Diffractals

Abstract

Diffractals are waves that have encountered fractals. Fractals are geometric objects with non-integral Hasudorff-Besicovitch dimension D; they have structure down to arbitrarily fine scales. Diffractals are a new wave regime characterised by a short-wave limit in which ever finer levels of structure are explored and geometrical optics is never applicable. The diffractal studied here is the wave psi (x,z) at distance z beyond a one-dimensional random phase screen that deforms an initially plane wavefront at z=0 into a random fractal curve h(x) with power law spectrum and dimension D (between 1 and 2), after which the wave propagates freely. Some averages of psi are calculated. These are ( psi ), ( psi (x,z) phi *(x+X,z)), the spectrum of psi and the spectrum of the intensity fluctuations and, most important, the second intensity moment I2 identical to ( mod psi mod 4). It is proved that the intensity fluctuations are non-Gaussian. A variety of scaling laws is derived, all involving D. For the 'Brownian' diffractal (D=1.5) all averages are expressed exactly in closed form.