Optimization of Lens Adjustment in Semiconductor Lithography Equipment Using Quadratically Constrained and Second-Order Cone Programming

Abstract

The present paper considers optimization of lens adjustment in semiconductor lithography equipment. For improving productivity, the laser irradiation power of recent semiconductor lithography equipment has been boosted, which causes significant aberration due to heating during exposure. The aberration of the equipment must be measured or estimated in order to adjust the positions and orientations of the lenses. Since this adjustment is performed sequentially during exposure, the optimization problem to obtain optimal lens adjustment should be solved within a time as short as 100 ms. Although the problem of calculating the optimal lens adjustment can be naturally formulated as a convex minimization problem, in such a formulation the objective function is convex but includes several nondifferentiable points. Hence, optimization methods based on derivatives cannot be applied. Other approaches using derivative-free optimization or meta-heuristic methods cannot guarantee that the obtained solutions are truly optimal. Therefore, we formulate the optimization problem as quadratically constrained and second-order cone programming problems, which can be handled by solvers using an interior point method. Using the proposed formulations, computational experiments demonstrate that the optimal lens adjustment is obtained in a practical computational time, which is much less than 100 ms.

Keywords

• semiconductor lithography
• aberration
• optimum design
• system engineering
• optical instrument
• precision instrument
• second order cone programming
• quadratically constrained programming
• engineering optimization