampling of white-light interferograms,” Opt. Lett. 18, 1462–1464 (1993).
    [CrossRef] [PubMed]
  6. K. G. Larkin, “Efficient nonlinear algorithm for envelope detection in white-light interferometry,” J. Opt. Soc. Am. A 4, 832–843 (1996).
    [CrossRef]
  7. P. Sandoz, R. Devillers, A. Plata, “Unambiguous profilometry by fringe-order identification in white-light phase-shifting interferometry,” J. Mod. Opt. 44, 519–534 (1997).
    [CrossRef]
  8. P. J. Caber, S. J. Martinek, R. J. Niemann, “A new interferometric profiler for smooth and rough surfaces,” Laser Dimensional Metrology: Recent Advances for Industrial Application, M. J. Downs, ed., Proc. SPIE2088, 195–203 (1993).
    [CrossRef]
  9. A. Harasaki, J. C. Wyant, “Fringe modulation skewing effect in white-light vertical scanning interferometry,” Appl. Opt. 39, 2101–2106 (2000).
    [CrossRef]
  10. A. Pförtner, J. Schwider, “Dispersion error in white-light Linnik interferometers and its implications for evaluation procedures,” Appl. Opt. 40, 6223–6228 (2001).
    [CrossRef]
  11. P. de Groot, L. Deck, “Surface profiling by analysis of white-light interferograms in the spatial frequency domain,” J. Mod. Opt. 42, 389–401 (1995).
    [CrossRef]
  12. P. de Groot, “Method and apparatus for surface topography measurement by spatial-frequency analysis of interferograms,” U.S. patent5,398,113 (14March1995).
  13. B. L. Danielson, C. Y. Boisrobert, “Absolute optical ranging using low coherence interferometry,” Appl. Opt. 30, 2975–2979 (1991).
    [CrossRef] [PubMed]
  14. L. Deck, “Method and apparatus for the rapid acquisition of data in coherence scanning interferometry,” U.S. patent5,402,234 (28March1995).
  15. J. Schwider, L. Zhou, “Dispersive interferometric profilometer,” Opt. Lett. 19, 995–997 (1994).
    [CrossRef] [PubMed]
  16. D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, New York, 1998), p. 122.
  17. P. de Groot, J. Biegen, J. Clark, X. Colonna de Lega, D. Grigg, “Optical interferometry for measuring the geometric dimensions of industrial parts,” Appl. Opt. 41, 3853–3860 (2002).
    [CrossRef] [PubMed]
  18. A. Harasaki, J. Schmit, J. C. Wyant, “Improved vertical-scanning interferometry,” Appl. Opt. 13, 2107–2115 (2000).
    [CrossRef]
  19. L. Deck, S. Chakmakjian, “Method and apparatus for automatically and simultaneously determining best focus and orientation of objects to be measured by broad-band interferometric means,” U.S. patent5,784,164 (20March1997).
  20. Improved frequency-domain analysis and phase-gap analysis are the topics of multiple U.S. and foreign patents pending.
  21. S.-W. Kim, G.-H. Kim, “Thickness-profile measurement of transparent thin-film layers by white-light scanning interferometry,” Appl. Opt. 38, 5968–5973 (1999).
    [CrossRef]

2002 (1)

2001 (1)

2000 (2)

A. Harasaki, J. C. Wyant, “Fringe modulation skewing effect in white-light vertical scanning interferometry,” Appl. Opt. 39, 2101–2106 (2000).
[CrossRef]

A. Harasaki, J. Schmit, J. C. Wyant, “Improved vertical-scanning interferometry,” Appl. Opt. 13, 2107–2115 (2000).
[CrossRef]

1999 (1)

1997 (1)

P. Sandoz, R. Devillers, A. Plata, “Unambiguous profilometry by fringe-order identification in white-light phase-shifting interferometry,” J. Mod. Opt. 44, 519–534 (1997).
[CrossRef]

1996 (1)

K. G. Larkin, “Efficient nonlinear algorithm for envelope detection in white-light interferometry,” J. Opt. Soc. Am. A 4, 832–843 (1996).
[CrossRef]

1995 (1)

P. de Groot, L. Deck, “Surface profiling by analysis of white-light interferograms in the spatial frequency domain,” J. Mod. Opt. 42, 389–401 (1995).
[CrossRef]

1994 (1)

1993 (1)

1992 (1)

1991 (1)

1990 (1)

1987 (1)

M. Davidson, K. Kaufman, I. Mazor, “The coherence probe microscope,” Solid State Technol. 30, 57–59 (1987).

Biegen, J.

Boisrobert, C. Y.

Caber, P. J.

P. J. Caber, S. J. Martinek, R. J. Niemann, “A new interferometric profiler for smooth and rough surfaces,” Laser Dimensional Metrology: Recent Advances for Industrial Application, M. J. Downs, ed., Proc. SPIE2088, 195–203 (1993).
[CrossRef]

Chakmakjian, S.

L. Deck, S. Chakmakjian, “Method and apparatus for automatically and simultaneously determining best focus and orientation of objects to be measured by broad-band interferometric means,” U.S. patent5,784,164 (20March1997).

Chim, S. S. C.

Clark, J.

Colonna de Lega, X.

Danielson, B. L.

Davidson, M.

M. Davidson, K. Kaufman, I. Mazor, “The coherence probe microscope,” Solid State Technol. 30, 57–59 (1987).

de Groot, P.

P. de Groot, J. Biegen, J. Clark, X. Colonna de Lega, D. Grigg, “Optical interferometry for measuring the geometric dimensions of industrial parts,” Appl. Opt. 41, 3853–3860 (2002).
[CrossRef] [PubMed]

P. de Groot, L. Deck, “Surface profiling by analysis of white-light interferograms in the spatial frequency domain,” J. Mod. Opt. 42, 389–401 (1995).
[CrossRef]

P. de Groot, L. Deck, “Three-dimensional imaging by sub-Nyquist sampling of white-light interferograms,” Opt. Lett. 18, 1462–1464 (1993).
[CrossRef] [PubMed]

P. de Groot, “Method and apparatus for surface topography measurement by spatial-frequency analysis of interferograms,” U.S. patent5,398,113 (14March1995).

Deck, L.

P. de Groot, L. Deck, “Surface profiling by analysis of white-light interferograms in the spatial frequency domain,” J. Mod. Opt. 42, 389–401 (1995).
[CrossRef]

P. de Groot, L. Deck, “Three-dimensional imaging by sub-Nyquist sampling of white-light interferograms,” Opt. Lett. 18, 1462–1464 (1993).
[CrossRef] [PubMed]

L. Deck, S. Chakmakjian, “Method and apparatus for automatically and simultaneously determining best focus and orientation of objects to be measured by broad-band interferometric means,” U.S. patent5,784,164 (20March1997).

L. Deck, “Method and apparatus for the rapid acquisition of data in coherence scanning interferometry,” U.S. patent5,402,234 (28March1995).

Devillers, R.

P. Sandoz, R. Devillers, A. Plata, “Unambiguous profilometry by fringe-order identification in white-light phase-shifting interferometry,” J. Mod. Opt. 44, 519–534 (1997).
[CrossRef]

Dresel, T.

Ghiglia, D. C.

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, New York, 1998), p. 122.

Grigg, D.

Harasaki, A.

A. Harasaki, J. Schmit, J. C. Wyant, “Improved vertical-scanning interferometry,” Appl. Opt. 13, 2107–2115 (2000).
[CrossRef]

A. Harasaki, J. C. Wyant, “Fringe modulation skewing effect in white-light vertical scanning interferometry,” Appl. Opt. 39, 2101–2106 (2000).
[CrossRef]

Häusler, G.

Kaufman, K.

M. Davidson, K. Kaufman, I. Mazor, “The coherence probe microscope,” Solid State Technol. 30, 57–59 (1987).

Kim, G.-H.

Kim, S.-W.

Kino, G. S.

Larkin, K. G.

K. G. Larkin, “Efficient nonlinear algorithm for envelope detection in white-light interferometry,” J. Opt. Soc. Am. A 4, 832–843 (1996).
[CrossRef]

Martinek, S. J.

P. J. Caber, S. J. Martinek, R. J. Niemann, “A new interferometric profiler for smooth and rough surfaces,” Laser Dimensional Metrology: Recent Advances for Industrial Application, M. J. Downs, ed., Proc. SPIE2088, 195–203 (1993).
[CrossRef]

Mazor, I.

M. Davidson, K. Kaufman, I. Mazor, “The coherence probe microscope,” Solid State Technol. 30, 57–59 (1987).

Niemann, R. J.

P. J. Caber, S. J. Martinek, R. J. Niemann, “A new interferometric profiler for smooth and rough surfaces,” Laser Dimensional Metrology: Recent Advances for Industrial Application, M. J. Downs, ed., Proc. SPIE2088, 195–203 (1993).
[CrossRef]

Pförtner, A.

Plata, A.

P. Sandoz, R. Devillers, A. Plata, “Unambiguous profilometry by fringe-order identification in white-light phase-shifting interferometry,” J. Mod. Opt. 44, 519–534 (1997).
[CrossRef]

Pritt, M. D.

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, New York, 1998), p. 122.

Sandoz, P.

P. Sandoz, R. Devillers, A. Plata, “Unambiguous profilometry by fringe-order identification in white-light phase-shifting interferometry,” J. Mod. Opt. 44, 519–534 (1997).
[CrossRef]

Schmit, J.

A. Harasaki, J. Schmit, J. C. Wyant, “Improved vertical-scanning interferometry,” Appl. Opt. 13, 2107–2115 (2000).
[CrossRef]

Schwider, J.

Venzke, H.

Wyant, J. C.

A. Harasaki, J. C. Wyant, “Fringe modulation skewing effect in white-light vertical scanning interferometry,” Appl. Opt. 39, 2101–2106 (2000).
[CrossRef]

A. Harasaki, J. Schmit, J. C. Wyant, “Improved vertical-scanning interferometry,” Appl. Opt. 13, 2107–2115 (2000).
[CrossRef]

J. C. Wyant, “How to extend interferometry for rough-surface tests,” Laser Focus World (Sept.1993), 131–135 (1993).

Zhou, L.

Appl. Opt. (8)

J. Mod. Opt. (2)

P. de Groot, L. Deck, “Surface profiling by analysis of white-light interferograms in the spatial frequency domain,” J. Mod. Opt. 42, 389–401 (1995).
[CrossRef]

P. Sandoz, R. Devillers, A. Plata, “Unambiguous profilometry by fringe-order identification in white-light phase-shifting interferometry,” J. Mod. Opt. 44, 519–534 (1997).
[CrossRef]

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

K. G. Larkin, “Efficient nonlinear algorithm for envelope detection in white-light interferometry,” J. Opt. Soc. Am. A 4, 832–843 (1996).
[CrossRef]

Opt. Lett. (2)

Solid State Technol. (1)

M. Davidson, K. Kaufman, I. Mazor, “The coherence probe microscope,” Solid State Technol. 30, 57–59 (1987).

Other (7)

J. C. Wyant, “How to extend interferometry for rough-surface tests,” Laser Focus World (Sept.1993), 131–135 (1993).

P. J. Caber, S. J. Martinek, R. J. Niemann, “A new interferometric profiler for smooth and rough surfaces,” Laser Dimensional Metrology: Recent Advances for Industrial Application, M. J. Downs, ed., Proc. SPIE2088, 195–203 (1993).
[CrossRef]

L. Deck, S. Chakmakjian, “Method and apparatus for automatically and simultaneously determining best focus and orientation of objects to be measured by broad-band interferometric means,” U.S. patent5,784,164 (20March1997).

Improved frequency-domain analysis and phase-gap analysis are the topics of multiple U.S. and foreign patents pending.

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, New York, 1998), p. 122.

P. de Groot, “Method and apparatus for surface topography measurement by spatial-frequency analysis of interferograms,” U.S. patent5,398,113 (14March1995).

L. Deck, “Method and apparatus for the rapid acquisition of data in coherence scanning interferometry,” U.S. patent5,402,234 (28March1995).

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Figures (8)

Fig. 1
Fig. 1

White-light microscope with a Mirau interference objective for profiling surface heights h with respect to datum plane H by scanning the objective in the ζ direction: PZT, piezoelectric transducer.

Fig. 2
Fig. 2

Conceptual illustration of the mathematical decomposition of a single-pixel white-light interference pattern into multiple-constituent single-wave-number patterns for frequency-domain analysis. Wave number k and corresponding phase ϕ characterize each of the patterns.

Fig. 3
Fig. 3

Results of a Fourier transform of a white-light interference pattern. The phase evolution as a function of wave number is summarized by the linear fit, weighted by amplitudes p.

Fig. 4
Fig. 4

Frequency-domain portrait of a white-light interference pattern for a single image pixel, showing how coherence profile Θ and phase profile θ relate to surface height h. Note dispersion and phase offsets τ and γ and phase gap A.

Fig. 5
Fig. 5

Intensity-domain portrait of a white-light interference pattern for a single image pixel, showing the meaning of the symbols Θ, θ, h, τ, and γ and of phase gap A to aid in the interpretation of Fig. 4.

Fig. 6
Fig. 6

Example phase-gap analysis of a tilted silicon carbide flat viewed by a SWLI microscope. (a) Wrapped phase profile θ″(x) and (b) coherence profile Θ(x) in units of phase at k 0; (c) wrapped phase gap A″(x) and (d) filtered, connected phase gap α′(x); (e) fringe-order map M′(x) and (f) final surface height k 0 h′(x), also in units of phase. For the surface height, one phase cycle corresponds to 280 nm.

Fig. 7
Fig. 7

Example phase-gap analysis of a sphere viewed by a SWLI microscope. After processing, wrapped experimental phase gap A″(x) becomes connected approximate phase gap α′(x) (center). The deviation of α′(x) from perfectly flat is possibly attributable to slope-dependent dispersion τ(x) in the optical system. The final surface height k 0 h′(x) has units of phase at k 0 = 2π cycle/280 nm.

Fig. 8
Fig. 8

Comparison of coherence profiling (i.e., fringe localization or envelope peak) with phase profiling by use of coherence only to identify fringe order. The lower, phase-data profile has lower noise and is free of diffraction effects. The sample is patterned silicon.

Tables (1)

Tables Icon

Table 1 Possible Appearance and Interpretation of Phase Gap A″(x) after Phase Connection of Pixels

Equations (30)

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pk, x=FTIζ, x2,
ϕk, x=argFTIζ, x.
σxdϕ/dk|x
Axϕk=0, x.
ϕk, x=khx-ζstartx+k-k0τx+γx.
hx=θx-γx/k0,
θx=θx-2πMx,
θx=k0ζstartx+k0σx+Ax
hx=Θx/k0-τx,
Θx=k0ζstartx+k0σx.
Ax=θx-Θx.
Ax=γx-k0τx.
2πMx=Ax-Ax.
αavg=arctan 2C¯, S¯,
Sx=sinAx,
Cx=cosAx.
Mx=IntAx-αx2π,
hx=1/k0θx-γx-2πMx.
δΘx, y=Θx, y-smoothΘx, y,
Θsmx, y=Θx, y-δΘx, y.
Ax=connectAx.
αFITx, y=c0+c1x+c2y+c3Θx, y.
c0c1c2c3=1xyΘxx2xyxΘyxyy2yΘΘxΘyΘΘ2-1AxAyAΘA,
f=x, y Wxfx.
αx=αFITx+Ax-αFITxWx2.
hx=hx-2πM0k0,
M0=Mx-Mx.
γx=γpartx+γsysx,
τx=τpartx+τsysx.
M0=Intα¯-γ¯-k0τ¯2π.

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