Systematic corrections to quadratic approximations for power-law structure functions: the δ expansion

Abstract

A new perturbation expansion for optical propagation in turbulence is presented. The method consists of expanding the power of the (Markov) refractive-index structure function about the trivial value 2. By providing a method of obtaining systematic corrections to results obtained by using a quadratic approximation to the structure function, the expansion permits quantitative estimation of the errors in these results. It is shown that this expansion necessarily introduces an arbitrary length, even in zeroth order (quadratic approximation). This length may be adjusted to improve the convergence of the expansion, thereby giving insight into the most-important-length scale of the problem. Essential differences are noted between optical processes in which the effects of overall wave-front tilts cancel and processes in which they do not. In the latter, the δ expansion gives (in low order) excellent approximations for all turbulence strengths, and the most important length decreases in increasing turbulence. In a tilt-canceling process, however, only moderate improvement over a weak-turbulence expansion is obtained, and the most important length appears to increase with the strength of the turbulence.