A fast convolution method using basis expansion for highly efficient intensity calculation in mask optimization

Abstract

Finer grid representation is inevitable to describing mask patterns more accurately in inverse lithography technology (ILT), thus resulting in large-size mask representation and heavy computational cost. In this work we proposed a fast convolution method called convolution using basis expansion (CBE) method to resolve computational issues caused by intensive convolutions. The CBE method process can be elaborated as: 1) Project mask and kernel matrices from fine grid representation to coarse grid representation under certain basis functions, which is similar to DCT or wavelet transformations. This matrix formed by the expansion coefficient can be considered as the projection of the original large matrix on coarse grid; 2) Perform mask and kernel convolutions on coarse grids; 3) the convolution result on fine grids is restored by interpolation method. The selection of the basis set can be arbitrary. In this paper, we compare the convolution accuracy and computational cost using 1) linear basis function; 2)discrete cosine basis function; 3) basis function based on K-L transform for different fine and coarse matrix size ratios <i>n</i> in both 1-D and 2-D conditions. Also, the quantitative interpolation error of cubic spline interpolation function is discussed. In numerical verification of aerial image calculation, this new method provides almost the same effectiveness and 10X~20X running speed improvement comparing to traditional convolution method. The CBE method will show its large effectiveness and efficiency in mask optimization.