Wave transport in two-dimensional random media: the ballistic to diffusive transition and the extrapolation length

By using a first-principles approach based on the Bethe-Salpeter equation, we study the behavior of wave propagation through a two-dimensional random slab as a function of thickness, L , in the region where L is much smaller than the localization length. A general two-dimensional vertex function for the ladder diagrams is derived from the Ward identity. We calculate both the static and the time-resolved transmitted intensities as functions of L/l , where l is the mean free path. When L is comparable to l , we study the ballistic to diffusive transition. A sharp crossover is observed when L(c) approximately = 6l, which is significantly larger than the crossover thickness of L(c) approximately = 3l found in three dimensions. When L>>l , we obtain the extrapolation length in two dimensions, i.e., z2De approximately = 0.82l, which is noticeably larger than the previously used value of z2De = pi/4 obtained by an analytical approach.