Wave propagation through Cantor-set media: Chaos, scaling, and fractal structures

TL;DR

This paper analyzes wave propagation through Cantor-set media using renormalization-group methods, revealing chaotic behavior, scaling laws, and conditions for complete transmission or reflection.

Contribution

It introduces a novel analysis of wave transmission in fractal media, linking chaos, scaling, and fractal structures with analytical and numerical results.

Findings

Transmission coefficients follow the logistic map at specific wave numbers.

Chaotic dependence of transmission on system parameters is observed.

Scaling behavior near complete transmission is analytically derived and numerically confirmed.

Abstract

Propagation of waves through Cantor-set media is investigated by renormalization-group analysis. For specific values of wave numbers, transmission coefficients are shown to be governed by the logistic map, and in the chaotic region, they show sensitive dependence on small changes of parameters of the system such as the index of refraction. For other values of wave numbers, our numerical results suggest that light transmits completely or reflects completely by the Cantor-set media ${\rm C}_{\infty}$. It is also shown that transmission coefficients exhibit a local scaling behavior near complete transmission if the complete transmission is achieved at a wave number $\kappa=\kappa^*$ with a rational $\kappa^*/\pi$. The scaling function is obtained analytically by using the Euler's totient function, and the local scaling behavior is confirmed numerically.