Systematics of the design shapes in the optical merit function landscape

Abstract

In this paper we describe new properties of the design landscape that could lead in the future to a new way to determine good starting points for subsequent local optimization. While in optimization the focus is usually only on local minima, here we show that points selected in the vicinity of other types of critical points (i.e. points where the merit function gradient vanishes) can be very useful starting points. We study here a problem that is simple enough to be analyzed in detail, the design landscape of triplets with variable curvatures. We show here how representatives of all triplet design shapes observed in global optimization runs can be obtained in a simple and systematic way by locally optimizing for each design shape one starting point obtained with the new method. Good approximations of these special starting points are also computed analytically with two theoretical models. We have found a one-to-one correspondence between the possible triplet design shapes and the critical points resulting from a model based on third-order spherical aberration within the framework of thin-lens theory. The same number and properties of critical points are predicted by a second model, which is even simpler and mathematically more general.

Finding starting points analytically for optical system optimization

“…In a study of a thin-film design landscape (that will be discussed in detail elsewhere) it was found that minima form complex, almost random patterns, but critical points with indices higher than 1 form regular patterns that are useful for understanding the topology of the landscape and for finding the minima systematically. For the triplet problem under investigation here, a set of critical points having indices 0, 1 and 2 called fundamental critical points have a remarkable property that shows the existence of order in the design landscape 6 .…”

“…6 we have presented two analytical models that can be used to calculate simple approximations for the fundamental critical points of the triplet landscape. When the stop is at the lens, spherical aberration is the most nonlinear of the Seidel aberrations, and determines the non-convex character of the landscape.…”

“…Earlier, we have defined the concept of fundamental design shape 6,8 and both numerical experiments and analytical models 7 indicate that for triplets there are 22 fundamental design shapes. We have performed several triplet global searches with different specifications (aperture, field, transverse magnification, etc.)…”

“…For the specifications used in Ref. 6, the 22 fundamental design shapes are the local minima M 1 -M 22 shown in Fig.1. Remarkably, there exist 22 critical points, that can be obtained with SPC, one with index 0, 6 with index 1 and 15 with index 2, denoted in Fig 1 by C 1 -C 22 , that strongly resemble the fundamental design shapes.…”

“…All conclusions are also valid in the much simpler problem of doublets. We have shown recently that for these systems a one-to-one correspondence can be found between the possible design shapes and the critical points resulting from a simplified model based on third-order spherical aberration within the framework of thin-lens theory 6 . Here, we develop this idea further, based on new insight resulting from the study of the underlying analytic model 7 .…”