# Global optimization of masks, including film stack design to restore TM contrast in high NA TCC's

## Abstract

We provide an expanded description of the global algorithm for mask optimization introduced in our earlier papers, and discuss auxiliary optimizations that can be carried out in the problem constraints and film stack. Mask optimization tends inherently to be a problem with non-convex quadratic constraints, but for small problems we can mitigate this difficulty by exploiting specialized knowledge that applies in the lithography context. If exposure latitude is approximated as maximization of edge slope between image regions whose intensities must print with opposite polarity, we show that the solution space can be approximately divided into regions that contain at most one local minimum. Though the survey of parameter space to identify these regions requires an exhaustive grid search, this search can be accelerated using heuristics, and is not the rate-limiting step at SRAM scale or below. We recover a degree of generality by using a less simplified objective function when we actually assess the local minima. The quasi-binary specialization of lithographic targets is further exploited by searching only in the subspace formed by the dominant joint eigenvectors for dark region intensity and bright region intensity, typically reducing problem dimensionality to less than half that of the full set of frequency-domain variables (i.e. collected diffraction orders). Contrast in this subspace across the bright/dark edge will approximately reflect exposure latitude when we apply the standard fixed edge-placement constraints of lithography. However, during an exploratory stage of optimization we can define preliminary tolerances which more explicitly reflect constraints on devices, e.g. as is done with compactor codes for design migration. Our algorithm can handle vector imaging in a general way, but for the special case of unpolarized illumination and a lens having radial symmetry (but arbitrary source shape) we show that the bilinear function which describes vector interference within the film stack can be expressed in terms of three generic radial functions, enabling rapid numerical evaluation of the Hopkins kernel. By inspection these functions show that one can in principle recover classical scalar-like imaging even at high NA by exposing a very thin layer spaced above a reflective substack. The reflected image largely restores destructive interference in TM polarized fringes, if proper phasing is achieved. With an ideal reflector, the first-order azimuthal contrast loss term vanishes in all TCC components, and complete equivalence to scalar imaging is obtained in classical two-beam imaging.