Proceedings Article | 17 October 2008
KEYWORDS: Reconstruction algorithms, Optical proximity correction, Photomasks, System on a chip, Projection systems, Optical lithography, Imaging systems, Lithographic illumination, Microscopy, Signal processing
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 wave-length which is used in commercially-available
photolithography tools. This is an important point, because masks dimensions are continuously shrinking.
As for the current masks, several billions of segments have to be moved, and also, several iterations are needed
to reach convergence. Therefore, fast and accurate algorithms are mandatory to perform OPC on a mask in a
reasonably short time for industrial purposes.
As imaging with an optical lithography system is similar to microscopy, the theory used in MBOPC is drawn
from the works originally conducted for the theory of microscopy. Fourier Optics was first developed by Abbe to
describe the image formed by a microscope and is often referred to as Abbe formulation. This is one of the best
methods for optimizing illumination and is used in most of the commercially available lithography simulation
packages.
Hopkins method, developed later in 1951, is the best method for mask optimization. Consequently, Hopkins
formulation, widely used for partially coherent illumination, and thus for lithography, is present in most of the
commercially available OPC tools. This formulation has the advantage of a four-way transmission function independent
of the mask layout. The values of this function, called Transfer Cross Coefficients (TCC), describe
the illumination and projection pupils.
Commonly-used algorithms, involving TCC of 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. These techniques that use numerical approximation and empirical determination
of the number of eigenvectors taken into account, could not match reality and lead to an information loss.
They also remain highly runtime consuming.
We propose an original technique, inspired from tensor signal processing tools. Our aim is to improve the simulation
results and to obtain a faster algorithm runtime. We consider multiway array called tensor data T CC. Then,
in order to compute an aerial image, we develop a lower-rank tensor approximation algorithm based on the signal
subspaces. For this purpose, we propose to replace SVD by the Higher Order SVD to compute the eigenvectors
associated with the different modes of TCC. Finally, we propose a new criterion to estimate the optimal number
of leading eigenvectors required to obtain a good approximation while ensuring a low information loss.
Numerical results we present show that our proposed approach is a fast and accurate for computing aerial
images.
