Optimum Adjustment for Distortion in Semiconductor Lithography Equipment

Fukagawa, Youzou; Shinano, Yuji; Nakamori, Mario

doi:10.1299/jamdsm.2.378

Cited by 2 publications

(2 citation statements)

References 2 publications

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“…Application of linear programming enables quick adjustment of lenses if only the linear characteristic values are considered (3), (4) . However, this procedure fails when the quadratic characteristic values should also be examined.…”

Section: Introductionmentioning

confidence: 99%

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

Shinano

Yoshihara

Miyashiro

et al. 2010

JAMDSM

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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.

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Section: Introductionmentioning

confidence: 99%

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

Shinano

Yoshihara

Miyashiro

et al. 2010

JAMDSM

Self Cite

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“…We have applied the mathematical technique of linear programming (LP) to optimize the adjustment for distortion in a short computation time (2) . This is a very significant development, since it has the potential to release the industry from time-consuming search algorithms.…”

Section: Introductionmentioning

confidence: 99%

Lens System Adjustment in Semiconductor Lithography Equipment -Optimization for Lens Groups Rotation-

Shinano

Fukagawa

Takano

et al. 2008

JAMDSM

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In this paper, we address the problem of lens aberration adjustment for lenses of semiconductor lithography equipment. The objective of aberration adjustment is to minimize the maximum value of aberration in any part of the images that are projected on the wafer. Formerly, an approximate solution method has been used, which is based on a brute force search for a subset of all cases. However, using that method, computational times are often long and a faster solution method is necessary. Nowadays, owing to significant progress in mathematical programming research, MIP(Mixed Integer Programming) solvers can reach an optimal solution quite fast. That is why, in this paper, we propose an exact solution method based on 0-1 MIP, which addresses the same "lens aberration" problem with the same objective of minimizing maximum absolute value of lens aberration. In real-world 9 instances, our solution method improves maximum aberration values by 18% to 56% over similar results from conventional solution methods.

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Optimum Adjustment for Distortion in Semiconductor Lithography Equipment

Cited by 2 publications

References 2 publications

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

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

Lens System Adjustment in Semiconductor Lithography Equipment -Optimization for Lens Groups Rotation-

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