Trapping in poor local minima is a common problem in lens design. Conventional lens design approaches only lead the designer to one solution each time, and, especially when the designer has only limited experience, often better solutions exist. Global optimization algorithms such as genetic algorithms and simulated annealing have been used to find alternatives. However, these algorithms usually require significant computational power, and the designer has not much control over the process. Saddle point construction (SPC) method has been developed as a technique for adding lenses to the original system or to systematically switch from an existing local minimum to a different one. Earlier research has shown that in idealized lens design and simple practical lens design problems, SPC is able to effectively switch through the network of minima and get out of the poor local minima. However, the effectivity of SPC in the complex optical system has not yet been studied. We show in this paper that how SPC can be used to switch between local minima in a system with moderate complexity. The result shows that with SPC, it is possible to switch from a poor local minimum to better systems. SPC is also applied to a lithographic system to show how the switching mechanism works in highly complex systems.
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.
The present research is part of an effort to develop tools that make the lens design process more systematic. In typical optical design tasks, the presence of many local minima in the optical merit function landscape makes design non-trivial. With the method of Saddle Point Construction (SPC) which was developed recently ([F. Bociort and M. van Turnhout, Opt. Engineering 48, 063001 (2009)]) new local minima are obtained efficiently from known ones by adding and removing lenses in a systematic way. To illustrate how SPC and special properties of the lens design landscape can be used, we will present the step-by-step design of a wide-angle pinhole lens and the automatic design of a 9-lens system which, after further development with traditional techniques, is capable of good performance. We also give an example that shows how to visualize the saddle point that can be constructed at any surface of any design of an imaging system that is a local minimum.