مرکز منطقه ای اطلاع رساني علوم و فناوري - A Wigner-distribution matrix for a stochastic medium

Abstract :

Wigner distribution functions W( $\\vec{r},t;\\vec{k},\\omega$ ) describe the distribution of the local energy density of waves over the propagation directions ( $\\over\\rightarrow{k}$ spectrum) and the frequencies $\\omega$ . "Transport equations" give the rate of change of these functions in the direction of the $\\over\\rightarrow{k}$ vector considered. The paper concerns a time dependent medium with a stochastic permittivity. Its Wigner functions fix a tensor the elements $W_{ik}$ 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 $W_{ik}$ \´S indicate resonance effects interpretable in terms of Bragg reflections against periodic structures generated by the travelling waves contained in the $\\over\\rightarrow{k}\\omega$ spectrum of the fluctuations. The vectorial treatment involves transport equations in which all $W_{ik}$ 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.