Efficient level-set based mask optimization with a vector imaging model

Abstract

With the extension of lithography simulations from low-NA to hyper-NA for immersion lithography and from scalar optics to vector optics including the polarization effect at source, mask, lens and resist film stack, inverse lithography technology (ILT) schemes based on the scalar imaging models which provide sufficient accuracy for NA less than 0.4, cannot track the whole process of polarized light propagation through the optical components of the lithography system rendering their impotence in hyper-NA immersion systems. In this paper, we address the mask synthesis problem by developing a optimization framework based on the vector imaging model fully describing the vector nature of electromagnetic fields in the aerial imaging formation and the stratified media model for the resist image, which is further reduced to a variational level-set based time-dependent model combining an internal energy term forcing the level set function close to a signed distance function and an extra energy term that drives the zero level set toward desired mask features minimizing the pattern difference between the printed wafer and the desired pattern, thereby optimizing the mask layout without the costly re- initialization procedure and ensuring the stability of the evolution of level surfaces. Represented by a partial differential equation, the time-dependent model is readily solved by finite difference schemes. Acceleration of over 30 times of the optimization procedure is achieved by computing convolution operations in the frequency domain with fast fourier transfrom (FFT) and the repeated usage of data, together with a 50% convergency performance improvement by means of applying Polak-Ribi`ere-Polyak (PRP) conjugate gradient (CG) to updating the normal velocity of the level-set function, which are merited by the experimental results.