Maarten van Turnhout

Saddle points in the merit function landscape of systems of thin lenses in contact

The merit function landscape of systems of thin lenses in contact, which are perhaps the simplest possible types of optical systems, shows remarkable regularities. It is easier to understand how the optimization parameter space of these simple systems is divided into basins of attraction for the various local minima if one focuses on the (Morse index 1) saddle points in the landscape rather than on the local minima themselves. The existence and the basic properties of these saddle points can be predicted by thin-lens theory, which is applied on a simplified model of the merit function containing only third-order spherical aberration. The predictions of this simplified model are confirmed by numerical results obtained with a typical merit function based on ray tracing.

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

Finding good new local minima in the merit function land- scape of optical system optimization is a difficult task, especially for com- plex design problems where many minima are present. Saddle-point construction SPC 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 unchanged, 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.

Using saddle points for challenging optical design tasks

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.

Predictability and unpredictability in optical system optimization

Local optimization algorithms, when they are optimized only for speed, have in certain situations an unpredictable behavior: starting points very close to each other lead after optimization to different minima. In these cases, the sets of points, which, when chosen as starting points for local optimization, lead to the same minimum (the so-called basins of attraction), have a fractal-like shape. Before it finally converges to a local minimum, optimization started in a fractal region first displays chaotic transients. The sensitivity to changes in the initial conditions that leads to fractal basin borders is caused by the discontinuous evolution path (i.e. the jumps) of local optimization algorithms such as the damped-least-squares method with insufficient damping. At the cost of some speed, the fractal character of the regions can be made to vanish, and the downward paths become more predictable. The borders of the basins depend on the implementation details of the local optimization algorithm, but the saddle points in the merit function landscape always remain on these borders.