Fast pixel-based mask optimization for inverse lithography

Mask-filtering-based inverse lithography

“…The gradient-based method considers a mask as a raster image constituted by pixels directly, where it is synthesized pixel-by-pixel in an iterative direction of the steepest descent, conjugate gradient, and so on. 8,[10][11][12][13][14][15] It is probably the most popular technique in the literature due to its high flexibility, ease of understanding, and implementation.…”

“…In the literature, they are usually combined with certain proportions λ 1 and λ 2 to be stated as a single-objective minimization problem: [10][11][12][13][14][15] Finding M Ã ðrÞ to minimize: JðMÞ þ λ 1 R Q ðMÞþ λ 2 R TV ðMÞ where: λ 1 , λ 2 ≥ 0, subject to: 0 ≤ M ≤ 1.…”

“…[6][7][8][10][11][12][13][14][15] The pattern error employs an approximated and continuous resist model, and it is defined as a square of the L 2 norm of the difference between the output pattern of the input mask and the desired feature, which causes pattern error to be continuous and differentiable with respect to the input mask explicitly. 10,11 However, pattern error is a dimensionless quantity and highly depends on mask feature and simulation parameters, such as simulation area and simulation resolution. For this reason, pattern error is not popular in the industry.…”

“…[10][11][12][13][14][15] To address these problems, regularization approaches are introduced to guarantee the synthesized mask to be binary and less complex. In the literature, almost all the regularization approaches take the regularization terms as penalty functions incorporated into a cost function with corresponding weighted parameters [10][11][12][13][14][15] as…”

“…Various computation techniques have been proposed to deal with this inverse problem in the literature, such as the level-set method, [5][6][7][8] the discrete cosine transform (DCT)-based method, 9 and the gradient-based method. [10][11][12][13][14][15] The level-set method treats a mask as a sophisticated continuum, [5][6][7][8] and consequently, the boundary of the mask is iteratively evolved according to an optimization algorithm. The DCT-based method transforms a mask to the frequency space using a two-dimensional DCT.…”