Engineering equations for the filamentation collapse distance in lossy, turbulent, nonlinear media

The propagation of high peak-power laser beams in real atmospheres has been an active research area for a couple of decades. Atmospheric turbulence and loss will induce decreases in the filamentation self-focusing collapse distance as the refractive index structure parameter and volume extinction coefficient, respectively, increase. This paper provides a validated analytical method for predicting the filamentation onset distance in lossy, turbulent, nonlinear media. It is based on a modification of Petrishchev's and Marburger theories. It postulates that the ratio of the peak power to critical power at range in turbulence is modified by a multiplicative, rather than additive, gain factor. Specifically, this new approach modifies the Petrishchev's turbulence equation to create the required multiplicative factor. This is necessary to create the shortened filamentation onset distance that occurs when a laser beam propagates through the cited nonlinear medium. This equation then is used with the Marburger distance and the Karr et al loss equations to yield the filamentation onset distance estimate in lossy, turbulent, nonlinear environment. Theory validation is done against two independent sets of computer simulation results. One comes from the NRL's HELCAP software and the other from MZA's Wave Train modeling software package. This paper also shows that once the zero-turbulence onset distance is set based on link loss, the addition of turbulence creates essentially the same PDFs at similar median distances for each loss case. This result had not been previously reported. This is the first quantitative comparison between closed form equations and computer simulation results characterizing filament generation in a lossy, turbulent, nonlinear medium.