Computational algorithms for constructing reflectors

Abstract

Consider a point source of light O radiating with intensity I(m) in direction m. Let R be a perfectly reflecting smooth surface. Suppose that a light ray emitted by the source O in direction m hits the surface R and reflects off it in direction y. Denote by γ:my the corresponding map defined by the surface R which transforms the set of rays incident on R into a set of reflected directions. If J(γ(·)) is the Jacobian of this map then the light intensity in direction y=γ(m) is given by I(m)/|J(γ(m))|. The problem consists of constructing a reflecting surface R such that for given sets of input and output directions (input and output apertures, respectively) and given input and output light intensities I and L, the surface R maps the input aperture into the output aperture and relates the input and output intensities via the energy conservation law of geometrical optics, that is, I(γ-1(y))|J(γ-1(y))|=L(y). This equation can be re-written as a second order equation of Monge–Ampere type. Theoretical results regarding its solvability have been established in [2] and a numerical method for calculating its solutions was proposed in [1]. In this paper we investigate a numerical procedure which in combination with the method in [1] provides a faster computational scheme for numerical solution of this problem.