Investigation of branch-point density using traditional wave-optics techniques

This study uses traditional wave-optics techniques, such as the split-step beam propagation method with angular-spectrum propagation, to explore the number of branch points as a function of the numerical grid size (i.e., the branch-point density) with increasing strengths of turbulence. To help quantify the strength of the turbulence, the analysis makes use of the log-amplitude variance for a spherical wave. Given a point-source beacon and horizontal-propagation paths, this parameter gives a straightforward gauge for the amount of scintillation, and therefore the number of branch points in the phase function. As such, the goal throughout is to investigate the branch-point density in terms of a two-step process. The first step is to increase the numerical grid size to have an ever increasing number of grid points for a given instance of turbulence; particularly, with a log-amplitude variance for a spherical wave above 0.25, because this is where branch points start to arise in the phase function. In turn, the second step is to utilize a Monte-Carlo averaging scheme with the resultant branch-point density for many instances of turbulence and turbulence strengths. Using this two-step process, the initial results show that the branch-point density grows without bound. Such results seem unphysical and could have direct implications for wave-optics studies that involve wavefront sensing in the presence of deep turbulence.