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Generalized inverse problem for partially coherent projection lithography

In this paper, we will outline general mathematical techniques applied to the solution of the inverse problem for partially coherent lithographic imaging. The forward imaging problem is reviewed and its solution is discussed within the framework of 2D sampling and matrix coherence theory. The intensity distribution on the wafer is shown to be a bilinear functional in the sampled mask transmission values, and represents a continuous sparse set of variables for optimization. We review various iterative techniques to optimize the sampled mask transmission, called a tau-map. From the optimal tau-map, a procedure is required to construct a pixelated mask representation with restricted transmission values. This mask representation is not unique since the problem is ill-posed, and leads to multiple mask solutions for a single optimal tau-map. Various procedures based on spectral techniques and principle component analysis to quantize the mask are reviewed.