Propagation factors of Hermite–Gaussian beams in turbulent atmosphere

Based on the extended Huygens–Fresnel integral and the second-order moments of the Wigner distribution function, an analytical formulae for the propagation factors (M2-factors) of coherent and partially coherent one-dimensional Hermite–Gaussian beams in a turbulent atmosphere are derived. Evolution properties of the M2-factor of the Hermite–Gaussian beam in a turbulent atmosphere are studied numerically in detail. Our results show that the M2-factor of the Hermite–Gaussian beam increases upon propagation in a turbulent atmosphere. The M2-factor of the Hermite–Gaussian beam with larger beam order (or lower coherence) increases slower that of the Hermite–Gaussian beam with smaller beam order (or higher coherence) in a turbulent atmosphere, which means that the Hermite–Gaussian beam with a larger beam order and lower coherence is less affected by a turbulent atmosphere. Our results will be useful in long-distance free-space optical communications.