A Scaling Law for Truncated Gaussian Beams Propagating from Weak to Strong Turbulent Atmosphere

We have developed a scaling law from both theoretical and numerical perspectives to evaluate the far-field encircled radius of a truncated Gaussian beam propagating through atmospheric turbulence. First, a convolution-based approach is proposed to derive the scaling function. Second, the scaling coefficients in the formula are determined through numerical simulations. Finally, to enhance the evaluation accuracy under strong turbulence, a hybrid method combining the scaling law with an artificial intelligence (AI) model is developed. The results demonstrate that the scaling law provides reliable evaluations under weak-to-moderate turbulence conditions and yields reasonable estimates even in strong turbulence. The hybrid AI-scaling law method achieves superior accuracy specifically under strong turbulence. Furthermore, we find that the beam spreading may exhibit a saturation effect under strong turbulence conditions, which is similar to scintillation saturation.