A Wigner-distribution matrix for a stochastic medium

Wigner distribution functions W( ) describe the distribution of the local energy density of waves over the propagation directions ( spectrum) and the frequencies . "Transport equations" give the rate of change of these functions in the direction of the vector considered. The paper concerns a time dependent medium with a stochastic permittivity. Its Wigner functions fix a tensor the elements of which concern the product of two special electric-field components. After having derived wave equations for the corresponding correlation functions, a statistical averaging is performed, based on two assumptions: a normal distribution of the permittivity fluctuations, and a coherence scale much smaller than the mean free path for scattering. By a Fourier transform the resulting equations pass into corresponding ones for the Wigner tensor. The latter are simplified by forward-scattering approximations, while the medium fluctuations should be slow enough. The final equations thus obtained for the \´S indicate resonance effects interpretable in terms of Bragg reflections against periodic structures generated by the travelling waves contained in the spectrum of the fluctuations. The vectorial treatment involves transport equations in which all elements appear coupled. The neglect of all cross correlations between the field components reduces the main transport equations into a single one that has been presented in (M.S. Howe, Phil. Trans. Roy. Soc., 274A, 523-549, 1973), using a scalar treatment for a stochastic medium depending on a Helmholtz equation instead of our Maxwell equations.