Optimization of beam geometry for focusing through turbulence

Beam spread and beam wandering are the most perceptible effects of atmospheric turbulence on propagating laser beams. The width of the mean irradiance profile is typically used to characterize the beam spread. This so-called Long-Term (LT) statistic allows for a relatively simple theoretical description. The LT beam size is not a very practical measure of the beam spread because its measurements are sensitive to the movements of the source and detector, and to the large-scale variations of the refractive index that are not associated with turbulence. The Short-Term (ST) beam spread is measured relative to the instantaneous position of the beam center and is free of these drawbacks, but has not been studied as thorough as the LT spread. We use a Markov approximation-based theoretical model for the ST beam irradiance that is valid for the wide range of turbulent conditions. Additional approximations are invoked to allow introduction of the isoplanatic ST Point Spread Function (PSF). Unlike the LT PSF, the ST PSF depends on the overall beam geometry. Adjustments of the initial beam width and focal distance make it possible to increase the contribution of the LT beam spread that is attributed to the beam wander and minimize the ST beam size at the observation plane for any given turbulence level. Analytical calculations of the optimal beam geometry are presented for the simple case of the coherent Gaussian beam, and Kolmogorov turbulence. We present the results of direct numerical simulation of beam wave propagation that confirm the existence of the optimal beam geometry.