Manifold Optics

TL;DR

This paper introduces a novel framework linking three-dimensional manifolds with gradient media in transformation optics using Ricci scalar curvature, extending the geometric approach to 3D spaces and validating it with conformal optical lenses.

Contribution

It establishes an intrinsic connection between 3D manifolds and gradient media in transformation optics via the Yamabe problem and Ricci scalar curvature, expanding the geometric optics theory.

Findings

Proves Ricci scalar invariance under conformal mappings.

Validates the framework with analysis of conformal optical lenses.

Extends geometric optics to three-dimensional manifolds.

Abstract

Transformation optics establishes an equivalence relationship between gradient media and curved space, unveiling intrinsic geometric properties of gradient media. However, this approach based on curved spaces is concentrated on two-dimensional manifolds, namely curved surfaces. In this Letter, we establish an intrinsic connection between three-dimensional manifolds and three-dimensional gradient media in transformation optics by leveraging the Yamabe problem and Ricci scalar curvature, a measure of spatial curvature in manifolds. The invariance of the Ricci scalar under conformal mappings is proven. Our framework is validated through the analysis of representative conformal optical lenses.