Probability distribution of irradiance fluctuations of beam waves in a weakly turbulent atmosphere

The integral expression for the Nth-order moment of irradiance fluctuations of a finite beam wave traversing a weakly turbulent atmosphere is derived by the parabolic-equation method. The expression for the Nth-order moment, evaluated for the weak scintillation limit, is the same form as the Nth-order moment of the log-normal distribution. This expression provides a necessary condition for the log normality of the irradiance fluctuation of beam wave. For the plane- and spherical-wave limits, the result is valid at all distances ρ¯ from the axis of the beam less than the propagation distances z and the distribution function depends upon ρ¯. The values of the variance coincide with the results of the usual first-order perturbation theory, and log-normality also holds.