Review of the theory of the irradiance distribution function in turbulent media with a particular emphasis on analytical methods


  • K. Furutsu


The recent theories on the irradiance distribution function of waves in turbulent media are reviewed with a particular emphasis on their physical and mathematical bases. The Rice-Nakagami distribution is first investigated in connection with the analytical singularity of its characteristic function and the derivation of the latter. The distribution function P(I) of the irradiance I is then represented in terms of an integral over the order variable v of 〈Iv〉, v = 0, 1, 2, .... Here, the situation is rather similar to the Watson-Sommerfeld transformation for waves diffracted by a spherical surface. It turns out that the moment of irradiance 〈Iv〉 precisely provides the characteristic function of the log-irradiance E = log I with v as the variable of the characteristic function. This fact inevitably leads to the cumulant expansion of 〈Iv〉, and further to the distribution which is close to the log-normal distribution with a threshold value for irradiances giving nonvanishing probability. It is then shown that the same irradiance distribution can also be derived explicitly from the moment equations of wave functions (which are essentially based on the forward scattering approximation) as the result of a cluster approximation. In the special case of a gaussian turbulence spectrum, the exact solutions of the moment equations can be obtained for all orders and they lead to the Rice-Nakagami distribution with respect to the logarithm of irradiance; this distribution precisely describes the distribution when the wave beam is purely wandering or is in the state of “spot dancing,” as is expected from the physical model of the gaussian spectrum. Conversely, the latter fact strongly suggests that the moment equations are highly reliable for all orders to apply to various problems. The cluster approximation is also discussed with regard to its condition of applicability.