You do not need 1 nm contours for curvilinear shapes: pixel-based computing is the answer

Abstract

Enabled by multibeam mask writing, curvilinear free-form inverse lithography technology (ILT), and graphics processing unit acceleration, curvilinear masks are quickly becoming the norm in leading edge masks, whether for 193i or for extreme ultra violet (EUV), particularly for contact and via layers. An industry standard for compactly representing curvilinear shapes is being developed for Semiconductor Equipment and Materials International through an industry working group. Bezier and B-spline "Multigon" formats are proposed to augment the piecewise linear polygons that are supported today. Whether these infinite-resolution curvilinear formats are used or piecewise linear polygons are used, there is a question of what constitutes a high enough vertex density to be of some predefined accuracy requirement. With these infinite-resolution curvilinear formats, the vertex density would be lower than with piecewise linear polygons for a particular accuracy requirement. But it is still useful to know what density is theoretically sufficient. We explore the concept of rasterization and the mathematical dual between contours and pixel dose arrays given a particular known resolution limit. We further argue that curvilinear ILT, practically speaking, is all computed in the pixel domain. And all curvilinear masks, with the notable exception of MWCO masks for 193i, are written with multibeam machines using pixel dose arrays. We further argue that all images taken of the resulting masks, whether for inspection, disposition, or for metrology are pictures taken as pixel dose arrays of some resolution with some image processing afterward. Information theory is a branch of computer science that, among other things, gives insight on how much data are sufficient to represent any particular information content. Generally, the field covers the idea of digitizing the analog world to some known limit of resolution. Rasterization is digitalization of images that converts from contours, be it piecewise linear polygons, or some infinite resolution curves, to pixel doses of some pixel size and dose range. Contouring is the converse, going from pixel doses to geometric space. By understanding information theory, how curvilinear mask shapes are computed, and how curvilinear mask shapes are generated on the mask, we compute the theoretical limit of how much data are required to represent 193i and EUV curvilinear masks.