OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 3 — Jan. 31, 2011
  • pp: 2165–2180

Pixel-based OPC optimization based on conjugate gradients

Xu Ma and Gonzalo R. Arce  »View Author Affiliations

Optics Express, Vol. 19, Issue 3, pp. 2165-2180 (2011)

View Full Text Article

Enhanced HTML    Acrobat PDF (928 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



Optical proximity correction (OPC) methods are resolution enhancement techniques (RET) used extensively in the semiconductor industry to improve the resolution and pattern fidelity of optical lithography. In pixel-based OPC (PBOPC), the mask is divided into small pixels, each of which is modified during the optimization process. Two critical issues in PBOPC are the required computational complexity of the optimization process, and the manufacturability of the optimized mask. Most current OPC optimization methods apply the steepest descent (SD) algorithm to improve image fidelity augmented by regularization penalties to reduce the complexity of the mask. Although simple to implement, the SD algorithm converges slowly. The existing regularization penalties, however, fall short in meeting the mask rule check (MRC) requirements often used in semiconductor manufacturing. This paper focuses on developing OPC optimization algorithms based on the conjugate gradient (CG) method which exhibits much faster convergence than the SD algorithm. The imaging formation process is represented by the Fourier series expansion model which approximates the partially coherent system as a sum of coherent systems. In order to obtain more desirable manufacturability properties of the mask pattern, a MRC penalty is proposed to enlarge the linear size of the sub-resolution assistant features (SRAFs), as well as the distances between the SRAFs and the main body of the mask. Finally, a projection method is developed to further reduce the complexity of the optimized mask pattern.

© 2011 Optical Society of America

OCIS Codes
(100.3190) Image processing : Inverse problems
(110.4980) Imaging systems : Partial coherence in imaging
(110.5220) Imaging systems : Photolithography

ToC Category:
Image Processing

Original Manuscript: November 10, 2010
Revised Manuscript: December 25, 2010
Manuscript Accepted: December 26, 2010
Published: January 20, 2011

Xu Ma and Gonzalo R. Arce, "Pixel-based OPC optimization based on conjugate gradients," Opt. Express 19, 2165-2180 (2011)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. A. K. Wong, Resolution Enhancement Techniques (SPIE Press, 2001), Vol. 1. [CrossRef]
  2. X. Ma and G. R. Arce, Computational Lithography, Wiley Series in Pure and Applied Optics, 1st ed. (JohnWiley and Sons, 2010). [CrossRef]
  3. S. A. Campbell, The Science and Engineering of Microelectronic Fabrication, 2nd ed. (Publishing House of Electronics Industry, 2003).
  4. F. Schellenberg, “Resolution enhancement technology: the past, the present, and extensions for the future,” Proc. SPIE 5377, 1–20 (2004). [CrossRef]
  5. F. Schellenberg, Resolution Enhancement Techniques in Optical Lithography (SPIE Press, 2004).
  6. L. Liebmann, S. Mansfield, A. Wong, M. Lavin, W. Leipold, and T. Dunham, “TCAD development for lithography resolution enhancement,” IBM J. Res. Develop. 45, 651–665 (2001). [CrossRef]
  7. U. Okoroanyanwu, Chemistry and Lithography, 1st ed. (in review, SPIE Press, 2010).
  8. S. Sherif, B. Saleh, and R. Leone, “Binary image synthesis using mixed integer programming,” IEEE Trans. Image Process. 4(9), 1252–1257 (1995). [CrossRef]
  9. Y. Liu and A. Zakhor, “Binary and phase shifting mask design for optical lithography,” IEEE Trans. Semicond. Manuf. 5(2), 138–152 (1992). [CrossRef]
  10. A. Erdmann, R. Farkas, T. Fühner, B. Tollkühn, and G. Kokai, “Towards automatic mask and source optimization for optical lithography,” Proc. SPIE 5377, 646–657 (2004). [CrossRef]
  11. Y. Granik, “Solving inverse problems of optical microlithography,” Proc. SPIE 5754, 506–526 (2004). [CrossRef]
  12. Y. Granik, “Fast pixel-based mask optimization for inverse lithography,” J. Microlithogr., Microfabr, Microsyst. 5, 043002 (2006). [CrossRef]
  13. A. Poonawala, and P. Milanfar, “Fast and low-complexity mask design in optical microlithography–an inverse imaging problem,” IEEE Trans. Image Process. 16(3), 774–788 (2007). [CrossRef] [PubMed]
  14. X. Ma and G. R. Arce, “Binary mask optimization for inverse lithography with partially coherent illumination,” J. Opt. Soc. Am. A 25(12), 2960–2970 (2008). [CrossRef]
  15. X. Ma and G. R. Arce, “Binary mask optimization for forward lithography based on boundary layer model in coherent systems,” J. Opt. Soc. Am. A 26(7), 1687–1695 (2009). [CrossRef]
  16. X. Ma and G. R. Arce, “Pixel-based simultaneous source and mask optimization for resolution enhancement in optical lithography,” Opt. Express 17(7), 5783–5793 (2009). [CrossRef] [PubMed]
  17. J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer, 2006).
  18. B. E. A. Saleh and M. Rabbani, “Simulation of partially coherent imagery in the space and frequency domains and by modal expansion,” Appl. Opt. 21(15), 2770–2777 (1982). [CrossRef] [PubMed]
  19. X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express 15, 15066–15079 (2007). [CrossRef] [PubMed]
  20. X. Ma and G. R. Arce, “PSM design for inverse lithography with partially coherent illumination,” Opt. Express 16(24), 20126–20141 (2008). [CrossRef] [PubMed]
  21. K. Kato, K. Nishizawa, and T. Inoue, “Advanced mask rule check (MRC) tool,” Proc. SPIE 6283, 62830O (2006). [CrossRef]
  22. K. Kato, Y. Taniguchi, K. Nishizawa, and M. Endo, “Mask rule check using priority information of mask patterns,” Proc. SPIE 6730, 67304F (2007). [CrossRef]
  23. B. Salik, J. Rosen, and A. Yariv, “Average coherent approximation for partially cohernet optical systems,” J. Opt. Soc. Am. A 13, 2086–2090) (1996). [CrossRef]
  24. M. Born and E. Wolfe, Principles of Optics (Cambridge University Press, 1999).
  25. R. Wilson, Fourier Series and Optical Transform Techniques in Contemporary Optics (John Wiley and Sons, 1995).
  26. M. R. Hestenes and E. Stiefel, “Methods of conjugate gradients for solving linear systems,” J. Res. Natl. Bur. Stand. 49, 409–436 (1952).
  27. R. Fletcher and C. M. Reeves, “Function minimization by conjugate gradients,” Comput. J. 7, 149–154 (1964). [CrossRef]
  28. H. P. Crowder and P. Wolfe, “Linear convergence of the conjugate gradient method,” IBM J. Res. Develop. 16, 431–433 (1972). [CrossRef]
  29. A. Cohen, “Rate of convergence of several conjugate gredient algorithms,” SIAM J. Numer. Anal. 9, 248–259 (1972). [CrossRef]
  30. K. Ritter, “On the rate of superlinear convergence of a class of variable metric methods,” NumerischeMathematik 35, 293–313 (1980). [CrossRef]
  31. C. Vogel, Computational Methods for Inverse Problems (SIAM Press, 2002). [CrossRef]
  32. M. P. McLoughlin and G. R. Arce, “Deterministic properties of the recursive separable median filter,” IEEE Trans. Acoust. Speech Signal Process. 35, 98–106 (1987). [CrossRef]
  33. S. Kalluri and G. R. Arce, “Fast algorithms for weighted myriad computation by fixed-point search,” IEEE Trans. Signal Process. 48(1), 159–171 (2000). [CrossRef]
  34. K. Barner and G. R. Arce, “Permutation filters - a class of nonlinear filters based on set permutations,” IEEE Trans. Signal Process. 42, 782–798 (1994). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited