Parametric curvilinear mask optimization with rigorous gradient

A general framework is proposed for the parametric curvilinear mask optimization with rigorous gradient. As an example of parametric curve, the piecewise cubic Bézier curve is employed to represent the curvilinear contour of mask. By optimizing only the normal displacements of online knots and reconstructing the offline knots, the number of design variables is reduced while maintaining the smoothness of curve. Rigorous gradient of the objective function is derived, in which continuum-based formulations of design velocity and indicator function provide important links between the variation of design variable and the variation of mask transmittance. Gradient based optimization with simple strategy of adaptive step size is implemented. In addition, the methods of rasterization and target smoothing are presented. The proposed framework can be employed for any parametric curvilinear mask and even for Manhattan-style mask. Experiments with masks whose average critical dimension is 87 nm and area is 2.36 µm 2 demonstrate that the optimization can be completed in around 5 s and the average of maximum edge placement error is below 3 nm.