Two-scale method in the theory of scattering by fractal structures: One-dimensional regular problems.

Abstract

In a recent paper one of the present authors [V. V. Konotop, Phys. Rev. A 44, 1352 (1991)] put forth an analytical approach for the description of one-dimensional scattering by a Weierstrass-like layer. The method consists of splitting the fractal permittivity into two effective parts, the scales of which are long and short in comparison with a wavelength. The dependence of a splitting point on an incident wavelength is given. In the present paper we generalize such an approach, called a two-scale method (TSM) for a wide class of structures having irregularity, and provide complete analytical and numerical investigations of two particular cases: layers with Weierstrass-type and singular permittivities. A good correspondence between analytical predictions and numerical solutions of the corresponding Riccati equation, as well as qualitative agreement with outcomes published elsewhere [D. C. Jaggard and X. Sun, IEEE Trans. AP-37, 1511 (1989)], is observed. In all cases the scattering data have a strongly pronounced resonant character that is well described by the WKB approximation. Increasing the box dimension leads to growth of the internal scattering and, as a consequence, to the inapplicability of the TSM. To investigate the physical nature of this phenomena we study both the above problems in the Born approximation as well. The main feature of fractal layers consists of a noninteger-power-law dependence of scattering data on slab length. Application of the TSM allows one to find a number of harmonics (in the trigonometric-series expansion), enough to provide adequate numerical analysis of fractal layers. Validity limits of the theory developed are stated. They turn out to be approximately the same as those for the WKB approximation for the effective long-scale part of the fractal permittivity.