Reflector design for two-variable beam shaping in the hyperbolic case

Abstract

The design of a reflector capable of producing a generalized two-variable beam shape when illuminated by a point source is investigated under the geometric-optics approximation. The problem is formulated as a mapping problem between points on a unit sphere in which areas are related by energy considerations. The solution of a resulting set of nonlinear partial differential equations is required. An investigation of the hyperbolic form of these equations shows that they can be reduced to a set of quasi-linear first-order partial differential equations which can be solved subject to initial conditions.