Geometric and variational methods in optical design of reflecting surfaces with prescribed irradiance properties

摘要

Numerous optical and electromagnetic applications require numerical design of reflecting surfaces in 3D with capabilities to redirect the input energy flow and reshape the energy radiation intensity of a source into a prescribed output irradiance distribution over a specified target surface. In the geometrical optics approximation, a systematic application of the ray tracing equations and energy conservation law reduces the problem, in many cases, to finding numerical solutions to nonlinear, second order partial differential equations. If the severe limitation of rotational symmetry is not assumed then the resulting equations are very far from being standard and require significant efforts for their theoretical investigation and reliable numerical solution. In recent years a quite general approach combining geometric techniques with methods from calculus of variations has been developed and applied to a rigorous and unified investigation of several classes of such equations. Moreover, this approach allows implementations in provably convergent numerical algorithms. In this paper I outline this approach in the problem of designing a reflecting surface capable of redirecting the energy flow from a point source so that the reflected rays have directions specified in advance as a subset on the far-sphere and the output irradiance density is also pre-specified in advance as a function of the reflected direction. A numerical example illustrating the solution is also presented.