M. Van Turnhout

Instabilities and fractal basins of attraction in optical system optimization

Many optical design programs use various forms of the damped least-squares method for local optimization. In this paper, we show that damped least-squares algorithms, with maximized computational speed, can create sensitivity with respect to changes in initial conditions. In such cases, starting points, which are very close to each other, lead to different local minima after optimization. Computations of the fractal capacity dimension show that sets of these starting points, which lead to the same minimum (the basins of attraction for that minimum), have a fractal structure. Introducing more damping makes the optimization process stable.

Looking for order in the optical design landscape

In present-day optical system design, it is tacitly assumed that local minima are points in the merit function landscape without relationships between them. We will show however that there is a certain degree of order in the design landscape and that this order is best observed when we change the dimensionality of the optimization problem and when we consider not only local minima, but saddle points as well. We have developed earlier a computational method for detecting saddle points numerically, and a method, then applicable only in a special case, for constructing saddle points by adding lenses to systems that are local minima. The saddle point construction method will be generalized here and we will show how, by performing a succession of one-dimensional calculations, many local minima of a given global search can be systematically obtained from the set of local minima corresponding to systems with fewer lenses. As a simple example, the results of the Cooke triplet global search will be analyzed. In this case, the vast majority of the saddle points found by our saddle point detection software can in fact be obtained in a much simpler way by saddle point construction, starting from doublet local minima.

Obtaining new local minima in lens design by constructing saddle points.

We show that in the lens design landscape saddle points exist that are closely related to local minima of simpler problems. On the basis of this new theoretical insight we develop a systematic and efficient saddle-point method that uses a-priori knowledge for obtaining new local minima. In contrast with earlier saddle-point methods, the present method can create both positive and negative lenses. As an example, by successively using the method a good-quality local minimum is obtained from a poor-quality one. The method could also be applicable in other global optimization problems that satisfy the requirements discussed in this paper.

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

In lens design, damped least-squares methods are typically used to find the nearest local minimum to a starting configuration in the merit function landscape. In this paper, we explore the use of such a method for a purpose that goes beyond local optimization. The merit function barrier, which separates an unsatisfactory solution from a neighboring one that is better, can be overcome by using low damping and by allowing the merit function to temporarily increase. However, such an algorithm displays chaos, chaotic transients and other types of complex behavior. A successful escape of the iteration trajectory from a poor local minimum to a better one is associated with a crisis phenomenon that transforms a chaotic attractor into a chaotic saddle. The present analysis also enables a better understanding of peculiarities encountered with damped least-squares algorithms in conventional local optimization tasks.