Nonlinear Dependence of the Phase Screen Structure Function on the Atmospheric Layer Thickness

The phase structure function accumulated by two parallel rays after transmission through a layer of turbulent air is best known by a proportionality to the 5/3rd power of the lateral distance in the aperture, derived from an isotropic Kolmogorov spectrum of the refractive index. For a von-Karman spectrum of the refractive index, a dependence involving a modified Bessel function of the ratio of the distance over the outer scale is also known. A further standard proposition is a proportionality to the path length through the atmospheric layer. The manuscript modifies this factor through a refined calculation of an integral representation of the structure function. The correction establishes a sub-linearity as the lateral distance grows in proportion to the layer thickness; it is more important for large than for small outer scales.