Proceedings Article | 1 December 2022
KEYWORDS: Photomasks, Semiconducting wafers, Manufacturing, Information theory, Extreme ultraviolet, Image resolution, Reticles, Optical proximity correction, Linear filtering, Inspection
Enabled by multi-beam mask writing [1], curvilinear free-form ILT [2], and GPU acceleration [3], curvilinear masks are quickly becoming the norm in leading edge masks, whether for 193i or for EUV, particularly for contact and via layers. An industry standard for compactly representing curvilinear shapes is being developed for SEMI through an industry working group. In it, Bezier, and B-spline "Multigon" formats are proposed to augment the piecewise linear polygons that are supported today [4]. 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 pre-defined 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. This paper explores the concept of rasterization and the mathematical dual between contours and pixel dose arrays given a particular known resolution limit. The paper further argues 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 [5], are written with multi-beam machines using pixel dose arrays. The paper further argues 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 afterwards. Information theory is a branch of computer science that, among other things, gives insight on how much data is sufficient to represent any particular information content [6,7,8]. More 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
