Bundle separation, obstructions to perfect imaging, and other qualitative aspects of simultaneous multiple surface design

Abstract

We investigate the qualitative aspects of the simultaneous multiple surface (SMS) design method. The SMS method is a powerful and complex tool, whose abilities and limitations we feel are not yet fully understood. Here, we view the SMS method as an iterative dynamical system, consider the physical boundaries on which the method applies and show that in some cases, attractors occur on the boundaries of these regions. One consequence of our viewpoint is numerical evidence of the impossibility of perfectly imaging three points to three points using only two reflectors. Another consequence is that SMS may be viewed as a means for designing a primary component that essentially separates the bundles in such a way that a continuous secondary component exists that images the two separate bundles appropriately. In other words, a clever choice of the primary component decouples the problem.