Wave structure function of an underwater optical Gaussian beam including attenuation

This paper presents a closed-form analytical derivation of the wave structure function (WSF) for an underwater optical Gaussian beam propagating through oceanic turbulence with attenuation caused by absorption and scattering. Using the extended first-order Rytov approximation together with the oceanic turbulence optical power spectrum (OTOPS) model, attenuation is incorporated into the propagation path integral, leading to compact hypergeometric-function solutions. Numerical results show that, at a propagation distance of 20 m, the Gaussian-beam WSF increases by 32.08%, 52.61%, and 65.33% for wavelengths of 400, 450, and 532 nm, respectively, when compared with no-attenuation scenarios, while at 40 m, these values increase to 55.76%, 80.36%, and 90.51%. These results demonstrate that attenuation significantly modifies the effective weighting of turbulence-induced phase perturbations as propagation distance increases, with shorter wavelengths exhibiting stronger proportional growth in the predicted WSF. The proposed model extends existing turbulence-only formulations and provides a unified analytical framework for underwater optical wireless communication systems under combined turbulence and attenuation effects.