Abstract

Model Based Optical Proximity Correction (MBOPC) is since a decade a widely used technique that permits to achieve resolutions on silicon layout smaller than the wavelength used in commercially-available photolithography tools. This is an important point, because patterns dimensions on masks are continuously shrinking. Commonly-used algorithms, involving Transfer Cross Coefficients (TCC) drawn from Hopkins formulation to compute aerial images during MBOPC treatment are based on TCC decomposition into its eigenvectors using matricization and the well known Singular Value Decomposition (SVD) tool. This technique remains highly runtime consuming. We propose in this paper to extend a fast fixed point algorithm to estimate an a priori fixed number of leading eigenvectors required to obtain a good approximation while ensuring a low information loss for computing aerial images.

© 2008 Optical Society of America

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References

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  1. E. Hecht, Optics (Addison-Wesley Publishing, Reading, Massachussetts, 1987).
  2. J. W. Goodman, Introduction to Fourier optics (McGraw-Hill, New York, 1996).
  3. H. H. Hopkins, “The concept of partial coherence in optics,” Proc. Royal Soc. Series A 208, 263–277 (1951).
    [Crossref]
  4. H. H. Hopkins, “On the diffraction theory of optical images,” Proc. Royal Soc. Series A 217, 408–432 (1952).
    [Crossref]
  5. M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1980).
  6. P. D. Flanner, Two-dimensional optical imaging for photolithography simulation (Technical Report Memorandum, UCB ERL M86 57, 1986).
  7. N. Cobb, Fast Optical and Process Proximity Correction Algorithms for Integrated Circuit Manufacturing (PhD Thesis, University of California at Berkeley, 1998).
  8. B. E. A. Saleh and M. Rabbani, “Simulation of partially coherent imagery in the space and frequency domains by modal expansion,” IEEE Trans. Electron. Devices 12, 1828–1836 (1982).
  9. A. Hyvarinen and E. Oja, “A fast-fixed point algorithm for independent component analysis,” Neural comput. 9, 1483–1492 (1997).
    [Crossref]

1997 (1)

A. Hyvarinen and E. Oja, “A fast-fixed point algorithm for independent component analysis,” Neural comput. 9, 1483–1492 (1997).
[Crossref]

1982 (1)

B. E. A. Saleh and M. Rabbani, “Simulation of partially coherent imagery in the space and frequency domains by modal expansion,” IEEE Trans. Electron. Devices 12, 1828–1836 (1982).

1952 (1)

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. Royal Soc. Series A 217, 408–432 (1952).
[Crossref]

1951 (1)

H. H. Hopkins, “The concept of partial coherence in optics,” Proc. Royal Soc. Series A 208, 263–277 (1951).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1980).

Cobb, N.

N. Cobb, Fast Optical and Process Proximity Correction Algorithms for Integrated Circuit Manufacturing (PhD Thesis, University of California at Berkeley, 1998).

Flanner, P. D.

P. D. Flanner, Two-dimensional optical imaging for photolithography simulation (Technical Report Memorandum, UCB ERL M86 57, 1986).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier optics (McGraw-Hill, New York, 1996).

Hecht, E.

E. Hecht, Optics (Addison-Wesley Publishing, Reading, Massachussetts, 1987).

Hopkins, H. H.

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. Royal Soc. Series A 217, 408–432 (1952).
[Crossref]

H. H. Hopkins, “The concept of partial coherence in optics,” Proc. Royal Soc. Series A 208, 263–277 (1951).
[Crossref]

Hyvarinen, A.

A. Hyvarinen and E. Oja, “A fast-fixed point algorithm for independent component analysis,” Neural comput. 9, 1483–1492 (1997).
[Crossref]

Oja, E.

A. Hyvarinen and E. Oja, “A fast-fixed point algorithm for independent component analysis,” Neural comput. 9, 1483–1492 (1997).
[Crossref]

Rabbani, M.

B. E. A. Saleh and M. Rabbani, “Simulation of partially coherent imagery in the space and frequency domains by modal expansion,” IEEE Trans. Electron. Devices 12, 1828–1836 (1982).

Saleh, B. E. A.

B. E. A. Saleh and M. Rabbani, “Simulation of partially coherent imagery in the space and frequency domains by modal expansion,” IEEE Trans. Electron. Devices 12, 1828–1836 (1982).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1980).

IEEE Trans. Electron. Devices (1)

B. E. A. Saleh and M. Rabbani, “Simulation of partially coherent imagery in the space and frequency domains by modal expansion,” IEEE Trans. Electron. Devices 12, 1828–1836 (1982).

Neural comput. (1)

A. Hyvarinen and E. Oja, “A fast-fixed point algorithm for independent component analysis,” Neural comput. 9, 1483–1492 (1997).
[Crossref]

Proc. Royal Soc. Series A (2)

H. H. Hopkins, “The concept of partial coherence in optics,” Proc. Royal Soc. Series A 208, 263–277 (1951).
[Crossref]

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. Royal Soc. Series A 217, 408–432 (1952).
[Crossref]

Other (5)

M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1980).

P. D. Flanner, Two-dimensional optical imaging for photolithography simulation (Technical Report Memorandum, UCB ERL M86 57, 1986).

N. Cobb, Fast Optical and Process Proximity Correction Algorithms for Integrated Circuit Manufacturing (PhD Thesis, University of California at Berkeley, 1998).

E. Hecht, Optics (Addison-Wesley Publishing, Reading, Massachussetts, 1987).

J. W. Goodman, Introduction to Fourier optics (McGraw-Hill, New York, 1996).

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Figures (3)

Fig. 1.
Fig. 1.

General optical lithography process diagram.

Fig. 2.
Fig. 2.

Computational times (in sec.) as a function of the number of rows and columns.

Fig. 3.
Fig. 3.

Reconstruction error comparison between SVD and Fixed Point Algorithm.

Tables (1)

Tables Icon

Table 1. Some numerical values corresponding with Fig. 2.

Equations (3)

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

I ( u , v ) = ( 1 2 π ) 2 i 1 I 1 i 2 I 2 i 3 I 3 i 4 I 4 TCC ( i 1 , i 2 , i 3 , i 4 ) E ( i 1 , i 2 ) E * ( i 3 , i 4 ) e i { ( i 1 i 3 ) u + ( i 2 i 4 ) v }
TCC ( i 1 , i 2 , i 3 , i 4 ) = 2 π + Γ ( x , y ) F ( x + i 1 , y + i 2 ) F * ( x + i 3 , y + i 4 ) dx dy
T = k = 1 I 2 λ k u k u k H

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