Abstract

We propose a fast surface-profiling algorithm based on white-light interferometry by use of sampling theory. We first provide a generalized sampling theorem that reconstructs the squared-envelope function of the white-light interferogram from sampled values of the interferogram and then propose the new algorithm based on the theorem. The algorithm extends the sampling interval to 1.425 µm when an optical filter with a center wavelength of 600 nm and a bandwidth of 60 nm is used. The sampling interval is 6–14 times wider than those used in conventional systems. The algorithm has been installed in a commercial system that achieved the world’s fastest scanning speed of 80 µm/s. The height resolution of the system is of the order of 10 nm for a measurement range of greater than 100 µm.

© 2002 Optical Society of America

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References

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  1. G. S. Kino, S. S. C. Chim, “Mirau correlation microscope,” Appl. Opt. 29, 3775–3783 (1990).
    [Crossref] [PubMed]
  2. S. S. C. Chim, G. S. Kino, “Phase measurements using the Mirau correlation microscope,” Appl. Opt. 30, 2197–2201 (1991).
    [Crossref] [PubMed]
  3. S. S. C. Chim, G. S. Kino, “Three-dimensional image realization in interference microscopy,” Appl. Opt. 31, 2550–2553 (1992).
    [Crossref] [PubMed]
  4. P. J. Caber, “Interferometric profiler for rough surfaces,” Appl. Opt. 32, 3438–3441 (1993).
    [Crossref] [PubMed]
  5. P. J. Caber, S. J. Martinek, R. J. Niemann, “New interferometric profiler for smooth and rough surfaces,” in Laser Dimensional Metrology: Recent Advances for Industrial Application, M. J. Downs, ed., Proc. SPIE2088, 195–203 (1993).
    [Crossref]
  6. Veeco Product Brochure NTS-1-0500, Wyko NT SeriesUltrafast Optical Profilers, Tucson, AZ. (2000).
  7. P. de Groot, L. Deck, “Three-dimensional imaging by sub-Nyquist sampling of white-light interferograms,” Opt. Lett. 18, 1462–1464 (1993).
    [Crossref] [PubMed]
  8. 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]
  9. Data are available at http://www.zygo.com .
  10. R-J. Recknagel, G. Notni, “Analysis of white light interferograms using wavelet methods,” Opt. Commun. 148, 122–128 (1998).
    [Crossref]
  11. A. Hirabayashi, H. Ogawa, K. Kitagawa, “Fast surface profiler by white-light interferometry using a new algorithm, the SEST algorithm,” in Optical Manufacturing and Testing IV, H. P. Stahl, ed., Proc. SPIE4451, 356–367 (2001).
    [Crossref]
  12. Data are available at http://www.scn.tv/corp/torayins/index-eng.html .
  13. A. Kohlenberg, “Exact interpolation of band-limited functions,” J. Appl. Phys. 24, 1432–1436 (1953).
    [Crossref]
  14. A. J. Jerri, “The Shannon sampling theorem—its various extensions and applications: a tutorial review,” Proc. IEEE 65, 1565–1596 (1977).
    [Crossref]
  15. R. J. Marks, Introduction to Shannon Sampling and Interpolation Theory (Springer-Verlag, New York, 1991).
    [Crossref]
  16. R. G. Vaughan, N. L. Scott, D. R. White, “The theory of bandpass sampling,” IEEE Trans. Signal Process. 39, 1973–1984 (1991).
    [Crossref]
  17. A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1987).
  18. I. Someya, Waveform Transmission (Shukyosha, Tokyo, Japan, 1949).
  19. C. E. Shannon, “Communications in the presence of noise,” Proc. IRE 37, 10–21 (1949).
    [Crossref]

1998 (1)

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

1993 (2)

1992 (1)

1991 (2)

S. S. C. Chim, G. S. Kino, “Phase measurements using the Mirau correlation microscope,” Appl. Opt. 30, 2197–2201 (1991).
[Crossref] [PubMed]

R. G. Vaughan, N. L. Scott, D. R. White, “The theory of bandpass sampling,” IEEE Trans. Signal Process. 39, 1973–1984 (1991).
[Crossref]

1990 (1)

1977 (1)

A. J. Jerri, “The Shannon sampling theorem—its various extensions and applications: a tutorial review,” Proc. IEEE 65, 1565–1596 (1977).
[Crossref]

1953 (1)

A. Kohlenberg, “Exact interpolation of band-limited functions,” J. Appl. Phys. 24, 1432–1436 (1953).
[Crossref]

1949 (1)

C. E. Shannon, “Communications in the presence of noise,” Proc. IRE 37, 10–21 (1949).
[Crossref]

Caber, P. J.

P. J. Caber, “Interferometric profiler for rough surfaces,” Appl. Opt. 32, 3438–3441 (1993).
[Crossref] [PubMed]

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

Chim, S. S. C.

de Groot, P.

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]

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]

Hirabayashi, A.

A. Hirabayashi, H. Ogawa, K. Kitagawa, “Fast surface profiler by white-light interferometry using a new algorithm, the SEST algorithm,” in Optical Manufacturing and Testing IV, H. P. Stahl, ed., Proc. SPIE4451, 356–367 (2001).
[Crossref]

Jerri, A. J.

A. J. Jerri, “The Shannon sampling theorem—its various extensions and applications: a tutorial review,” Proc. IEEE 65, 1565–1596 (1977).
[Crossref]

Kino, G. S.

Kitagawa, K.

A. Hirabayashi, H. Ogawa, K. Kitagawa, “Fast surface profiler by white-light interferometry using a new algorithm, the SEST algorithm,” in Optical Manufacturing and Testing IV, H. P. Stahl, ed., Proc. SPIE4451, 356–367 (2001).
[Crossref]

Kohlenberg, A.

A. Kohlenberg, “Exact interpolation of band-limited functions,” J. Appl. Phys. 24, 1432–1436 (1953).
[Crossref]

Marks, R. J.

R. J. Marks, Introduction to Shannon Sampling and Interpolation Theory (Springer-Verlag, New York, 1991).
[Crossref]

Martinek, S. J.

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

Niemann, R. J.

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

Notni, G.

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

Ogawa, H.

A. Hirabayashi, H. Ogawa, K. Kitagawa, “Fast surface profiler by white-light interferometry using a new algorithm, the SEST algorithm,” in Optical Manufacturing and Testing IV, H. P. Stahl, ed., Proc. SPIE4451, 356–367 (2001).
[Crossref]

Papoulis, A.

A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1987).

Recknagel, R-J.

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

Scott, N. L.

R. G. Vaughan, N. L. Scott, D. R. White, “The theory of bandpass sampling,” IEEE Trans. Signal Process. 39, 1973–1984 (1991).
[Crossref]

Shannon, C. E.

C. E. Shannon, “Communications in the presence of noise,” Proc. IRE 37, 10–21 (1949).
[Crossref]

Someya, I.

I. Someya, Waveform Transmission (Shukyosha, Tokyo, Japan, 1949).

Vaughan, R. G.

R. G. Vaughan, N. L. Scott, D. R. White, “The theory of bandpass sampling,” IEEE Trans. Signal Process. 39, 1973–1984 (1991).
[Crossref]

White, D. R.

R. G. Vaughan, N. L. Scott, D. R. White, “The theory of bandpass sampling,” IEEE Trans. Signal Process. 39, 1973–1984 (1991).
[Crossref]

Appl. Opt. (4)

IEEE Trans. Signal Process. (1)

R. G. Vaughan, N. L. Scott, D. R. White, “The theory of bandpass sampling,” IEEE Trans. Signal Process. 39, 1973–1984 (1991).
[Crossref]

J. Appl. Phys. (1)

A. Kohlenberg, “Exact interpolation of band-limited functions,” J. Appl. Phys. 24, 1432–1436 (1953).
[Crossref]

J. Mod. Opt. (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]

Opt. Commun. (1)

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

Opt. Lett. (1)

Proc. IEEE (1)

A. J. Jerri, “The Shannon sampling theorem—its various extensions and applications: a tutorial review,” Proc. IEEE 65, 1565–1596 (1977).
[Crossref]

Proc. IRE (1)

C. E. Shannon, “Communications in the presence of noise,” Proc. IRE 37, 10–21 (1949).
[Crossref]

Other (8)

R. J. Marks, Introduction to Shannon Sampling and Interpolation Theory (Springer-Verlag, New York, 1991).
[Crossref]

A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1987).

I. Someya, Waveform Transmission (Shukyosha, Tokyo, Japan, 1949).

A. Hirabayashi, H. Ogawa, K. Kitagawa, “Fast surface profiler by white-light interferometry using a new algorithm, the SEST algorithm,” in Optical Manufacturing and Testing IV, H. P. Stahl, ed., Proc. SPIE4451, 356–367 (2001).
[Crossref]

Data are available at http://www.scn.tv/corp/torayins/index-eng.html .

Data are available at http://www.zygo.com .

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

Veeco Product Brochure NTS-1-0500, Wyko NT SeriesUltrafast Optical Profilers, Tucson, AZ. (2000).

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Figures (8)

Fig. 1
Fig. 1

Basic setup of an optical system used for surface profiling by white-light interferometry.

Fig. 2
Fig. 2

Example of a white-light interferogram g(z) and its sampled values.

Fig. 3
Fig. 3

Interferogram f(z) shown in Fig. 2 and its envelope function m(z).

Fig. 4
Fig. 4

Examples of the interval [-ω c - ω b , -ω c + ω b ] ∪ [ω c - ω b , ω c + ω b ]. If I = 0, 1, and 2, then the interval becomes [-2ω b , 2ω b ], [-4ω b , -2ω b ] ∪ [2ω b , 4ω b ], and [-6ω b , -4ω b ] ∪ [4ω b , 6ω b ], respectively.

Fig. 5
Fig. 5

Sampling intervals Δ belonging to these ten blocks can be used for reconstruction of r(z) from sampled values of f(z) when the optical filter of λ c = 600 nm and λ b = 30 nm is used. In this case, the Nyquist interval of f(z) is 0.1425 µm.

Fig. 6
Fig. 6

Photograph of a surface profiler in which the SEST algorithm has been installed.

Fig. 7
Fig. 7

Three-dimensional image of bumps on an integrated circuit obtained by the surface profiler in Fig. 6.

Fig. 8
Fig. 8

Three-dimensional image of a step-height standard of 9.947 µm obtained by the surface profiler in Fig. 6.

Tables (1)

Tables Icon

Table 1 Specifications of the System Shown in Fig. 6

Equations (48)

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bλ=0 0<λ<λc-λb, λ>λc+λb.
k=2πλ.
ak=b2πk.
kl=2πλc+λb, ku=2πλc-λb,
ak=0  0<k<kl, k>ku.
ψk=2ak2qokqrkk>00k0.
ψk=0 k<kl, k>ku.
gx, y, z=f x, y, z+C,
fx, y, z=klku ψkcos 2k z-zpx, ydk,
C=klkuak2qok2+qrk2dk,
gz=f z+C,
fz=klku ψkcos 2k z-zpdk.
gz<g zp.
fsz=klku ψksin 2kz-zpdk.
mz=fz2+fsz21/2.
rz=fz2+fsz2.
rz<r zp.
fˆω=- fzexp-iωzdz.
fˆω=π2exp-iωzpψω2.
ωl=2kl=4πλc+λb, ωu=2ku=4πλc-λb.
rˆω=π2exp-iωzpωlωu ψω2ψ-ω-ω2dω.
ωu-ωl=2.10 1/μm, ωl=19.95 1/μm.
ωc=2I+1ωb
ωc-ωbωl, ωuωc+ωb,
0Iωlωu-ωl
ωu2ωb I=0, ωu2I+1ωbωl2I I0.
Δ=π2ωb
zn=nΔ.
sincz=sin πzπzz01z=0
ϕnz=sincωbz-znπcos ωcz-zn.
fz=n=- fznϕnz.
φnz=sincωbz-znπsin ωcz-zn.
fsz=1π-fzz-zdz,
φnz=1π-ϕnzz-zdz.
fsz=n=- fznφnz.
rzj=fzj2+4π2n=-fzj+2n+12n+12.
rz=2Δ2π21-cosπzΔn=-fz2nz-z2n2+1+cosπzΔn=-fz2n+1z-z2n+12.
fz=fzjz=zjcos ωczsin ωbzωbn=-fz2nz-z2n--1I sin ωczcos ωbzωbn=-fz2n+1z-z2n+1otherwise.
fsz=2-1Iπn=-fzj+2n+12n+1z=zjsin ωczsin ωbzωbn=-fz2nz-z2n+-1I cos ωczcos ωbzωbn=-fz2n+1z-z2n+1otherwise.
0Iλc-λb2λb,
0<Δλc-λb4  I=0, λc+λb4IΔλc-λb4I+1  I0
Δmax=λc-λb4λc+λb2λb,
fn=gzn-Cˆ,
Cˆ=1Nn=0N-1gzn.
rNzj=fj2+4π2n=- j+1/2N-j-2/2fj+2n+12n+12.
rNz=2Δ2π21-cosπzΔn=0N-1/2f2nz-z2n2+1+cosπzΔn=0N/2-1f2n+1z-z2n+12.
Iωl2ωb.
ωu2I+1ωl2I.

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