Network search method in the design of extreme ultraviolet lithographic objectives

2015

SPIE Proceedings

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A major challenge in lens design is the presence of many local minima in the optimization landscape. However, unlike other global optimization problems, the lens design landscape has an additional structure, that can facilitate the design process: many local minima are closely related to minima of simpler problems. For discussing this property, in addition to local minima other critical points in the landscape must also be considered. Usually, in a global optimization problem with M variables one has to perform M-dimensional searches in order to find minima that are different from the known ones. We discuss here simple examples where, due to the special structure that is present, all types of local minima found by other methods can be obtained by a succession of one-dimensional searches. Replacing M-dimensional searches by a set of one-dimensional ones has very significant practical advantages. If the ability to reach solutions by decomposing the search in simple steps will survive generalization to more complex systems, new design tools using this property could have a significant impact on lens design.

Section: Saddle-point Constructionmentioning

confidence: 99%

Reducible complexity in lens design

Hóu

2015

SPIE Proceedings

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“…2͒ The saddle points are therefore special points on the boundary between the basins of attraction that correspond to the two adjacent local minima. 16 ͑The basin of attraction is the set of starting points that, after local optimization, lead to the same minimum.͒ As has been shown earlier, 11,15 if a local minimum is known, new local minima can be found by detecting Morse index 1 saddle points in the vicinity of the known minimum and then by optimizing the configurations on the other side of these saddle points. A drawback of this method is that detecting Morse index 1 saddle points without a priori information about them is computationally more expensive than finding local minima.…”

Section: Introductionmentioning

confidence: 99%

“…[10][11][12][13][14][15] Minima, saddle points, and maxima are all critical points, i.e., the gradient of the merit function vanishes at these points. An important property of ͑nondegenerate͒ critical points is the so-called Morse index.…”

Section: Introductionmentioning

confidence: 99%

Finding new local minima in lens design landscapes by constructing saddle points

2009

Opt. Eng

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Abstract. Finding good new local minima in the merit function landscape of optical system optimization is a difficult task, especially for complex design problems where many minima are present. Saddle-point construction ͑SPC͒ is a method that can facilitate this task. We prove that, if the dimensionality of the optimization problem is increased in a way that satisfies certain mathematical conditions ͑the existence of two independent transformations that leave the merit function unchanged͒, then a local minimum is transformed into a saddle point. With SPC, lenses are inserted in an existing design in such a way that subsequent optimizations on both sides of the saddle point result in two different system shapes, giving the designer two choices for further design. We present a simple and efficient version of the SPC method. In spite of theoretical novelty, the practical implementation of the method is very simple. We discuss three simple examples that illustrate the essence of the method, which can be used in essentially the same way for arbitrary systems.

“…points where the merit function gradient vanishes) can be very useful as an intermediate step in the process of finding good lens designs. Saddle-point detection (SPD) is a thorough search method that works without any a-priori knowledge for most optimization problems with continuous variables 1,2,3 . The algorithm looks at every known local minimum for index-1 saddle-points around it, and for each detected saddle-point it finds a new minimum on the other side of the "saddle".…”

Section: Introductionmentioning

confidence: 99%

Finding starting points analytically for optical system optimization

Grol

2012

SPIE Proceedings

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Understanding the structure of the design space in optical system optimization is difficult, because common human intuition fails when it encounters the challenge of high dimensionality, resulting from the many optimization parameters of lens systems. However, a deep mathematical idea, that critical points structure the properties of the space around them, is fruitful in lens design as well. Here we discuss simple systems, triplets with curvatures as variables, for which the design space is still simple enough to be studied in detail, but complex enough to be non-trivial. A one-to-one correspondence between the possible design shapes and the critical points resulting from a simplified model based on third-order spherical aberration within the framework of thin-lens theory could lead in the future to a new way to determine good starting points for subsequent local optimization.