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

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