Theory of irradiance distribution function in turbulent media—cluster approximation

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Abstract
The equations of the correlation functions of irradiance in turbulent media are formally solved for arbitrary order by use of an operator method. A cluster approximation is then applied to the fourth and higher order moments of irradiance to express those in terms of the lower order moments. The values of moments predicted by this approximation show a very good agreement with the recent experimental values observed up to the fourth order moment of irradiance. The irradiance distribution function is then analytically derived from the obtained expression of the moments of irradiance and is found to have the following features: (1) It is close to the Gaussian distribution with respect to the logarithm of irradiance, (2) it has a threshold value for irradiances giving nonvanishing probability, and (3) it has a very small but sharp distribution of the form of the δ function at the threshold value. The spectrum of medium as well as the conditions of the initial wave, e.g., of whether it is a plane wave or is a beam wave do not directly enter in the expression but appear only through the first three moments of irradiance. The condition of applicability of the cluster approximation is also discussed in some details based on the Kolmogorov spectrum of turbulence.