Abstract

The local model fitting (LMF) method is a useful single-shot surface profiling algorithm that features fast measurement speed and robustness against vibration. However, the measurement range of the LMF method (i.e., measurable height difference between two neighboring pixels) is limited up to a quarter of the light source wavelength. To cope with this problem, the multiwavelength-matched LMF (MM-LMF) method was proposed, where the plain LMF method is first applied individually to interference images obtained from multiple light sources with different wavelengths, and then the LMF solutions are matched to obtain a range-extended solution. Although the MM-LMF method was shown to provide high measurement accuracy under moderate noise, phase unwrapping errors can occur if individual LMF solutions are erroneous. In this paper, we propose the multiwavelength-integrated LMF (MI-LMF) method, which directly computes a range-extended solution from multiple interference images in an integrated way. The effectiveness of the proposed MI-LMF method is demonstrated through simulations and actual experiments.

© 2012 Optical Society of America

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References

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  1. S. W. Kim, High-speed 3d inspection for densely packed semiconductor chips, http://spie.org/x24235.xml?ArticleID=x24235 .
  2. J. H. Brunning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wave front measuring interferometer for testing optical surface and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef]
  3. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  6. Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43, 2695–2702 (2004).
    [CrossRef]
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    [CrossRef]
  8. N. Brock, J. Hayes, B. Kimbrough, J. Millerd, M. North-Morris, M. Novak, and J. C. Wyant, “Dynamic interferometry,” Proc. SPIE 5875, 58750F (2005).
    [CrossRef]
  9. M. Sugiyama, H. Ogawa, K. Kitagawa, and K. Suzuki, “Single-shot surface profiling by local model fitting,” Appl. Opt. 45, 7999–8005 (2006).
    [CrossRef]
  10. K. Kitagawa, “Fast surface profiling by multi-wavelength single-shot interferometry,” Int. J. Optomechatronics 4, 136–156 (2010).
    [CrossRef]
  11. M. Takeda and T. Abe, “Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: A comparative study,” Opt. Eng. 35, 2345–2351 (1996).
    [CrossRef]
  12. In this paper, we use the following method: we first apply the plain LMF method to each interference image and obtain estimates of the bias aj(x,y) and amplitude bj(x,y). Then we apply a 7×7 pixel median filter to the entire images of aj(x,y) and bj(x,y) and use the obtained values as a^j(x,y) and b^j(x,y).
  13. See http://www.scn.tv/user/torayins/ for details.
  14. In this experiment, we were interested in the sharpness of the steps along the x axis. A practical heuristic for accurate measurement is to introduce a spatial carrier orthogonal to the direction of interest, i.e., along the y axis. Following this heuristic, we decided to use a rectangular-shaped local area along the y axis.

2010 (1)

K. Kitagawa, “Fast surface profiling by multi-wavelength single-shot interferometry,” Int. J. Optomechatronics 4, 136–156 (2010).
[CrossRef]

2006 (1)

2005 (1)

N. Brock, J. Hayes, B. Kimbrough, J. Millerd, M. North-Morris, M. Novak, and J. C. Wyant, “Dynamic interferometry,” Proc. SPIE 5875, 58750F (2005).
[CrossRef]

2004 (1)

1997 (1)

1996 (1)

M. Takeda and T. Abe, “Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: A comparative study,” Opt. Eng. 35, 2345–2351 (1996).
[CrossRef]

1991 (1)

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147–150 (1991).
[CrossRef]

1986 (1)

1982 (1)

1974 (1)

Abe, T.

M. Takeda and T. Abe, “Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: A comparative study,” Opt. Eng. 35, 2345–2351 (1996).
[CrossRef]

Banyard, J. E.

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147–150 (1991).
[CrossRef]

Brangaccio, D. J.

Brock, N.

N. Brock, J. Hayes, B. Kimbrough, J. Millerd, M. North-Morris, M. Novak, and J. C. Wyant, “Dynamic interferometry,” Proc. SPIE 5875, 58750F (2005).
[CrossRef]

Brunning, J. H.

Gallagher, J. E.

Hayes, J.

N. Brock, J. Hayes, B. Kimbrough, J. Millerd, M. North-Morris, M. Novak, and J. C. Wyant, “Dynamic interferometry,” Proc. SPIE 5875, 58750F (2005).
[CrossRef]

Herriott, D. R.

Ina, H.

Iwaasa, Y.

Kato, J.

Kemao, Q.

Kimbrough, B.

N. Brock, J. Hayes, B. Kimbrough, J. Millerd, M. North-Morris, M. Novak, and J. C. Wyant, “Dynamic interferometry,” Proc. SPIE 5875, 58750F (2005).
[CrossRef]

Kitagawa, K.

K. Kitagawa, “Fast surface profiling by multi-wavelength single-shot interferometry,” Int. J. Optomechatronics 4, 136–156 (2010).
[CrossRef]

M. Sugiyama, H. Ogawa, K. Kitagawa, and K. Suzuki, “Single-shot surface profiling by local model fitting,” Appl. Opt. 45, 7999–8005 (2006).
[CrossRef]

Kobayashi, S.

Kuwashima, S.

Millerd, J.

N. Brock, J. Hayes, B. Kimbrough, J. Millerd, M. North-Morris, M. Novak, and J. C. Wyant, “Dynamic interferometry,” Proc. SPIE 5875, 58750F (2005).
[CrossRef]

Nakamura, T.

Nassar, N. S.

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147–150 (1991).
[CrossRef]

North-Morris, M.

N. Brock, J. Hayes, B. Kimbrough, J. Millerd, M. North-Morris, M. Novak, and J. C. Wyant, “Dynamic interferometry,” Proc. SPIE 5875, 58750F (2005).
[CrossRef]

Novak, M.

N. Brock, J. Hayes, B. Kimbrough, J. Millerd, M. North-Morris, M. Novak, and J. C. Wyant, “Dynamic interferometry,” Proc. SPIE 5875, 58750F (2005).
[CrossRef]

Ogawa, H.

Rosenfeld, D. P.

Sugiyama, M.

Suzuki, K.

Takeda, M.

M. Takeda and T. Abe, “Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: A comparative study,” Opt. Eng. 35, 2345–2351 (1996).
[CrossRef]

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

Toyooka, S.

Virdee, M. S.

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147–150 (1991).
[CrossRef]

White, A. D.

Williams, D. C.

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147–150 (1991).
[CrossRef]

Wyant, J. C.

N. Brock, J. Hayes, B. Kimbrough, J. Millerd, M. North-Morris, M. Novak, and J. C. Wyant, “Dynamic interferometry,” Proc. SPIE 5875, 58750F (2005).
[CrossRef]

Yamaguchi, I.

Appl. Opt. (5)

Int. J. Optomechatronics (1)

K. Kitagawa, “Fast surface profiling by multi-wavelength single-shot interferometry,” Int. J. Optomechatronics 4, 136–156 (2010).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

M. Takeda and T. Abe, “Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: A comparative study,” Opt. Eng. 35, 2345–2351 (1996).
[CrossRef]

Opt. Laser Technol. (1)

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147–150 (1991).
[CrossRef]

Proc. SPIE (1)

N. Brock, J. Hayes, B. Kimbrough, J. Millerd, M. North-Morris, M. Novak, and J. C. Wyant, “Dynamic interferometry,” Proc. SPIE 5875, 58750F (2005).
[CrossRef]

Other (4)

S. W. Kim, High-speed 3d inspection for densely packed semiconductor chips, http://spie.org/x24235.xml?ArticleID=x24235 .

In this paper, we use the following method: we first apply the plain LMF method to each interference image and obtain estimates of the bias aj(x,y) and amplitude bj(x,y). Then we apply a 7×7 pixel median filter to the entire images of aj(x,y) and bj(x,y) and use the obtained values as a^j(x,y) and b^j(x,y).

See http://www.scn.tv/user/torayins/ for details.

In this experiment, we were interested in the sharpness of the steps along the x axis. A practical heuristic for accurate measurement is to introduce a spatial carrier orthogonal to the direction of interest, i.e., along the y axis. Following this heuristic, we decided to use a rectangular-shaped local area along the y axis.

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Figures (4)

Fig. 1.
Fig. 1.

Simulations for spiky bumps.

Fig. 2.
Fig. 2.

Profile of the error criterion J(z) defined by Eq. (19) at point (25, 25) for the interference image shown in Fig. 1(b). Note that the horizontal axis of this graph corresponds to the vertical axis (i.e., the height) in Figs. 1 and 3.

Fig. 3.
Fig. 3.

Simulations for super-spiky bumps.

Fig. 4.
Fig. 4.

Actual measurement results.

Equations (20)

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g(x,y)a(x,y)+b(x,y)cos(4πz(x,y)λ+2πpx+2πqy),
g¯(x,y)a+bcos(4πzλ+2πp^x+2πq^y),
g¯(x,y)=a+ξφ(x,y)+ζψ(x,y),
ξbcos(4πzλ),
ζbsin(4πzλ),
φ(x,y)cos(2πp^x+2πq^y),
ψ(x,y)sin(2πp^x+2πq^y).
(a^,ξ^,ζ^)argmin(a,ξ,ζ)i=1n(gig¯(xi,yi))2.
(a^,ξ^,ζ^)=(AA)1Ag,
A(1φ(x1,y1)ψ(x1,y1)1φ(xn,yn)ψ(xn,yn))andg(g1gn).
z^(k)λ4πarctan(ξ^ζ^)+λk2,
b^=ξ^2+ζ^2.
gj(x,y)aj(x,y)+bj(x,y)cos(4πz(x,y)λj+2πpjx+2πqjy),
z^j(kj)λj4πarctan(ξ^jζ^j)+λjkj2,
(k^1,,k^m)argmin(k1,,km)(max(z^1(k1),,z^m(km))min(z^1(k1),,z^m(km))).
z^1mj=1mz^j(k^j).
g¯j(x,y)a^j(x,y)+b^j(x,y)cos(4πzλj+2πp^jx+2πq^jy),
z^argminzJ(z),
J(z)j=1m1cji=1n(gi,jg¯j(xi,yi))2
cj1ni=1nb^j(xi,yi)2

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