The propagation of an electromagnetic wave in a one-dimensional fractalobject, the Cantor set, is studied. The transfer matrix of the wave amplitudeis formulated and its renormalization transformation is analyzed. The focus ison 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, wideregions of nearly total reflection appear with sharp peaks of the transmissioncoefficient beside them. It is also revealed that the electromagnetic wave isstrongly enhanced and localized in the cavity of the Cantor set near theresonant frequency. The enhancement factor of the wave amplitude at theresonant frequency is approximately $6/|\eta_\mathrm{r}|$, where$\eta_\mathrm{r}$ is the imaginary part of the corresponding resonanteigenvalue. For example, a resonant state of the lifetime$\tau_\mathrm{r}=4.3$ms and of the enhancement factor $M=7.8\times10^7$ isfound at the resonant frequency $\omega_\mathrm{r}=367$GHz for the Cantor setof the fourth generation of length L=10cm made of a medium of the dielectricconstant $\epsilon=10$.
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