Wave transport through thin slabs of random media with internal reflection: Ballistic to diffusive transition

The static and dynamic properties of wave transport through thin slabs of random media in the presence of internal reflection are investigated by performing first-principles calculations. These results are compared with results from time-independent and time-dependent diffusion equations, respectively, where the effects due to internal reflection are incorporated into an average extrapolation length in the boundary conditions. For the static properties, we find an abrupt transition from ballistic to diffusive behavior when sample thickness is about three mean free paths, i.e., L ≈ 3l. The diffusion approximation is valid when L>31, independent of the amount of internal reflection. For the dynamic properties, both the peak arrival time at short times and the diffusion constant at long times of the transmitted pulse indicate that there is a region of anomalous diffusion when 3l<L<L _C. The diffusion constant in this region increases with decreasing L. It also increases with the amount of internal reflection. The physical origin of the existence of such an anomalous region is the resonance-induced wave focusing effect. Due to the presence of internal reflection, the wave energy tends to concentrate in the forward direction at output boundary. It makes direction randomization difficult in the scattered waves. A similar wave focusing effect has been found in resonant tunneling systems of electrons in the presence of elastic scattering. The diffusion approximation is valid when L>L_C. The value of L_c is about ten times the average extrapolation length, i.e., L ≈ 10z_e, where z_e is a fast increasing function of the amount of internal reflection.