TITLE:
Eliminating Residual Non‐Binary Intensities in Pixelated EUV Source Optimization
AUTHORS:
Athanasios Batgidis, Eytan Barouch, Michael Yeung
KEYWORDS:
EUV Lithography, Quadratic Programming, Source Optimization, Constraint Optimization, Numerical Precision
JOURNAL NAME:
Modeling and Numerical Simulation of Material Science,
Vol.16 No.2,
April
30,
2026
ABSTRACT: Recently, pixelated source optimization (SO) for EUV lithography has been posed as a convex quadratic program (QP) whose decision variables are pupil pixel intensities. The result of this QP generated an almost binary illuminator (~95%), where optimal solutions are composed of pixels lying almost exclusively at the set boundaries (0 or 1), while a small subset remains strictly interior. Although these pixels are few, they complicate hardware realization and raise the question of whether they reflect intrinsic properties of the lithography system or are simply numerical artifacts. In this paper, we studied the stability of non-binary pixels in convex QP source optimization by systematically varying three parameters that directly affect the optimization scheme: the total source power constraint W, the placement of measurement point pairs used to form the objective function, and the numerical precision used to build and solve the QP. Using a primal-dual interior-point method (PDIPM) and high-fidelity aerial image simulation, we quantified non-binary intensities and examined how this metric changed under controlled perturbations of the problem statement. Across multiple test patterns and both circular and annular pixelated pupils, we observed that increasing W yields a noisy reduction in the number of non-binary pixels, consistent with increased constraint activity under a sum equality constraint. In contrast, varying measurement pairs led to non-monotonic changes in non-binary pixels without an underlying trend. We also showed that an equal number (p = 100) of point pairs placed randomly across trials yielded noticeably different optimal sources and aerial images, while preserving the same exact number of non-binary pixels. Finally, repeating identical optimization trials in 64-bit and 128-bit arithmetic yielded systematically fewer non-binary pixels in 128-bit while preserving identical illuminator shapes and aerial image metrics (NILS, line intensity) in both circular and annular illuminators. We also showed that manually rounding these intensities to the nearest integer value does not affect the aerial image in a significant manner. These results indicate that non-binary pixels are the result of a numerically sensitive QP, and that higher-precision arithmetic primarily reduces active-set ambiguity rather than altering the physical optimum.