Ray mapping approach for the efficient design of continuous freeform surfaces

TL;DR

This paper presents a novel ray-mapping method for designing continuous freeform surfaces that efficiently transform a source into a target intensity, utilizing optimal mass transport and linear advection equations for practical surface construction.

Contribution

It introduces a new integrability condition based on the law of reflection/refraction and optimal mass transport, enabling efficient freeform surface design.

Findings

Mapping can be computed via optimal mass transport.

Surface construction reduces to solving a linear advection equation.

Method is effective for a wide range of distances.

Abstract

The efficient design of continuous freeform surfaces, which transform a given source into an arbitrary target intensity, remains a challenging problem. A popular approach are ray-mapping methods, where first a ray mapping between the intensities is calculated and in a subsequent step the surface is constructed. The challenging part hereby is the to find an integrable mapping ensuring a continuous surface. Based on the law of reflection/refraction and the well-known integrability condition, we derive a general condition for the surface and ray mapping for a collimated input beam. It is shown that in a small-angle approximation a proper mapping can be calculated via optimal mass transport. We show that the surface can be constructed by solving a linear advection equation with appropriate boundary conditions. The results imply that the optimal mass transport mapping is approximately integrable over a wide range of distances between the freeform and the target plane and offer an efficient way to construct the surface by solving standard integrals. The efficiency is demonstrated by applying it to two challenging design examples.