Reflection and refraction problems for metasurfaces related to Monge-Amp\`ere equations

TL;DR

This paper derives Monge-Ampère PDEs governing phase discontinuities on metasurfaces to achieve specific reflection and refraction effects, and proves the existence of solutions.

Contribution

It establishes a mathematical framework linking metasurface design to Monge-Ampère equations and proves solution existence for these PDEs.

Findings

Derivation of Monge-Ampère equations for metasurface phase functions

Proof of existence of solutions to these PDEs

Framework for designing metasurfaces with desired optical properties

Abstract

A metasurface is a surface, tipically a plane, on which a function called phase discontinuity is chosen so that the metasurface produces a desired reflection or refraction job. We derive the equations that the phase discontinuity function must satisfy, they are Monge-Amp\`ere partial differential equations, and we prove existence of solutions.