Abstract

The local model fitting (LMF) method is a single-shot interferometric surface profiling algorithm that possesses nondestructive, fast, accurate, and robust measurement capabilities. To extend the measurement range of LMF, extensions based on multiwavelength light sources such as the multiwavelength-matched LMF (MM-LMF) method and the multiwavelength-integrated LMF (MI-LMF) method were proposed recently. MM-LMF is computationally efficient but it tends to suffer from phase unwrapping errors, whereas MI-LMF tends to be accurate but it is computationally expensive. In this paper, we improve the computational efficiency of MI-LMF by combining it with MM-LMF via local information sharing. Through actual experiments, we demonstrate that the proposed method is approximately 10 times faster than the original MI-LMF method, with measurement accuracy kept comparable.

© 2013 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,” (2008), 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]
  4. J. Kato, I. Yamaguchi, T. Nakamura, and S. Kuwashima, “Video-rate fringe analyzer based on phase-shifting electronic moiré patterns,” Appl. Opt. 36, 8403–8412 (1997).
    [CrossRef]
  5. M. Sugiyama, H. Ogawa, K. Kitagawa, and K. Suzuki, “Single-shot surface profiling by local model fitting,” Appl. Opt. 45, 7999–8005 (2006).
    [CrossRef]
  6. Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43, 2695–2702 (2004).
    [CrossRef]
  7. 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]
  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. K. Kitagawa, “Fast surface profiling by multi-wavelength single-shot interferometry,” Int. J. Optomechatron. 4, 136–156 (2010).
    [CrossRef]
  10. Toray Engineering Co., Ltd., “MW-500,” (2011), http://www.scn.tv/user/torayins/ .
  11. A. Yamashita, M. Sugiyama, K. Kitagawa, and H. Kobayashi, “Multiwavelength-integrated local model fitting method for interferometric surface profiling,” Appl. Opt. 51, 6700–6707 (2012).
    [CrossRef]
  12. 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]

2012 (1)

2010 (1)

K. Kitagawa, “Fast surface profiling by multi-wavelength single-shot interferometry,” Int. J. Optomechatron. 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]

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.

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.

Kobayashi, H.

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]

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.

Yamashita, A.

Appl. Opt. (5)

Int. J. Optomechatron. (1)

K. Kitagawa, “Fast surface profiling by multi-wavelength single-shot interferometry,” Int. J. Optomechatron. 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 (2)

Toray Engineering Co., Ltd., “MW-500,” (2011), http://www.scn.tv/user/torayins/ .

S. W. Kim, “High-speed 3D inspection for densely packed semiconductor chips,” (2008), http://spie.org/x24235.xml?ArticleID=x24235 .

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Figures (10)

Fig. 1.
Fig. 1.

Measurement apparatus for the plain LMF method. In the MM-LMF, MI-LMF, and proposed methods, a multiwavelength light source consisting of red, green, and blue is used.

Fig. 2.
Fig. 2.

Artificial object.

Fig. 3.
Fig. 3.

RGB-mixed fringe image for the artificial object.

Fig. 4.
Fig. 4.

Measurement results for the artificial object. Left: Three-dimensional reconstructions. Right: Cross sections at y=45. Bottom: The number of pixels that contain phase unwrapping errors (#UE), the root mean squared error (RMSE) between true and estimated surface profiles at pixels without phase unwrapping errors, the number of initial points for gradient search (#IP), the number of gradient steps (#GS), and computation time for the entire measurement process (CT). Numbers in brackets are reduction rates from the MI-LMF method to the proposed method.

Fig. 5.
Fig. 5.

Four choices of vicinity areas for local information sharing. The black and gray squares denote the target pixel and vicinity pixels, respectively.

Fig. 6.
Fig. 6.

Measurement results for the artificial object with inclinations. Left: Three-dimensional reconstructions. Right: Cross sections at y=45. Bottom: The number of pixels that contain phase unwrapping errors (#UE), the root mean squared error (RMSE) between true and estimated surface profiles at pixels without phase unwrapping errors, the number of initial points for gradient search (#IP), the number of gradient steps (#GS), and computation time for the entire measurement process (CT). Numbers in brackets are reduction rates from the MI-LMF method to the proposed method.

Fig. 7.
Fig. 7.

Measurement results for the artificial object with rough surfaces. Left: Three-dimensional reconstructions. Right: Cross sections at y=45. Bottom: The number of pixels that contain phase unwrapping errors (#UE), the root mean squared error (RMSE) between true and estimated surface profiles at pixels without phase unwrapping errors, the number of initial points for gradient search (#IP), the number of gradient steps (#GS), and computation time for the entire measurement process (CT). Numbers in brackets are reduction rates from the MI-LMF method to the proposed method.

Fig. 8.
Fig. 8.

MW-500 developed by Toray Engineering Co., Ltd. [10].

Fig. 9.
Fig. 9.

RGB-mixed fringe image obtained by MW-500.

Fig. 10.
Fig. 10.

Actual measurement results. Left: Three-dimensional reconstructions. Right: Cross sections at y=45. Bottom: The number of initial points for gradient search (#IP), the number of gradient steps (#GS), and computation time for the entire measurement process (CT). Numbers in brackets are reduction rates from the MI-LMF method to the proposed method.

Tables (1)

Tables Icon

Table 1. Performance of the Proposed Method for Different Choices of the Vicinity Area Illustrated in Fig. 5a

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+2πpx+2πqy),
g¯(x,y)=a+ξφ(x,y)+ζψ(x,y),
ξbcos(4πzλ),
ζbsin(4πzλ),
φ(x,y)cos(2πpx+2πqy),
ψ(x,y)sin(2πpx+2πqy).
(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(ζ^ξ^)+λk2,
b^=ξ2^+ζ2^,
gj(x,y)aj(x,y)+bj(x,y)cos(4πz(x,y)λj+2πpjx+2πqjy),
z^j(kj)λj4πarctan(ζj^ξj^)+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)aj^(x,y)+bj^(x,y)cos(4πzλj+2πpjx+2πqjy).
z^argminzJ(z),
J(z)j=1m1cji=1n(gi,jg¯j(xi,yi))2
cj1ni=1nb^j(xi,yi)2.

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