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

Full Article  |  PDF Article

References

  • View by:
  • |

  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, E. Bernabeu, �??Design of zero reference codes by means of a global optimization method�?? Op. Ex. 13, 195-201 (2005), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-195.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-195.</a>
    [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 and Holmstrom, 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).

Advanced Modeling and Optimization (1)

Bjorkman, Mattias and Holmstrom, Kenneth. �??Global Optimization Using the DIRECT Algorithm in Matlab�?? Advanced Modeling and Optimization, 1, 17-37 (1999).

Appl. Opt. (1)

Encyclopedia of Optimization (1)

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

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, E. Bernabeu, �??Design of zero reference codes by means of a global optimization method�?? Op. Ex. 13, 195-201 (2005), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-195.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-195.</a>
[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).

Optimization Online (1)

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

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 (1)

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


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)

Equations on this page are rendered with MathJax. Learn more.

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 ,

Metrics