A Novel Approach for Exploring the Light Traveling Path in the Medium with a Spherically Symmetric Refractive Index

TL;DR

This paper introduces an analogy-based method to analytically solve the ray path in a medium with a spherically symmetric refractive index, simplifying a traditionally complex PDE problem.

Contribution

It proposes a novel analogy between ray refraction and inverse-square central force systems, enabling analytical solutions for inhomogeneous media.

Findings

Analytical relation between ray paths and inverse-square force systems.

Numerical simulations confirm the validity of the analogy method.

Theoretical analysis links Fermat's and Hamilton's principles.

Abstract

A unique perspective approach based on an analogy method is presented to solve the ray equation in a model of a continuous inhomogeneous medium, which has a spherically symmetric distribution. Basically, in the standard undergraduate physics teaching, the curved ray path caused by refraction in a medium with a continuously varying refractive index has always been a relatively difficult problem to solve. The equation is usually expressed in terms of partial differential equations (PDEs), which cannot be solved by analytical methods. Based on the analogy method, this work proposes the correspondence between ray refraction in an established medium model and the inverse-square central force system, succinctly obtaining their relation equations mathematically. We also verify the correctness of the method by qualitative and quantitative analysis. In terms of theoretical validation, we analyse the relation between Fermat's principle and Hamilton's principle, which lays a theoretical foundation for the analogy method. In addition, ray paths in the medium model were also simulated by numerical calculations based on COMSOL Multiphysics, and the results are in perfect agreement with the conclusions.