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.
Contrary to the frequent tacit assumption that the local minima of a merit function are points scattered more or less
randomly over the design landscape, we have found that, at least for simple imaging systems (doublets with three and
triplets with five variables) all design shapes we have observed thus far form a strictly ordered set of points, the
"fundamental network". The design shapes obtained for practical specifications with global optimization algorithms are a
subset of the set of local minima in the fundamental network and are organized in a way that can be understood on the
basis of the fundamental network.
Finding good new local minima in the merit function landscape of optical system optimization is a difficult task, especially for complex 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.
Saddle-point construction (SPC) is a new method to insert lenses into an existing design. With SPC, by inserting and
extracting lenses new system shapes can be obtained very rapidly, and we believe that, if added to the optical designer's
arsenal, this new tool can significantly increase design productivity in certain situations. Despite the fact that the theory
behind SPC contains mathematical concepts that are still unfamiliar to many optical designers, the practical
implementation of the method is actually very easy and the method can be fully integrated with all other traditional
design tools. In this work we will illustrate the use of SPC with examples that are very simple and illustrate the essence
of the method. The method can be used essentially in the same way even for very complex systems with a large number
of variables, in situations where other methods for obtaining new system shapes do not work so well.
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.
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.
Finding multiple local minima in the merit function landscape of optical system optimization is a difficult task, especially for complex designs that have a large number of variables. We discuss here a method that enables a rapid generation of new local minima for optical systems of arbitrary complexity. We have recently shown that saddle points known in mathematics as Morse index 1 saddle points can be useful for global optical system optimization. In this work we show that by inserting a thin meniscus lens (or two mirror surfaces) into an optical design with N surfaces that is a local minimum, we obtain a system with N+2 surfaces that is a Morse index 1 saddle point. A simple method to compute the required meniscus curvatures will be discussed. Then, letting the optimization roll down on both sides of the saddle leads to two different local minima. Often, one of them has interesting special properties.
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.