A neutral Kähler surface with applications in geometric optics

Abstract

The space L of oriented lines, or rays, in Euclidean 3-space E3 is a 4-dimensional space with an abundance of natural geometric structure. In particular, it boasts a neutral Kähler metric which is closely related to the Euclidean metric on E3. In this article we review recent work on this Kähler structure and consider its applications to geometric optics in a homogeneous isotropic medium. In particular, we discuss the complex geometry of reflection in a surface in E3 and the computation of focal sets of bundles of rays. To illustrate the method, we compute the focal set of the kth reflection of a point source off the inside of a cylinder. The focal sets, which we explicitly parameterize, exhibit unexpected symmetries, and are found to fit well with observable phenomena. We conclude with generalizations of the geometric construction.