The Refractor Problem with Loss of Energy and Monge-Ampere Type Equations

Abstract

In this dissertation we study The Refractor Problem and its analytic formulation which leads to Monge-Ampere type equation. This problem can be described as follows: Suppose that A and B are two domains of the unit sphere in n dimensions and g and f are two positive functions integrable on A and B respectively. Consider two homogeneous, isotropic media; medium I and medium II, which have different optical densities and assume that from a point O inside medium I, light emanates with intensity g(x); where x is in A. When an incident ray of light hits an interface between two media with different indices of refraction, it splits into two rays a reflected ray that propagates back into medium I and a refracted ray that proceeds into medium II. Consequently, the incident ray loses some of its energy as it proceeds into medium II. By using Fresnel equations, which are consequences of Maxwell's Equations, one can determine precisely how much of the energy is lost due to internal reflection. The problem is to take into account this loss and construct a surface such that all rays emitted from a point O in the first medium, with directions in A are refracted by the surface into media II with directions in B and the prescribed illumination intensity received in the direction m, where m is in B is f(m). We propose a model to this problem. We introduce weak solutions for the problem and prove their existence by using approximation by ellipsoids or hyperboloids depending on whether n1 is less than n2 or n1 is greater than n2. We will also prove that a solution of the problem satisfies a Monge-Ampere type of PDE.