Analytical study on the upward laser beam propagation in the turbulent atmosphere

The upward laser beam propagation in the turbulent atmosphere is studied analytically, where the nonlinear self-focusing, the atmospheric turbulence, and the atmospheric extinction effects are all taken into account. The analytical propagation expression of intensity is derived. The initial beam defocusing and the optimal beam power are two methods to improve the beam quality at the target. The analytical expressions of the focal length considering the initial beam defocusing and the corresponding target intensity are derived. Moreover, the analytical expressions of optimal beam power and the corresponding target intensity are also derived, and the target intensity is independent of the atmospheric extinction. In particular, the criterion for which method (the initial beam defocusing or the optimal beam power) to use is derived to optimize the target beam quality. The choice of the two methods depends on the laser beam parameters and the turbulent atmosphere parameters.