Abstract

The fractal concept, introduced by Mandelbrot in the 1970s, complements Euclidean geometry when used to describe irregular or variegated structures such as those that occur in nature. This concept embodies ideas of both dilation symmetry and nondifferentiability and introduces a fractional dimension. The fractional dimension, denoted the fractual dimension D, exceeds the topological dimension d of the structure of interest. For example, the fractal dimension can vary from one for smooth curves to two for irregular area-filling curves while the topological dimension remains unity. These concepts can be generalized to higher dimensional structures where the topological dimension is two or three. Here we introduce the fractal concept and review its application to several wave problems in scattering, diffraction, and reflection. Of particular interest are the following topics that have been under investigation at the University of Pennsylvania: scattering from fractal fibers; diffraction by fractal screens and surfaces; applications to turbulence; reflection from fractal structures. The models for these applications are based on the concept of bandlimited fractal structures which provide the appropriate tool for analysis.

© 1987 Optical Society of America