Transmission Properties of Electromagnetic Wave in Pre-Cantor Bar: Scaling and Double-Exponetiality

Abstract

Pre-Cantor bar, the one-dimensional fractal media, consists of two kinds of materials. Using the transmission-line theory we will explain the double-exponential behavior of the minimum of the transmittance as a function of the stage number n, and obtain formulae of two kinds of scaling behaviors of the transmittance. From numerical calculations for n=1 to 5 we will find that the maximum of field amplitudes of resonance which increases double-exponentially with n is well estimated by the theoretical upper bound. We will show that after sorting field amplitudes for resonance frequencies of the 5th stage their distribution is a staircase function of the index.