Inverse lithography in polar coordinates

The introduction of inverse lithography technology (ILT) into public domain in 2005 publications established a new paradigm in computational lithography. ILT offers fundamental advantages over traditional OPC by enabling complex mask shapes and using robust gradient-based optimization. This paper proposes polar ask representation and analyses its impact on ILT performance using various optimization algorithms. We compare polygonal, Bezier multigonal, and level set with our novel polar coordinate representation. While the mask can be converted from one representation to another, often without loss of information, these representations yield substantially different optimization paths and final masks due to the non-invariance of first-order gradient descent methods to transformation of variables, their number, and the structure of Hessian matrix. The polar coordinate approach expresses mask shapes as a series of expansion in polar coordinates. It demonstrates superior performance by converging faster, as well as reducing data volume while maintaining high accuracy. These improvements stem from inherent mathematical properties that naturally align with common lithographic patterns for contact and via layers. In summary, the polar approach provides a high-performance framework characterized by rapid convergence and a compact, efficient representation of smooth mask morphologies for contact-like layers.