Rigorous solution for electromagnetic waves propagating through pre-Cantor sets

We study one-dimensional systems constructed from a segment by employing the Cantor-set rule up to an arbitrary stage of self-similar patterns. The rigorous expression of the transfer matrix to describe the electromagnetic waves propagating through them is presented. As displayed by numerical demonstration, the transmission spectra change drastically with the increase of the stage. At rather high stages the periodicity hidden in the self-similarity comes out. This is the first theoretical description of sharp attenuation in the transmission spectrum of electromagnetic waves propagating through a fractal medium.