On the existence of dichromatic single element lenses

TL;DR

This paper investigates the theoretical possibility of designing single-element lenses that refract two different colors of light into a fixed direction, considering different source configurations and demonstrating existence or non-existence results.

Contribution

It provides a rigorous analysis of the conditions under which dichromatic single-element lenses can be designed, including non-existence and local existence results based on functional equations.

Findings

No such lens exists for parallel beams with a different final direction.

A trivial solution exists when the beam and final directions are aligned.

Local solutions exist when the desired direction matches an incident direction.

Abstract

Due to dispersion, light with different wavelengths, or colors, is refracted at different angles. Our purpose is to determine when is it possible to design a lens made of a single homogeneous material so that it refracts light superposition of two colors into a desired fixed final direction. Two problems are considered: one is when light emanates in a parallel beam and the other is when light emanates from a point source. For the first problem, and when the direction of the parallel beam is different from the final desired direction, we show that such a lens does not exist; otherwise we prove the solution is trivial, i.e., the lens is confined between two parallel planes. For the second problem we prove that is impossible to design such a lens when the desired final direction is not among the set of incident directions. Otherwise, solving an appropriate system of functional equations we show that a local solution exists.