Abstract

A great advantage of the white-light interferometry is that it can be used for profile measurement of objects with a rough surface. A speckle pattern that arises in the image plane allows one to observe the interference; however, this pattern is also the source of the measurement uncertainty. We derive the theoretical limits of the longitudinal uncertainty by virtue of the first-order statistics of the speckle pattern. It is shown that this uncertainty depends on the surface roughness of the measured object only; it does not depend on the setup parameters.

© 2003 Optical Society of America

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References

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  1. T. Dresel, G. Häusler, H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. 31, 919–925 (1992).
    [CrossRef] [PubMed]
  2. G. Häusler, P. Ettl, M. Schenk, C. Bohn, I. Laszlo, “Limits of optical range sensors and how to exploit them,” in International Trends in Optics and Photonics ICO IV, T. Asakura, ed., Vol. 74 of Springer Series in Optical Sciences, (Springer Verlag, Berlin, 1999), pp. 328–342.
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  5. L. Deck, P. de Groot, “High-speed noncontact profiler based on scanning white-light interferometry,” Appl. Opt. 33, 7334–7338 (1994).
    [CrossRef] [PubMed]
  6. K. G. Larkin, “Efficient nonlinear algorithm for envelope detection in white light interferometry,” J. Opt. Soc. Am. A 13, 832–843 (1996).
    [CrossRef]
  7. M. C. Park, S. W. Kim, “Direct quadratic polynomial fitting for fringe peak detection of white light scanning interferograms,” Opt. Eng. 39, 952–959 (2000).
    [CrossRef]
  8. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1980).
  9. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 9–75.
  10. I. Yamaguchi, A. Yamamoto, S. Kuwamura, “Speckle decorelation in surface profilometry by wavelength scanning interferometry,” Appl. Opt. 37, 6721–6728 (1998).
    [CrossRef]
  11. P. Lehmann, “Surface roughness measurement based on the intensity correlation function of scattered light under speckle pattern illumination,” Appl. Opt. 39, 1144–1152 (1999).
    [CrossRef]
  12. G. Häusler, G. Leuchs, “Physikalische Grenzen der optischen Formerfassung mit Licht,” Phys. Bl. 53, 417–422 (1997).
    [CrossRef]
  13. N. George, A. Jain, “Speckle reduction using multiple tones of illumination,” Appl. Opt. 12, 1202–1212 (1973).
    [CrossRef] [PubMed]
  14. M. Fleischer, R. Windecker, H. J. Tiziani, “Theoretical limits of scanning white-light interferometry signal evaluation algorithms,” Appl. Opt. 40, 2815–2820 (2001).
    [CrossRef]
  15. R. G. Dorsch, G. Häusler, J. M. Herrmann, “Laser triangulation: fundamental uncertainty in distance measurement,” Appl. Opt. 33, 1306–1314 (1994).
    [CrossRef] [PubMed]

2001 (1)

2000 (1)

M. C. Park, S. W. Kim, “Direct quadratic polynomial fitting for fringe peak detection of white light scanning interferograms,” Opt. Eng. 39, 952–959 (2000).
[CrossRef]

1999 (2)

P. Lehmann, “Surface roughness measurement based on the intensity correlation function of scattered light under speckle pattern illumination,” Appl. Opt. 39, 1144–1152 (1999).
[CrossRef]

Y. Teramura, K. Suzuki, F. Kannari, “Low-coherence interferometry with synthesis of coherence function,” Appl. Opt. 38, 5974–5980 (1999).
[CrossRef]

1998 (1)

1997 (1)

G. Häusler, G. Leuchs, “Physikalische Grenzen der optischen Formerfassung mit Licht,” Phys. Bl. 53, 417–422 (1997).
[CrossRef]

1996 (1)

1994 (2)

1993 (1)

1992 (1)

1973 (1)

Bohn, C.

G. Häusler, P. Ettl, M. Schenk, C. Bohn, I. Laszlo, “Limits of optical range sensors and how to exploit them,” in International Trends in Optics and Photonics ICO IV, T. Asakura, ed., Vol. 74 of Springer Series in Optical Sciences, (Springer Verlag, Berlin, 1999), pp. 328–342.
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1980).

Caber, P. J.

de Groot, P.

Deck, L.

Dorsch, R. G.

Dresel, T.

Ettl, P.

G. Häusler, P. Ettl, M. Schenk, C. Bohn, I. Laszlo, “Limits of optical range sensors and how to exploit them,” in International Trends in Optics and Photonics ICO IV, T. Asakura, ed., Vol. 74 of Springer Series in Optical Sciences, (Springer Verlag, Berlin, 1999), pp. 328–342.
[CrossRef]

Fleischer, M.

George, N.

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 9–75.

Häusler, G.

G. Häusler, G. Leuchs, “Physikalische Grenzen der optischen Formerfassung mit Licht,” Phys. Bl. 53, 417–422 (1997).
[CrossRef]

R. G. Dorsch, G. Häusler, J. M. Herrmann, “Laser triangulation: fundamental uncertainty in distance measurement,” Appl. Opt. 33, 1306–1314 (1994).
[CrossRef] [PubMed]

T. Dresel, G. Häusler, H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. 31, 919–925 (1992).
[CrossRef] [PubMed]

G. Häusler, P. Ettl, M. Schenk, C. Bohn, I. Laszlo, “Limits of optical range sensors and how to exploit them,” in International Trends in Optics and Photonics ICO IV, T. Asakura, ed., Vol. 74 of Springer Series in Optical Sciences, (Springer Verlag, Berlin, 1999), pp. 328–342.
[CrossRef]

Herrmann, J. M.

Jain, A.

Kannari, F.

Kim, S. W.

M. C. Park, S. W. Kim, “Direct quadratic polynomial fitting for fringe peak detection of white light scanning interferograms,” Opt. Eng. 39, 952–959 (2000).
[CrossRef]

Kuwamura, S.

Larkin, K. G.

Laszlo, I.

G. Häusler, P. Ettl, M. Schenk, C. Bohn, I. Laszlo, “Limits of optical range sensors and how to exploit them,” in International Trends in Optics and Photonics ICO IV, T. Asakura, ed., Vol. 74 of Springer Series in Optical Sciences, (Springer Verlag, Berlin, 1999), pp. 328–342.
[CrossRef]

Lehmann, P.

P. Lehmann, “Surface roughness measurement based on the intensity correlation function of scattered light under speckle pattern illumination,” Appl. Opt. 39, 1144–1152 (1999).
[CrossRef]

Leuchs, G.

G. Häusler, G. Leuchs, “Physikalische Grenzen der optischen Formerfassung mit Licht,” Phys. Bl. 53, 417–422 (1997).
[CrossRef]

Park, M. C.

M. C. Park, S. W. Kim, “Direct quadratic polynomial fitting for fringe peak detection of white light scanning interferograms,” Opt. Eng. 39, 952–959 (2000).
[CrossRef]

Schenk, M.

G. Häusler, P. Ettl, M. Schenk, C. Bohn, I. Laszlo, “Limits of optical range sensors and how to exploit them,” in International Trends in Optics and Photonics ICO IV, T. Asakura, ed., Vol. 74 of Springer Series in Optical Sciences, (Springer Verlag, Berlin, 1999), pp. 328–342.
[CrossRef]

Suzuki, K.

Teramura, Y.

Tiziani, H. J.

Venzke, H.

Windecker, R.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1980).

Yamaguchi, I.

Yamamoto, A.

Appl. Opt. (9)

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

Opt. Eng. (1)

M. C. Park, S. W. Kim, “Direct quadratic polynomial fitting for fringe peak detection of white light scanning interferograms,” Opt. Eng. 39, 952–959 (2000).
[CrossRef]

Phys. Bl. (1)

G. Häusler, G. Leuchs, “Physikalische Grenzen der optischen Formerfassung mit Licht,” Phys. Bl. 53, 417–422 (1997).
[CrossRef]

Other (3)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1980).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 9–75.

G. Häusler, P. Ettl, M. Schenk, C. Bohn, I. Laszlo, “Limits of optical range sensors and how to exploit them,” in International Trends in Optics and Photonics ICO IV, T. Asakura, ed., Vol. 74 of Springer Series in Optical Sciences, (Springer Verlag, Berlin, 1999), pp. 328–342.
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Schematic diagram of the white-light interferometer.

Fig. 2
Fig. 2

White-light correlogram and correlogram envelope function.

Fig. 3
Fig. 3

Scattering regions of the rough surface.

Equations (34)

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Izo-zr=I01+gzo-zrcos2k0zo-zr+α.
Sk= 12πΔkexp-k-k02Δk2,
Aˆk=j=1N1N aˆj= 1Nj=1N aj expi2kzj+α,
Aˆk=Mkexpi2kzo+iΘk+iα,
zo= 1Nj=1N zj.
Bˆk=B expi2kzr.
Iz=0 Sk×|Bˆk+Aˆk|2dk=0 Sk×B2dk+0 Sk×Mk2dk+2 0 Sk×B×Mk×cos2kzo-zr+Θk+αdk.
Mk=M0+Mk×k-k0+1/2Mkk×k-k02+,
Θk=Θ0+Θk×k-k0+1/2Θkk×k-k02+,
Iintf=2B 0 Sk×M0+Mk×k-k0×cos2kzo-zr+ Θk2+Θ0-Θkk0+αdk.
δz= 12Θk2-Θk21/2.
Cr=ReAˆrandk=k0= 1Nj=1N aj cos2k0hj,
Ci=ImAˆrandk=k0=- 1Nj=1N aj sin2k0hj,
Ckr=ReAˆrandkk=k0=- 2Nj=1N ajhj sin2k0hj,
Cki=ImAˆrandkk=k0=- 2Nj=1N ajhj cos2k0hj.
pCr, Ci, Ckr, Cki= 14π2σ2σ2×exp- Cr2+Ci22σ2×exp- Ckr2+Cki22σ2.
σ2=limN1Nj=1Naj22,
σ2=limN2Nj=1N aj2hj2.
σ2=limN2Nj=1N aj21Nj=1N hj24σ2σz2,
M0=Cr2+Ci21/2
Θ0=arctanCiCr,
Mk= 1M0CrCkr+CiCki,
Θk= 1M02CrCki-CiCkr.
pM0, Mk, Θ0, Θk= M0216π2σ4σz2exp- M022σ2- Mk28σ2σz2- M02Θk28σ2σz2.
pM0, ΘkM0= M0222πσ3σzexp- M022σ2- M02Θk28σ2σz2.
Θk=0,
Θk2= 4σ2M02 σz2.
Iobj=0 Sk×M0+Mkk-k02dkM02.
Iobj=M02=2σ2=limN1Nj=1N aj2,
pIobj, Θk= Iobj2πIobj3/2σz×exp- IobjIobj×exp- IobjΘk24Iobjσz2,
Θk2=2 IobjIobj σz2.
pIobj= 1Iobjexp- IobjIobj.
δz= 12IobjIobj1/2σz.
σz< lc4,

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