5.2 Field Calculation over Planar Topography

Assuming a planar resist the field calculation is considerably simplified. Various models with different levels of accuracy exist for a planar topography. The simplest and most basic one is the vertical propagation model, but it incorrectly predicts all lithographic behavior to be completely symmetric in the focus offset positions. The scaled defocus approach overcomes this drawback as it represents a first-order correction to the vertical propagation method. The transfer matrix algorithm additionally considers the impact of non-vertical propagation on thin-film interference effects and is thus the most advanced approach among the analytical models. These three methods are closely related to the analysis of a stratified medium presented in Appendix C. Thus all of them neglect scattering effects due to the lateral inhomogeneity of the resist layer. The beam propagation method overcomes this limitation. The EM field is modeled by a weakly amplitude-modulated wave traveling into the resist. Numerical schemes have to be employed to solve the partial differential equation involved. Although the computational effort is increased in comparison to the purely analytical models, the performance is much faster than that of the rigorous EM scattering theory that is required for a full nonplanar inhomogeneous resist simulation. In the following four sections we will briefly sketch each of the methods mentioned.