Finding order in the design landscape of simple optical systems

2015

A major challenge in lens design is the presence of many local minima in the optimization landscape. However, unlike other global optimization problems, the lens design landscape has an additional structure, that can facilitate the design process: many local minima are closely related to minima of simpler problems. For discussing this property, in addition to local minima other critical points in the landscape must also be considered. Usually, in a global optimization problem with M variables one has to perform M-dimensional searches in order to find minima that are different from the known ones. We discuss here simple examples where, due to the special structure that is present, all types of local minima found by other methods can be obtained by a succession of one-dimensional searches. Replacing M-dimensional searches by a set of one-dimensional ones has very significant practical advantages. If the ability to reach solutions by decomposing the search in simple steps will survive generalization to more complex systems, new design tools using this property could have a significant impact on lens design.

Section: Decomposing the High-dimensional Search For New Local Minimamentioning

confidence: 61%

Reducible complexity in lens design

Hóu

2015

“…The one-to-one correspondence should be therefore understood in the sense that every C i produces a new M i which was not previously obtained via the same correspondence from critical points having a lower MI. Other critical points also exist in the triplet landscape 1 , but because of this one-to one correspondence with the fundamental design shapes M i , C i will be called fundamental critical points.…”

Section: Triplet Design Shapesmentioning

confidence: 99%

“…The fundamental critical points C i have been obtained from local minima of simpler problems having the same aperture and field specifications by using our so-called "saddle-point construction" (SPC) method 1,7,8,9 . For more details on the general version of SPC we refer to these papers.…”

Section: Triplet Design Shapesmentioning

confidence: 99%

See 1 more Smart Citation

Systematics of the design shapes in the optical merit function landscape

Grol

2010

Self Cite

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.

“…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.)…”

Section: Critical -Point Projectionmentioning

confidence: 99%

Finding starting points analytically for optical system optimization

Grol

2012

Understanding the structure of the design space in optical system optimization is difficult, because common human intuition fails when it encounters the challenge of high dimensionality, resulting from the many optimization parameters of lens systems. However, a deep mathematical idea, that critical points structure the properties of the space around them, is fruitful in lens design as well. Here we discuss simple systems, triplets with curvatures as variables, for which the design space is still simple enough to be studied in detail, but complex enough to be non-trivial. A one-to-one correspondence 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 could lead in the future to a new way to determine good starting points for subsequent local optimization.