Theory of light propagation in arbitrary two-dimensional curved space

TL;DR

This paper develops a theoretical framework for analyzing light propagation along geodesics on arbitrary 2D curved surfaces, addressing previous singularity issues and enabling studies of complex geometries like black hole analogs.

Contribution

The authors introduce a new theoretical model based on the Huygens-Fresnel principle for light in 2D curved spaces, overcoming prior singularity problems and broadening analysis capabilities.

Findings

Resolved the 'infinite intensity' issue at artificial singularities.

Enabled analysis of light behavior on complex curved surfaces.

Applied theory to Flamm's paraboloid, modeling black hole geometry.

Abstract

As an analog model of general relativity, optics on some two-dimensional (2D) curved surfaces has been increasingly paid attention to in the past decade. Here, in light of Huygens-Fresnel principle, we propose a theoretical frame to study light propagation along arbitrary geodesics on any 2D curved surfaces. This theory not only enables us to solve the enigma of "infinite intensity" existed previously at artificial singularities on surfaces of revolution, but also makes it possible to study light propagation on arbitrary 2D curved surfaces. Based on this theory, we investigate the effects of light propagation on a typical surface of revolution, Flamm's paraboloid, as an example, from which one can understand the behavior of light in the curved geometry of Schwarzschild black holes. Our theory provides a convenient and powerful tool for investigations of radiation in curved space.