Abstract

Scanning white-light interferometry is widely used for the microstructure analysis of technical and biological specimens. For each pixel in the focal plane of the apparatus a white-light interferogram is acquired and evaluated by means of an algorithm. We discuss some properties of mathematically optimal evaluation methods and the best possible achievable resolution derived therefrom depending on the setup parameters. A comparison of the results to one of the algorithms described in the literature is given.

© 2001 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  4. K. G. Larkin, “Efficient nonlinear algorithm for envelope detection in white light interferometry,” J. Opt. Soc. Am. A 13, 4, 832–843 (1996).
    [CrossRef]
  5. 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]
  6. P. Sandoz, M. Jacquot, “Processing of white light correlograms: simultaneous phase and envelope measurements by wavelet transformation,” in Optical Inspection and Micromeasurements II, C. Gorecki, ed. Proc. SPIE3098, 73–82 (1997).
    [CrossRef]
  7. R. J. Recknagel, G. Notni, “Analysis of white light interferograms using wavelet methods,” Opt. Commun. 148, 122–128 (1998).
    [CrossRef]
  8. D. J. Aziz, “Interferometric measurement of surface roughness in engine cylinder walls,” Opt. Eng. 37, 1429–1434 (1998).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2000 (1)

1998 (2)

R. J. Recknagel, G. Notni, “Analysis of white light interferograms using wavelet methods,” Opt. Commun. 148, 122–128 (1998).
[CrossRef]

D. J. Aziz, “Interferometric measurement of surface roughness in engine cylinder walls,” Opt. Eng. 37, 1429–1434 (1998).
[CrossRef]

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)

1995 (1)

P. Hariharan, M. Roy, “White-light phase-stepping interferometry: measurement of the fractional interference order,” J. Mod. Opt. 42, 2357–2360 (1995).
[CrossRef]

1994 (1)

1993 (1)

1986 (1)

D. Hertz, “Time delay estimation by combining efficient algorithms and generalized cross-correlation,” IEEE Trans. Acoust. Speech Signal Process. 34, 1–7 (1986).
[CrossRef]

Aziz, D. J.

D. J. Aziz, “Interferometric measurement of surface roughness in engine cylinder walls,” Opt. Eng. 37, 1429–1434 (1998).
[CrossRef]

Caber, P. J.

de Groot, P.

Deck, L.

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]

Fleischer, M.

Hariharan, P.

P. Hariharan, M. Roy, “White-light phase-stepping interferometry: measurement of the fractional interference order,” J. Mod. Opt. 42, 2357–2360 (1995).
[CrossRef]

Hertz, D.

D. Hertz, “Time delay estimation by combining efficient algorithms and generalized cross-correlation,” IEEE Trans. Acoust. Speech Signal Process. 34, 1–7 (1986).
[CrossRef]

Jacquot, M.

P. Sandoz, M. Jacquot, “Processing of white light correlograms: simultaneous phase and envelope measurements by wavelet transformation,” in Optical Inspection and Micromeasurements II, C. Gorecki, ed. Proc. SPIE3098, 73–82 (1997).
[CrossRef]

Larkin, K. G.

Notni, G.

R. J. Recknagel, G. Notni, “Analysis of white light interferograms using wavelet methods,” Opt. Commun. 148, 122–128 (1998).
[CrossRef]

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]

Recknagel, R. J.

R. J. Recknagel, G. Notni, “Analysis of white light interferograms using wavelet methods,” Opt. Commun. 148, 122–128 (1998).
[CrossRef]

Roy, M.

P. Hariharan, M. Roy, “White-light phase-stepping interferometry: measurement of the fractional interference order,” J. Mod. Opt. 42, 2357–2360 (1995).
[CrossRef]

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]

P. Sandoz, M. Jacquot, “Processing of white light correlograms: simultaneous phase and envelope measurements by wavelet transformation,” in Optical Inspection and Micromeasurements II, C. Gorecki, ed. Proc. SPIE3098, 73–82 (1997).
[CrossRef]

Tiziani, H. J.

Windecker, R.

Appl. Opt. (3)

IEEE Trans. Acoust. Speech Signal Process. (1)

D. Hertz, “Time delay estimation by combining efficient algorithms and generalized cross-correlation,” IEEE Trans. Acoust. Speech Signal Process. 34, 1–7 (1986).
[CrossRef]

J. Mod. Opt. (2)

P. Hariharan, M. Roy, “White-light phase-stepping interferometry: measurement of the fractional interference order,” J. Mod. Opt. 42, 2357–2360 (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)

Opt. Commun. (1)

R. J. Recknagel, G. Notni, “Analysis of white light interferograms using wavelet methods,” Opt. Commun. 148, 122–128 (1998).
[CrossRef]

Opt. Eng. (1)

D. J. Aziz, “Interferometric measurement of surface roughness in engine cylinder walls,” Opt. Eng. 37, 1429–1434 (1998).
[CrossRef]

Other (1)

P. Sandoz, M. Jacquot, “Processing of white light correlograms: simultaneous phase and envelope measurements by wavelet transformation,” in Optical Inspection and Micromeasurements II, C. Gorecki, ed. Proc. SPIE3098, 73–82 (1997).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Setup of the white-light interferometer.

Fig. 2
Fig. 2

Typical signal for one pixel.

Fig. 3
Fig. 3

Weighing sample points.

Fig. 4
Fig. 4

Noise over coherence length for the phase evaluation.

Fig. 5
Fig. 5

Noise over coherence length for the envelope evaluation.

Equations (35)

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Si=Szi=Izi-zc+Nzi,  i=0n.
zout=AS0,,Sn.
AS0,,SnAS0S0,,SnS0++ASnS0,, SnSn+c=a0S0++anSn+c=c+i aiSi.
dzoutdzc=dAdzc=j ajdSjdzc=-j ajIzj-zc,  Iz=dIdz.
Nout=jASj Nj21/2=j aj2Nj21/2.
ρSN=dAdzcNout=-j ajIzj-zcj aj2Nj21/2.
ρSNai=aiNi2j ajIzj-zc-Izi-zcj aj2Nj2j aj2Nj23/2=0.
ai=j aj2Nj2Ni2j ajIzj-zc Izi-zc=CNi2 Izi-zc.
zout=AI0,, Inc+i Izi-zcIi.
Nout=ΔzcdAdzc,Δzc=NoutdAdzc=j I2zj-zcNj21/2j I2zj-zc.
Δzc=Nj I2zj-zc1/2
ΔzcN1Δz-+ I2z-zcdz1/2=N1Δz-+ I2zdz1/2.
Iz=I01+exp-16 ln 2zlc2cos4πzλ,
Iz=-I032z ln 2lc2cos4πzλ+4πλsin4πzλexp-16 ln 2zlc2.
Iz=-I04πλsin4πzλexp-16 ln 2zlc2.
ΔzcN1Δz-+ Iz2dz1/2=NI04πλ1Δz-+sin24πzλexp-32 ln 2zlc2dz1/2.
ΔzcNI04πλ12Δz-+exp-32 ln 2zlc2dz1/2=NI04πλ12Δzπlc232 ln 21/21/2.
ΔzcNI0λ2πΔzlc1/28 ln 2π1/4.
NIz=NPIz.
Δzc=NoutdAdzc=1Δz-+NPIz2dz1/21Δz-+ I2zdz=NPΔz11Δz-+ I2zdz1/2,
Δzc=NPΔz×1-+sin24πzλexp-32 ln 2zlc2dz1/2.
Δzc=NPΔz112-+exp-32 ln 2zlc2dz1/2=NPΔz112πlc232 ln 21/21/2,
Δzc=NPΔzlc1/2 28 ln 2π1/4.
Iz=Iˆzcos4πzλ+φ+Nz,
Iz=12Iˆzexp i4πzλ+φ+exp-i4πzλ+φ+Nz+Nz*,
I˜z=Iˆzexp i4πzλ+φ+Nz.
Iz=Ĩzexp-4πizλ=Iˆzexp iφ+Nz+iÑzexp-4πizλ.
|Iz|=Iˆz+Nz+iÑzexp -i4πzλ+φ|Iˆz|+N˜˜z,
Iz=I0 exp-16 ln 2zlc2.
Iz=-I032z ln 2lc2exp-16 ln 2zlc2.
ΔzcN1Δz-+ Iz2dz1/2=Nlc232 ln 2I01Δz-+ z2 exp-32 ln 2zlc2dz1/2.
0 xn exp-ax2=Γn+12an+12
ΔzcNlc232 ln 2I02ΔzΓ1.532 ln 2lc23/21/2,
ΔzcNI0Δzlc242π ln 21/4.
lc<172ΔzI0N λ2.

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