## Fast approximation of transfer cross coefficient for optical proximity correction

Optics Express, Vol. 16, Issue 19, pp. 15249-15253 (2008)

http://dx.doi.org/10.1364/OE.16.015249

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### 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

## 1. Introduction

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]

## 2. Overview on existing methods

### 2.1. Hopkins Formulation

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

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

*F*, and the complex transmission of the object,

*E*. Therefore, the optical intensity in the image plane is written as follows:

*I*

_{1}=

*I*

_{2}=

*I*

_{3}=

*I*

_{4}=I for non-astigmatic systems for symmetry reasons.

### 2.2. Hopkins model Decomposition

*optical kernels*. In 2D case, TCC is represented as a four indices array, consequently SVD can not directly be applied. To circumvent this restriction, Cobb proposed to unfold this multiway array as a matrix

**T**owing to a column stacking function.

*I*

^{2}×

*I*

^{2}matrix

**T**yielding the decomposition

*=*

_{k}*𝒮*

^{-1}(

**u**

*). This algorithm may become highly runtime consuming as I increases and computes all eigenvectors as only few of them are used in practical cases. The new algorithm proposed in this paper computes only the required leading eigenvectors and reduces therefore computational load.*

_{k}## 3. Runtime improvement: Proposed algorithm

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

*K*orthonormal basis vectors is to use Gram-Schmidt method (

**see Algorithm 1**).

*K*-1 basis vectors (orthonormal to the previously measured basis vectors) will be measured one by one in a reducing order of dominance. The previously measured (

*p*-1)

^{th}basis vectors will be utilized to find the

*p*

^{th}basis vector. The algorithm for

*p*

^{th}basis vector will converge when the new value

**u**

^{+}

*and old value*

_{p}**u**

*are such that*

_{p}**u**

^{+}

^{T}

_{p}**u**

*= 1. It is usually economical to use a finite tolerance error to satisfy the convergence criterion ‖*

_{p}**u**

^{+}

^{T}

_{p}**u**

*-1‖<*

_{p}*η*where

*η*is a prior fixed threshold. Let

**U**=[

**u**

_{1},

**u**

_{2},…,

**u**

*] be the matrix whose columns are the*

_{K}*K*orthonormal basis vectors. Then

**UU**

*is the projector onto the subspace spanned by the*

^{T}*K*eigenvectors associated with the

*K*largest eigenvalues. This subspace is also called “signal subspace”.

Algorithm 1 Fixed-point. |
---|

1. Choose K, the number of principal axes or eigenvectors required to estimate. Consider matrix T and set p←12. Initialize eigenvector u
of size _{p}d × 1, e. g. randomly; 3. Update u
as _{p}u
←_{p}Tu
p; 4. Do the Gram-Schmidt orthogonalization process u
←_{p}u
-∑_{p}=^{j}^{p-1}
=_{j}_{1}(u
^{T}_{p}u
)_{j}u
; _{j}5. Normalize u
by dividing it by its norm: _{p}6. If u
has not converged, go back to step 3. _{p}7. Increment counter p←p+1 and go to step 2 until p equals K. |

## 4. Experimental results

### 4.1. Runtime improvement

^{®}SVD algorithm is used. We chose to compute 10 eigenvectors for each matrix with fixed point algorithm. This value corresponds to an average value of eigenvectors number used in classical OPC models. However, as Fixed Point algorithm runtime is linear with eigenvectors number to compute, Table 1 shows that this number can be increased up to around 300 eigenvectors for 3000 × 3000 matrices and up to around 100 eigenvectors for 120 × 120 matrices.

### 4.2. Fixed Point algorithm reconstruction error

**T′**the reconstructed matrix. For constancy purposes with Fixed Point algorithm, only the 10 first eigenvectors computed with SVD are used for reconstruction.

^{®}on a 2.4GHz dual core Pentium with 4Go RAM under Windows XP

^{®}.

## 5. Conclusion

## References and links

1. | E. Hecht, |

2. | J. W. Goodman, |

3. | H. H. Hopkins, “The concept of partial coherence in optics,” Proc. Royal Soc. Series A |

4. | H. H. Hopkins, “On the diffraction theory of optical images,” Proc. Royal Soc. Series A |

5. | M. Born and E. Wolf, |

6. | P. D. Flanner, |

7. | N. Cobb, |

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 |

9. | A. Hyvarinen and E. Oja, “A fast-fixed point algorithm for independent component analysis,” Neural comput. |

**OCIS Codes**

(110.5220) Imaging systems : Photolithography

(110.1758) Imaging systems : Computational imaging

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: May 27, 2008

Revised Manuscript: August 18, 2008

Manuscript Accepted: August 30, 2008

Published: September 12, 2008

**Citation**

Romuald Sabatier, Caroline Fossati, Salah Bourennane, and Antonio Di Giacomo, "Fast approximation of transfer cross coefficient for optical proximity correction," Opt. Express **16**, 15249-15253 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-19-15249

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### References

- E. Hecht, Optics (Addison-Wesley Publishing, Reading, Massachussetts, 1987).
- J. W. Goodman, Introduction to Fourier optics (McGraw-Hill, New York, 1996).
- 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]
- 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 M8657, 1986).
- N. Cobb, Fast Optical and Process Proximity Correction Algorithms for Integrated Circuit Manufacturing (PhD Thesis, University of California at Berkeley, 1998).
- 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).
- A. Hyvarinen and E. Oja, "A fast-fixed point algorithm for independent component analysis," Neural comput. 9, 1483-1492 (1997). [CrossRef]

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