Strong Resonance of Light in a Cantor Set

TL;DR

This paper investigates electromagnetic wave behavior in a fractal Cantor set, revealing strong resonance effects, wave localization, and high enhancement factors near resonant frequencies, with implications for wave control in complex structures.

Contribution

It introduces a transfer matrix approach and analyzes resonant states in higher-generation Cantor sets, highlighting their impact on wave localization and transmission properties.

Findings

Resonant states approach the real wave number axis in higher generations.

Wide reflection regions occur with sharp transmission peaks.

Wave amplitude is strongly enhanced and localized near resonant frequencies.

Abstract

The propagation of an electromagnetic wave in a one-dimensional fractal object, the Cantor set, is studied. The transfer matrix of the wave amplitude is formulated and its renormalization transformation is analyzed. The focus is on resonant states in the Cantor set. In Cantor sets of higher generations, some of the resonant states closely approach the real axis of the wave number, leaving between them a wide region free of resonant states. As a result, wide regions of nearly total reflection appear with sharp peaks of the transmission coefficient beside them. It is also revealed that the electromagnetic wave is strongly enhanced and localized in the cavity of the Cantor set near the resonant frequency. The enhancement factor of the wave amplitude at the resonant frequency is approximately $6/|\eta_\mathrm{r}|$, where $\eta_\mathrm{r}$ is the imaginary part of the corresponding resonant eigenvalue. For example, a resonant state of the lifetime $\tau_\mathrm{r}=4.3$ms and of the enhancement factor $M=7.8\times10^7$ is found at the resonant frequency $\omega_\mathrm{r}=367$GHz for the Cantor set of the fourth generation of length L=10cm made of a medium of the dielectric constant $\epsilon=10$.