Abstract

A method to obtain the absolute measure of the position is by means of the autocorrelation of two zero reference marks. In one-axis measurement systems one dimensional mark are used and the design of these marks is relatively complex. When the movement is in two-axes, two dimensional reference marks are required and they are even harder to design. We report a method of global optimization to calculate the optimal two dimensional zero reference marks which generate the reference signal with the highest central peak. This method proves to be a powerful tool for solving this problem.

© 2005 Optical Society of America

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References

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  1. M. C. King and D. H. Berry, “Photolithografic mask alignment using moiré techniques” Appl. Opt. 11, 2455–2459 (1972).
    [Crossref] [PubMed]
  2. V. T. Chitnis and Y. Uchida, “Moiré signals in reflection” Optics Communications 54, 207–211 (1985).
    [Crossref]
  3. Xiangyang Yang and Chunyong Yin, “A new method for the design of zero reference marks for grating measurement systems” J. Phys. E Sci. Instrum. 19, 34–7 (1986).
    [Crossref]
  4. Li Yajun, “Autocorrelation function of a bar code system” J. Mod. Opt. 34, 1571–5 (1987).
    [Crossref]
  5. Li Yajun, “Optical valve using bar codes” Optik 79, 67–74 (1988).
  6. J. Sáez-Landete, J. Alonso, and E. Bernabeu, “Design of zero reference codes by means of a global optimization method” Op. Ex. 13, 195–201 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-195.
    [Crossref]
  7. Y. Chen, W. Huang, and X. Dang, “Design and analysis of two-dimensional zero-reference marks for alignment systems” Review of Scientific Instruments,  74, 3549–53 (2003).
    [Crossref]
  8. D. R. Jones, C. D. Perttunen, and B. E. Stuckman. “Lipschitzian Optimization without the Lipschitz Constant” J. Optim. Theory Appl. 79, 157–181 (1993).
    [Crossref]
  9. Donald R. Jones. DIRECT Global optimization algorithm. Encyclopedia of Optimization. (Kluwer Academic Publishers, Dordrecht, 2001).
  10. Bjorkman, Mattias, Holmstrom, and Kenneth. “Global Optimization Using the DIRECT Algorithm in Matlab” Advanced Modeling and Optimization,  1, 17–37 (1999).
  11. Daniel E. Finkel and C. T. Kelley. “Convergence analysis of the DIRECT algorithm” Optimization Online (2004).
  12. J. M. Gablonsky. DIRECT Version 2.0 User Guide. (CRSC Technical Report, Raleigh, 2001).

2005 (1)

J. Sáez-Landete, J. Alonso, and E. Bernabeu, “Design of zero reference codes by means of a global optimization method” Op. Ex. 13, 195–201 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-195.
[Crossref]

2003 (1)

Y. Chen, W. Huang, and X. Dang, “Design and analysis of two-dimensional zero-reference marks for alignment systems” Review of Scientific Instruments,  74, 3549–53 (2003).
[Crossref]

1999 (1)

Bjorkman, Mattias, Holmstrom, and Kenneth. “Global Optimization Using the DIRECT Algorithm in Matlab” Advanced Modeling and Optimization,  1, 17–37 (1999).

1993 (1)

D. R. Jones, C. D. Perttunen, and B. E. Stuckman. “Lipschitzian Optimization without the Lipschitz Constant” J. Optim. Theory Appl. 79, 157–181 (1993).
[Crossref]

1988 (1)

Li Yajun, “Optical valve using bar codes” Optik 79, 67–74 (1988).

1987 (1)

Li Yajun, “Autocorrelation function of a bar code system” J. Mod. Opt. 34, 1571–5 (1987).
[Crossref]

1986 (1)

Xiangyang Yang and Chunyong Yin, “A new method for the design of zero reference marks for grating measurement systems” J. Phys. E Sci. Instrum. 19, 34–7 (1986).
[Crossref]

1985 (1)

V. T. Chitnis and Y. Uchida, “Moiré signals in reflection” Optics Communications 54, 207–211 (1985).
[Crossref]

1972 (1)

Alonso, J.

J. Sáez-Landete, J. Alonso, and E. Bernabeu, “Design of zero reference codes by means of a global optimization method” Op. Ex. 13, 195–201 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-195.
[Crossref]

Bernabeu, E.

J. Sáez-Landete, J. Alonso, and E. Bernabeu, “Design of zero reference codes by means of a global optimization method” Op. Ex. 13, 195–201 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-195.
[Crossref]

Berry, D. H.

Bjorkman,

Bjorkman, Mattias, Holmstrom, and Kenneth. “Global Optimization Using the DIRECT Algorithm in Matlab” Advanced Modeling and Optimization,  1, 17–37 (1999).

Chen, Y.

Y. Chen, W. Huang, and X. Dang, “Design and analysis of two-dimensional zero-reference marks for alignment systems” Review of Scientific Instruments,  74, 3549–53 (2003).
[Crossref]

Chitnis, V. T.

V. T. Chitnis and Y. Uchida, “Moiré signals in reflection” Optics Communications 54, 207–211 (1985).
[Crossref]

Dang, X.

Y. Chen, W. Huang, and X. Dang, “Design and analysis of two-dimensional zero-reference marks for alignment systems” Review of Scientific Instruments,  74, 3549–53 (2003).
[Crossref]

Finkel, Daniel E.

Daniel E. Finkel and C. T. Kelley. “Convergence analysis of the DIRECT algorithm” Optimization Online (2004).

Gablonsky, J. M.

J. M. Gablonsky. DIRECT Version 2.0 User Guide. (CRSC Technical Report, Raleigh, 2001).

Holmstrom,

Bjorkman, Mattias, Holmstrom, and Kenneth. “Global Optimization Using the DIRECT Algorithm in Matlab” Advanced Modeling and Optimization,  1, 17–37 (1999).

Huang, W.

Y. Chen, W. Huang, and X. Dang, “Design and analysis of two-dimensional zero-reference marks for alignment systems” Review of Scientific Instruments,  74, 3549–53 (2003).
[Crossref]

Jones, D. R.

D. R. Jones, C. D. Perttunen, and B. E. Stuckman. “Lipschitzian Optimization without the Lipschitz Constant” J. Optim. Theory Appl. 79, 157–181 (1993).
[Crossref]

Jones, Donald R.

Donald R. Jones. DIRECT Global optimization algorithm. Encyclopedia of Optimization. (Kluwer Academic Publishers, Dordrecht, 2001).

Kelley, C. T.

Daniel E. Finkel and C. T. Kelley. “Convergence analysis of the DIRECT algorithm” Optimization Online (2004).

Kenneth,

Bjorkman, Mattias, Holmstrom, and Kenneth. “Global Optimization Using the DIRECT Algorithm in Matlab” Advanced Modeling and Optimization,  1, 17–37 (1999).

King, M. C.

Mattias,

Bjorkman, Mattias, Holmstrom, and Kenneth. “Global Optimization Using the DIRECT Algorithm in Matlab” Advanced Modeling and Optimization,  1, 17–37 (1999).

Perttunen, C. D.

D. R. Jones, C. D. Perttunen, and B. E. Stuckman. “Lipschitzian Optimization without the Lipschitz Constant” J. Optim. Theory Appl. 79, 157–181 (1993).
[Crossref]

Sáez-Landete, J.

J. Sáez-Landete, J. Alonso, and E. Bernabeu, “Design of zero reference codes by means of a global optimization method” Op. Ex. 13, 195–201 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-195.
[Crossref]

Stuckman, B. E.

D. R. Jones, C. D. Perttunen, and B. E. Stuckman. “Lipschitzian Optimization without the Lipschitz Constant” J. Optim. Theory Appl. 79, 157–181 (1993).
[Crossref]

Uchida, Y.

V. T. Chitnis and Y. Uchida, “Moiré signals in reflection” Optics Communications 54, 207–211 (1985).
[Crossref]

Yajun, Li

Li Yajun, “Optical valve using bar codes” Optik 79, 67–74 (1988).

Li Yajun, “Autocorrelation function of a bar code system” J. Mod. Opt. 34, 1571–5 (1987).
[Crossref]

Yang, Xiangyang

Xiangyang Yang and Chunyong Yin, “A new method for the design of zero reference marks for grating measurement systems” J. Phys. E Sci. Instrum. 19, 34–7 (1986).
[Crossref]

Yin, Chunyong

Xiangyang Yang and Chunyong Yin, “A new method for the design of zero reference marks for grating measurement systems” J. Phys. E Sci. Instrum. 19, 34–7 (1986).
[Crossref]

Advanced Modeling and Optimization (1)

Bjorkman, Mattias, Holmstrom, and Kenneth. “Global Optimization Using the DIRECT Algorithm in Matlab” Advanced Modeling and Optimization,  1, 17–37 (1999).

Appl. Opt. (1)

J. Mod. Opt. (1)

Li Yajun, “Autocorrelation function of a bar code system” J. Mod. Opt. 34, 1571–5 (1987).
[Crossref]

J. Optim. Theory Appl. (1)

D. R. Jones, C. D. Perttunen, and B. E. Stuckman. “Lipschitzian Optimization without the Lipschitz Constant” J. Optim. Theory Appl. 79, 157–181 (1993).
[Crossref]

J. Phys. E Sci. Instrum. (1)

Xiangyang Yang and Chunyong Yin, “A new method for the design of zero reference marks for grating measurement systems” J. Phys. E Sci. Instrum. 19, 34–7 (1986).
[Crossref]

Op. Ex. (1)

J. Sáez-Landete, J. Alonso, and E. Bernabeu, “Design of zero reference codes by means of a global optimization method” Op. Ex. 13, 195–201 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-195.
[Crossref]

Optics Communications (1)

V. T. Chitnis and Y. Uchida, “Moiré signals in reflection” Optics Communications 54, 207–211 (1985).
[Crossref]

Optik (1)

Li Yajun, “Optical valve using bar codes” Optik 79, 67–74 (1988).

Review of Scientific Instruments (1)

Y. Chen, W. Huang, and X. Dang, “Design and analysis of two-dimensional zero-reference marks for alignment systems” Review of Scientific Instruments,  74, 3549–53 (2003).
[Crossref]

Other (3)

Donald R. Jones. DIRECT Global optimization algorithm. Encyclopedia of Optimization. (Kluwer Academic Publishers, Dordrecht, 2001).

Daniel E. Finkel and C. T. Kelley. “Convergence analysis of the DIRECT algorithm” Optimization Online (2004).

J. M. Gablonsky. DIRECT Version 2.0 User Guide. (CRSC Technical Report, Raleigh, 2001).

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Figures (3)

Fig. 1.
Fig. 1.

Two-dimensional alignment system based on two-dimensional ZRCs

Fig. 2.
Fig. 2.

Height of the second maximum of the autocorrelation with n=10. The continuous graph is the reached with DIRECT, the dotted one is a lower bound calculated theoretically in Eq. (19) and the dash-dot one is the bound showed in Eq. (5).

Fig. 3.
Fig. 3.

Optimum reference signal for n=10 and n1=50.

Equations (23)

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c = [ c ij ] = [ c 11 c 1 n c n 1 c nn ] , c ij { 0 , 1 } ,
S kl = i = 1 n j = 1 n c ij c i + k , j + l ,
S 00 = i = 1 n j = 1 n c ij 2 = i = 1 n j = 1 n c ij = n 1 .
σ = max k 2 + l 2 0 [ S kl ]
σ n 1 ( n 1 + 3 ) ( n 1 1 ) 4 ( n 2 1 ) .
k = n + 1 n 1 l = n + 1 n 1 S kl = k = n + 1 n 1 l = n + 1 n 1 i = 1 n j = 1 n c ij c i + k , j + l = i = 1 n j = 1 n c ij k = n + 1 n 1 l = n + 1 n 1 c i + k , j + l = n 1 2
[ X 1 1 X ] l + 1 } k + 1
S kl ( n k ) ( n l ) .
S kl k 2 + l 2 0 σ .
l 1 = n σ n k 1 .
| k | > n σ n .
S kl { σ 0 l n σ n k ( n k ) ( n l ) n σ n k l < n
S kl = ( n k ) ( n l ) 0 l n 1 .
k = n + 1 n 1 l = n + 1 n 1 S kl = S 00 + 2 l = 1 n 1 S 0 l + 2 k = 1 n 1 l = n + 1 n 1 S kl
l = 1 n 1 S 0 l = l = 1 n σ n σ + l = n σ n + 1 n 1 n ( n l ) = 1 2 ( σ 2 n + σ ( 2 n 1 ) ) .
k = 1 n 1 l = n + 1 n 1 S kl = k = 1 n σ n l = n + 1 n 1 S kl + k = n σ n + 1 n 1 l = n + 1 n 1 S kl k = 1 n σ n l = n + 1 ( n σ n k ) ( n k ) ( n + l ) + k = 1 n σ n l = ( n σ n k ) + 1 n σ n k σ +
+ k = 1 n σ n l = n σ n k + 1 n 1 ( n k ) ( n l ) + k = n σ n + 1 n 1 l = n + 1 0 ( n k ) ( n + l ) + k = n σ n + 1 n 1 l = 1 n 1 ( n k ) ( n l ) =
= 1 2 ( 3 σ 2 + σ n ( 4 n 1 ) 2 σ k = 1 n σ n σ n k )
0 n 1 ( n 1 1 ) + σ ( 2 n 2 + n 1 ) σ 2 ( 1 + 1 n ) .
σ 1 σ ,
σ 1 = ( 2 n 2 + n 1 ) + ( 2 n 2 + n 1 ) 2 + 4 ( 1 + 1 n ) n 1 ( n 1 1 ) 2 ( 1 + 1 n ) .
min f ( c ) , c binary f ( c ) = max k 2 + l 2 0 { S kl } , S kl = i = 1 n k j = 1 n 1 c ij c i + k , j + l
i = 1 n j = 1 n c ij = n 1 ,

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