Fast calculation of images for high numerical aperture lithography

Many hitherto small effects will become numerically significant in lithography at 70nm and below. The simple assumptions of scalar imaging and uniformly-polarized sources will no longer be tenable. Contrast losses in the resist (e.g. by diffusion) will become appreciable. In addition, the elements of 157nm lenses will be intrinsically polarizing due to spatial dispersion in CaF2, and in general lenses will exhibit residual polarization aberrations. We show here that these effects can be accounted for in a fast "sum-of-coherent-systems" (SOCS) algorithm that is suitable for model-based optical proximity correction (MBOPC). First, we cast the classic equations of vector image formation in a new form that explicitly distinguishes scalar and vector field terms. Lens birefringence is then added to the model; in doing so we take into account the classic phenomenon of double refraction, wherein a given ray splits into two rays each time it passes through an element. In principle, each incident ray then gives rise to an extended family of rays in the exit pupil. However, we show that this coherent set of rays can be merged into a single plane-wave component of the image, allowing a Jones matrix pupil to be defined. Once the vector imaging equations are modified to accommodate customized polarization distributions in the source as well as matrix pupils in the lens, we show that tractable SOCS kernels can be obtained under a generalization of the thin-mask approximation. Such models can be extended to include non-optical effects like resist blur, along with empirical modeling terms. We also discuss computational efficiencies that can be achieved when calculating SOCS kernels, for example by iteratively refining kernels calculated from a reduced basis, and by exploiting system symmetry (radial, dipole, or quadrupole).