Intensity fluctuations in a multiple scattering medium. Solution of the fourth moment equation

The intensity fluctuations arising when a wave propagates through a medium containing weak random inhomogeneities of refractive index are described by a parabolic equation for the fourth moment of the wave field. The present paper obtains an analytical solution for this equation when an initially plane wave is normally incident on a half-space containing such a medium. The solution is in the form of a multiple convolution and is valid even for multiple scatter. The multiple convolution is evaluated to yield an expression for the spatial frequency spectrum of intensity fluctuations. This spectrum is valid for any autocorrelation function of refractive index irregularities. Media with a Gaussian autocorrelation function and a Kolmogorov-type autocorrelation function of refractive index irregularities are treated as examples. Finally the spatial frequency spectra of intensity fluctuations are integrated to give the scintillation index curves as functions of distance of propagation in the medium. The regions of validity of the different approximations are discussed and the limits of error associated with the solutions are given.