Wave Transmission through a One-Dimensional Cantor-Like Fractal Medium

Abstract

We consider the transmission of plane waves through a one-dimensional medium whose material parameters differ from those of free space on segments which form a Cantor set. By employing self-similarity, it is shown that the transmission coefficient is obtained as a solution of an infinite recursive relation. The latter is solved numerically by taking the small fractal length limit as an 4nitialn condition. The results show that the transmission coefficient is not monotonic with increasing slab thickness. For certain values of the normalized (over the wavelength) thickness the fractal medium is practically transparent and for others it totally reflects the incident wave. A physical interpretation of these effects is suggested.

Peculiarities of wave scattering by fat fractals

“…The qualitative behavior of R =~r z~is similar to that observed in the case of a Cantor-like fractal [1,2]: At all values k one can see that the regions of the total reflection alternate with transparency windows. Sometimes the last ones are very narrow and are placed rather 17 '' The constant here is of order 1.…”

“…In Refs. [1,3,5] problems in which slabs do not have the smallest scale were examined. In the last case the fractal behavior was observed in the small-wavelength limit.…”

“…Strictly speaking, in this sense the medium has to be finite in extent. This restriction can be overcome with the help of the appropriate combinations of numerical methods with the special symmetry of the structure (if any) [1,2]. Continuing (2) with p being a yardstick, i.e.…”

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

“…Wave properties in fractal [1,2,3,4,5,6,7,8,9,10,11] and quasi-periodic [12,13,14,15,16,17] structures in one dimension have been of theoretical and practical interest over the past two decades. They are typical examples of self-similar structures, and physical properties peculiar to them have been explored.…”

Elastic scattering by deterministic and random fractals: Self-affinity of the diffraction spectrum

“…Scattering from such fractals, as well as randomized versions of them, will be the subject of this paper. The few examples include Berry's [11] above-mentioned work; further, mainly in optics, calculations on wave transmission [19,20] and Fraunhofer diffraction [21][22][23][24], on Cantor-bars, Koch fractals or Sierpinski-carpet like media; in x-rays, numerical calculations on scattering by a Menger sponge [25], and measurements on diffraction from Cantor lattices [26]. The most extensive treatment is probably due to Allain and Cloitre [27][28][29].…”