A systematic analysis of the optical merit function landscape: Towards improved optimization methods in optical design

A major problem in optical system design is that the optical merit function landscape is usually very complicated, especially for complex design problems where many minima are present. Finding good new local minima is then a difficult task. We show however that a certain degree of order is present in the optical design space, which is best observed when we consider not only local minima, but saddle points as well. With a special method, which we call Saddle-Point Construction (SPC), saddle points can be constructed in a simple way. Via saddle points, new local minima can be obtained very rapidly. When using a local optimization method, the final design after optimization highly depends on the starting configuration. We can group the initial configurations that lead to a given local minimum after local optimization into a graphical region, which shape depends on the optimization method used. However, saddle points are critical points in the merit function landscape that always remain on the boundaries, independent of the used optimization method. When the local optimization process is not chaotic, the geometric decomposition of the space of initial configurations into discrete regions has boundaries given by simple curves. But when the optimization is chaotic, the curves separating the different regions are very complicated objects termed fractals. In such cases, starting configurations, which are very close to each other, lead to different local minima after optimization. A better understanding of these instabilities can be obtained by using low damping values in a damped least-squares method.Applied Science