Long distance propagation of wave beams in paraxial regime

This paper concerns the propagation of high frequency wave-beams in highly turbulent atmospheres. Using a paraxial model of wave propagation, we show in the long-distance weak-coupling regime that the wavefields are approximately described by a complex Gaussian field whose scintillation index is unity. This provides a model of the speckle formation observed in many practical settings. The main step of the derivation consists in showing that closed-form moment equations in the It\^o-Schr\"odinger regime are still approximately satisfied in the paraxial regime. The rest of the proof is then an extension of results derived in [Bal, G. and Nair, A., arXiv:2402.17107.]