Analytical form of light-ray tracing in invisibility cloaks

TL;DR

This paper derives analytical expressions for light-ray and wave-normal rays in ideal invisibility cloaks using transformation optics, clarifying the physical interpretation and mathematical properties of light propagation in such media.

Contribution

It provides explicit analytical formulas for light-ray tracing in spherical and cylindrical cloaks with arbitrary radial transformations, enhancing understanding of light behavior in invisibility cloaks.

Findings

Derived analytical expressions for light-ray and wave-normal rays.

Proved the ray-vector satisfies the ray-vector eikonal equation.

Identified transformation functions that make light-ray functions harmonic.

Abstract

In this paper, we review the methodology of transformation optics, which can construct invisibility cloak through the transformation of coordinates based on the form invariance of Maxwell's equations. Three different ways to define the components of electromagnetic fields are compared for removing some ambiguities. The analytical expressions of light-ray and wave-normal ray are derived in spherical and cylindrical ideal invisibility cloaks created with any continuous radial transformation functions, and their physical interpretation is also given. Using the duality principle in anisotropic media, we prove that light-ray vector satisfies "ray-vector eikonal equation" corresponding to the usual "wave-vector eikonal equation". The results interpret why the wave vector maps to the ray vector transferring from the virtual space to the physical space, but not the wave vector. As an application, we investigate the special transformation functions which make the light-ray function satisfy harmonic equation.