The Multi-marginal Monge Problem and an Application to Metasurfaces

TL;DR

This paper investigates the multi-marginal Monge problem in metric spaces, establishing existence and uniqueness of solutions for Lipschitz costs, and applies these results to design energy-preserving metalenses in optics.

Contribution

It provides new theoretical results on the multi-marginal Monge problem and demonstrates their application in designing advanced optical metasurfaces.

Findings

Proved existence and uniqueness of solutions for Lipschitz cost functions.

Applied the theoretical framework to design energy-preserving metalenses.

Enhanced understanding of optimal transport in optical device design.

Abstract

This paper studies the multi-marginal Monge problem in the setting of compact metric spaces proving existence and uniqueness of solutions when the cost function is Lipschitz. We apply the results obtained to solve an optics problem involving metalenses, that is, we design a refracting-reflecting metalens that preserves given energy distributions.