Analytical model for structured light propagation through a turbulent atmosphere

We develop a straightforward analytical framework for the propagation of spatial light modes through a turbulent atmosphere. Built upon the split-step approach with the mode-based optical field representation, it directly assesses how turbulence-induced phase fluctuations deplete the optical power in the original mode and re-distribute it into neighboring spatial modes. Importantly, this power transfer scales linearly with the propagation distance in a uniform channel, yielding a simple solution for arbitrary distances in the form of a matrix exponential. The transfer rate is determined by the spatial spectral overlap between the turbulence spectrum and the acceptance spectrum for a pair of interacting spatial modes. The model predicts the average power in each spatial mode and is exact when a single mode strongly dominates all others. Our predictions show reasonably good agreement with simulations up to medium-to-strong turbulence levels. The model also confirms the scalings with mode order previously known as empirical observations.