Autocorrelations of random fractal apertures and phase screens

TL;DR

This paper introduces a new mathematical representation for random fractal apertures and phase screens, deriving simple formulas for their correlations and demonstrating power-law scaling, with applications in optics and related fields.

Contribution

It presents a novel product representation for random fractal apertures and phase screens, enabling straightforward derivation of correlation functions and fractional Brownian behavior.

Findings

Derived a closed-form expression for ensemble-averaged correlations.

Established power-law scaling at short distances.

Provided methods for constructing objects with fractional Brownian behavior.

Abstract

We introduce a new product representation for general random binary fractal apertures defined by removing voids from Euclidean space, and use it to derive a simple closed-form expression for ensemble-averaged correlations. Power-law scaling at short distance follows almost immediately. Similar techniques provide easy constructions of objects with fractional Brownian short-distance behavior for phase screens and other applications.