Localization of Electromagnetic Waves in Three-Dimensional Fractal Cavities

Three-dimensional fractals called the Menger sponge, with a fractal dimension $D=\log 20/\log 3$, were fabricated from epoxy resin by stereolithography. Clear attenuation of both reflection and transmission intensity was observed at 12.8 GHz for a cubic specimen with an edge size of 27 mm that was constructed up to the third stage of the self-similar patterns. The electromagnetic field was found to be confined in the central part of the specimen at this frequency. The localization is not caused by the presence of a photonic band gap as in photonic crystals but should be attributed to a singular photon density of states realized in the fractal structure. This is the first report on such localization of electromagnetic waves in three-dimensional fractal cavities.