Abstract

This paper presents two novel approaches to the design of two-dimensional (2-D) optical zero reference signals (ZRSs) that are robust against diffraction effects based on the parametric minimum cross-entropy (PMCE) method. Grating alignment systems require a 2-D optical ZRS to perform absolute measurements. A common method of acquiring 2-D optical ZRSs involves illuminating two identical superimposed 2-D zero reference codes (ZRCs). The output signal is the 2-D optical ZRS and can be represented as the autocorrelation of the 2-D ZRC transmittance. In ultrahigh-resolution systems, diffraction distorts the shadow of the first 2-D ZRC, degrading the autocorrelation and greatly reducing the amplitude of the 2-D optical ZRS. To improve the robustness of 2-D optical ZRSs against diffraction effects, this paper formulates two combinatorial optimization problems for the design of 2-D ZRCs with minimum diffraction effects: one of which is a maximization problem, and the other a minimization problem. Aiming at solving the two problems, this study proposes two schemes based on the PMCE method to search for an optimal 2-D ZRC. Simulation results reveal that there are 3.36–8.34% increases in the slope of the central peak of a 2-D optical ZRS and that there are 16.12–20.90% increases in the sum of the slope of the central peak and the effective signal amplitude of a 2-D optical ZRS, as compared with those obtained by the recently proposed cross-entropy method. The proposed PMCE-based schemes prove to search for 2-D ZRCs more effectively than existing solutions, while requiring less computational complexity.

© 2012 IEEE

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  1. Y. Chen, W. Huang, X. Dang, "Design and analysis of two dimensional zero-reference marks for alignment systems," Rev. Sci. Instrum. 74, 3549-3553 (2003).
  2. J. Sáez-Landete, J. Alonso, E. Bernabeu, "Design of two-dimensional zero reference codes by means of a global optimization method," Opt. Exp. 13, 4230-4236 (2005).
  3. J. Sáez-Landete, S. Salcedo-Sanz, M. Rosa-Zurera, J. Eusebio, A. Bernabeu, "Design of two-dimensional zero reference codes with a genetic algorithm," Opt. Lett. 31, 1648-1650 (2006).
  4. J. Chen, "Design of two-dimensional zero reference codes with cross-entropy method," Appl. Opt. 49, 3560-3565 (2010).
  5. Y. Su, T. Wu, C. Wang, M. Chang, "Generation of two-dimensional optical reference signals based on parametric minimum cross entropy," IEEE Photon. Technol. Lett. 23, 813-815 (2011).
  6. C. Zhou, Y. Wang, Y. Chen, W. Huang, "Alignment measurement of two-dimensional zero-reference marks," Precis. Eng. 30, 238-241 (2006).
  7. J. Sáez-Landete, S. Salcedo-Sanz, F. Cruz-Roldán, P. Amo-López, M. Blanco-Velasco, "Design of two-dimensional optical alignment signals robust to diffractive effects," J. Lightw. Technol. 26, 1702-1707 (2008).
  8. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  9. J. Chen, S. Wang, M. Lee, C. Li, "Cross-entropy method for the optimization of optical alignment signals with diffractive effects," J. Lightw. Technol. 29, 2706-2714 (2011).
  10. R. Y. Rubinstein, A. Dolgin, "Fast parametric entropy-based and the parametric MinxEnt method for counting of #P-complete problems," , private communication (2007).

2011 (2)

Y. Su, T. Wu, C. Wang, M. Chang, "Generation of two-dimensional optical reference signals based on parametric minimum cross entropy," IEEE Photon. Technol. Lett. 23, 813-815 (2011).

J. Chen, S. Wang, M. Lee, C. Li, "Cross-entropy method for the optimization of optical alignment signals with diffractive effects," J. Lightw. Technol. 29, 2706-2714 (2011).

2010 (1)

J. Chen, "Design of two-dimensional zero reference codes with cross-entropy method," Appl. Opt. 49, 3560-3565 (2010).

2008 (1)

J. Sáez-Landete, S. Salcedo-Sanz, F. Cruz-Roldán, P. Amo-López, M. Blanco-Velasco, "Design of two-dimensional optical alignment signals robust to diffractive effects," J. Lightw. Technol. 26, 1702-1707 (2008).

2006 (2)

C. Zhou, Y. Wang, Y. Chen, W. Huang, "Alignment measurement of two-dimensional zero-reference marks," Precis. Eng. 30, 238-241 (2006).

J. Sáez-Landete, S. Salcedo-Sanz, M. Rosa-Zurera, J. Eusebio, A. Bernabeu, "Design of two-dimensional zero reference codes with a genetic algorithm," Opt. Lett. 31, 1648-1650 (2006).

2005 (1)

J. Sáez-Landete, J. Alonso, E. Bernabeu, "Design of two-dimensional zero reference codes by means of a global optimization method," Opt. Exp. 13, 4230-4236 (2005).

2003 (1)

Y. Chen, W. Huang, X. Dang, "Design and analysis of two dimensional zero-reference marks for alignment systems," Rev. Sci. Instrum. 74, 3549-3553 (2003).

Appl. Opt. (1)

J. Chen, "Design of two-dimensional zero reference codes with cross-entropy method," Appl. Opt. 49, 3560-3565 (2010).

IEEE Photon. Technol. Lett. (1)

Y. Su, T. Wu, C. Wang, M. Chang, "Generation of two-dimensional optical reference signals based on parametric minimum cross entropy," IEEE Photon. Technol. Lett. 23, 813-815 (2011).

J. Lightw. Technol. (1)

J. Sáez-Landete, S. Salcedo-Sanz, F. Cruz-Roldán, P. Amo-López, M. Blanco-Velasco, "Design of two-dimensional optical alignment signals robust to diffractive effects," J. Lightw. Technol. 26, 1702-1707 (2008).

J. Lightw. Technol. (1)

J. Chen, S. Wang, M. Lee, C. Li, "Cross-entropy method for the optimization of optical alignment signals with diffractive effects," J. Lightw. Technol. 29, 2706-2714 (2011).

Opt. Lett. (1)

J. Sáez-Landete, S. Salcedo-Sanz, M. Rosa-Zurera, J. Eusebio, A. Bernabeu, "Design of two-dimensional zero reference codes with a genetic algorithm," Opt. Lett. 31, 1648-1650 (2006).

Opt. Exp. (1)

J. Sáez-Landete, J. Alonso, E. Bernabeu, "Design of two-dimensional zero reference codes by means of a global optimization method," Opt. Exp. 13, 4230-4236 (2005).

Precis. Eng. (1)

C. Zhou, Y. Wang, Y. Chen, W. Huang, "Alignment measurement of two-dimensional zero-reference marks," Precis. Eng. 30, 238-241 (2006).

Rev. Sci. Instrum. (1)

Y. Chen, W. Huang, X. Dang, "Design and analysis of two dimensional zero-reference marks for alignment systems," Rev. Sci. Instrum. 74, 3549-3553 (2003).

Other (2)

R. Y. Rubinstein, A. Dolgin, "Fast parametric entropy-based and the parametric MinxEnt method for counting of #P-complete problems," , private communication (2007).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

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