Lens Design: Global Optimization with Escape Function

Abstract

The technique of using the 'escape function' for global optimization in lens design is described. This includes how to identify two solutions as independent; the threshold value for this criterion can be chosen to determine how to explore local solutions-rough or fine. Choice of appropriate values for two parameters in the escape function is most important, since this will affect the efficiency of the automatic global optimization process. There are two problems, i.e. giving default values at the beginning of the design, and determining a default rule to change them when the escape is unsuccessful. The latter was solved by assuming a 'saddle path' as the best route to escape. An exact solution for the former was not found, but a hint for finding a second best solution is shown based on the statistical study of local minima.

Chaotic behavior in an algorithm to escape from poor local minima in lens design

“…In the latter case, only seven out of eight distinct points can be easily distinguished in Fig. 3, because two points in the attracting set have approximately the same value for c 3 . Examining the values of curvature c 2 shows that these points are distinct.…”

“…A major challenge is to find good solutions among these minima. Increasingly elaborate and powerful global optimization methods [1,2,3,4,5,6,7,8] can provide a remedy when local optimization produces an unsatisfactory solution. However, simpler empirical techniques can also be effective for escaping from poor local minima, especially when the merit function barrier that must be overcome is low.…”

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

“…[1][2][3][4][5][6][7][8][9] For optical designs for which the complexity is not too high, present-day global optimization algorithms are valuable tools for finding a good ͑perhaps even the best͒ solution among the many local minima that are found in the merit function landscape. For moving an optical configuration from one local minimum to another, these methods rely almost exclusively on generally applicable mathematical algorithms, rather than on specific optical properties of the design landscape.…”

Network search method in the design of extreme ultraviolet lithographic objectives

“…Commercial optical design programs contain, nowadays, powerful global optimization algorithms, such as global synthesis [1][2][3][4], global explorer [5], simulated annealing [6], and genetic algorithms [7]. However, these algorithms give solutions as single points in the merit function space, without information about any relations between them.…”