Designing coupled free-form surfaces

Abstract: 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 solu- tions do not always exist. Here we present a method for the coupled design of two free-formreflective 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 geom- etry and makes use of the Cartan Kaehler theorem from exterior differential systems for proof of existence. The method can also be described using tra- ditional vectors and matrices, which we do. 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.