Topography of the merit function landscape in optical system design

We have shown recently that, when certain quite general conditions are satisfied, the set of local minima in the optical merit function space forms a network where they are all connected through optimization paths generated from saddle points having a Morse index of 1. A new global optimization method, that makes use of this linking network to systematically detect all minima, is presented. The central component of this new method, the algorithm for saddle point detection, is described in detail and we show that the initialization of this algorithm has a significant impact on the performance. For a simple global optimization search (Cooke triplet) several representation forms of the network of the corresponding set of local minima are presented. These representations, which can be visualized in two dimensions, are independent of the dimensionality of the design space so that they can provide insight into the topography of merit function landscapes of arbitrary dimensionality. In this paper, we give additional details about the method used to detect the saddle points. We also introduce several representation forms for the topography of the network structure of merit function spaces of arbitrary dimensions. These representation forms can be used to visualize the relationship between the minima while ignoring issues such as the dimensionality or local characteristics of the merit function space. These networks can become an important tool for the study of complex design problems encountered in optics. In section 2 it is shown that the relationship between the various minima can be derived only by considering the equimagnitude surfaces around minima and saddle points having a Morse index of one. In Section 3 we describe a new method for global optimization based on the detection of the network of minima. This method makes use of the network structure to systematically find all minima and saddle points having a Morse index of 1. The detection of saddle points which forms the central component of our method is discussed in section 4. We will show that saddle points having a Morse index of 1 can be detected by using only the local optimization engine of optical design software. We also show that the initialization of such an algorithm has a significant impact on the efficiency of the algorithm. In section 5, several representation forms of the networks of minima are introduced. These representations can be used as tools for the analysis of the topography of merit function spaces of arbitrary dimensions.