Designing coupled free-form surfaces

This is joint work with Andrew Hicks. It is an applied geometric optics talk aimed at a general mathematical audience.

The problem of designing optical systems that contain free-form surfaces (i.e. not rotationally symmetric) is a challenging one, even in the case of designing a single surface. Part of the reason for this is that solutions do not always exist. Here we present a method for the coupled design of two free-form reflective surfaces (i.e. mirrors) which will have a prescribed distortion. One should think for example of a child's periscope with curved mirrors so as to give a wider field of view. The method is motivated by viewing the problem in the language of distributions from differential geometry and makes use of the Cartan Kaehler theorem from exterior differential systems for proof of existence. We give example applications to the design of a mirror pair that increases the field of view of an observer, a similar mirror pair that also rotate the observers view, and a pair of mirrors that give the observer a traditional panoramic strip view of the scene.