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## Goals of the Project

Put very briefly, geometrical optics is the study of the behaviour of light rays before and after refraction or reflection at a surface or at a number of surfaces. More specifically, geometrical optics is about the deviations from a perfect image after transition through an optical system. Such a system may either consist of homogeneous media (in which a light ray is a straight line) or inhomogeneous media (in which the rays are curved). In the latter case, the determination of the ray within one medium is a problem in itself.

There are many ways to study deviations in an optical system, or aberrations as they are called in optics. In principle, however, all of these methods allow the exact solution in terms of the specifications of the aberrations up to a certain order. Some of these methods are mathematically more sophisticated than others and some are more amenable to computer algebra implementation. For two methods in particular, computer algebra could possibly be a great help. One was is the method of so-called eikonals, the other is the closely related method of Lie transformations.

In the summer of 1993, André Heck and Marc Biemond showed that computer algebra could indeed be useful for the computation of eikonals. This work served as point of departure for the optics project. Whereas the work of Heck and Biemond had made it clear that computer algebra could be used in the case of eikonal theory, it was to be main goal of the project to investigate to what extent this holds true for the method of Lie transformations.

In the classical literature, Lie transformations are usually referred to as contact transformations. These play an important role in theoretical mechanics in that they allow the description of the trajectories for any object. In a similar way, the transition of light rays through an optical system can be described as a contact transformation as well. Until the early 1980s, however, this kind of description had hardly been investigated. Only after Kurt Bernardo Wolf in Mexico and Alex Dragt in Maryland had drawn attention to what they referred to as Lie transformations, these transformations and their application to geometrical and electron optics were extensively studied.

Both Dragt and Wolf had already used computer algebra to deal with Lie transformations and had shown how useful symbolic computation could be in this area. In this sense, it was only natural that the CAGO project would concentrate on the continuation of their work.

It was decided that the main activity of the CAGO-project should take place in the last quarter of 1994. As for all RIACA projects, the research was to be pursued by guest researchers who would be invited for a one or two-month period. As part of the project, a workshop was to take place in November. Dr. E.J. Atzema was appointed as of September 1, 1993 to prepare and co-ordinate the project.

Figure 2.5: David Stes' Editor with a Stop and Three Lenses

Next: Results Up: The CAGO Project Previous: Participants to the

Andre Heck
Sun Apr 23 10:32:10 MDT 1995