THESIS
2006

xvii, 123 leaves : ill. ; 30 cm

**Abstract**
In this thesis, we have studied the effects of weak localization (WL) on the dy-namics of wave propagations in mesoscopic random media both in the diffusive regime and at the Anderson localization transition. In the diffusion regime, we investigated the wave transport with WL in quasi-1D disordered waveguides and disordered slabs in the time domain. We solve the Bethe-Salpeter equation with recurrent scattering included in a manner that satisfies the Ward Iden-tity, from which we obtain a time-dependent diffusion constant, D(t). The WL effects are treated in the framework of self-consistent localization theory. The spatially-averaged WL is included in a frequency-dependent vertex function, which renormalizes the mean free path. We consider cylindrical disordered samples with reflecting...[

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In this thesis, we have studied the effects of weak localization (WL) on the dy-namics of wave propagations in mesoscopic random media both in the diffusive regime and at the Anderson localization transition. In the diffusion regime, we investigated the wave transport with WL in quasi-1D disordered waveguides and disordered slabs in the time domain. We solve the Bethe-Salpeter equation with recurrent scattering included in a manner that satisfies the Ward Iden-tity, from which we obtain a time-dependent diffusion constant, D(t). The WL effects are treated in the framework of self-consistent localization theory. The spatially-averaged WL is included in a frequency-dependent vertex function, which renormalizes the mean free path. We consider cylindrical disordered samples with reflecting side walls and open ends. By changing the ratio of the longitudinal dimension L and the radius R, a continuous transition can be made between two key experimental geometries: a quasi-1D geometry with L >> R, which is commonly employed in microwave experiments, and a slab with L << R, which is the typical optical geometry. For the quasi-1D geometry, we find D(t)/D

_{0} ≃ A - (2B/π

^{2}gτ

_{D})t for t << gτ

_{D}, where D

_{0} is the Boltzmann diffusion constant, τ

_{D} = L

^{2}/π

^{2}D

_{0} is the diffusion time and g is the dimen-sionless conductance. The constant part, A, is universal, depending only on g, while B is nonuniversal and depends upon L/ℓ

_{0} as well as g, where ℓ

_{0} is the scattering mean free path. In the limit of g >> 1 and L/ℓ

_{0} >> 1, our results coincide with the supersymmetry calculations by A. D. Mirlin. For moderate values of g, g ≥ 5, our results are in agreement with the recent microwave experiment by A. A. Chabanov et al. For slab geometry, we found that D(t) decays slowly in time and approaches a nearly constant value at long time, i.e. D̃(L)/D

_{0} ≃ 1 - 1.02/(kℓ

_{0})

^{2}+3.72ℓ

_{0}/L̃(kℓ

_{0})

^{2}. This is also in agreement with the optical transmission measurements in slab geometry. In the limit of L/ℓ

_{0} >> 1, D̃(L) approaches D

_{0}(1 - 1.02/(kℓ

_{0})

^{2}). This is close to the result of WL theory in the infinite medium.

At the Anderson localization transition, we have studied the scaling behav-iors of both static and dynamic wave propagations. Our results show that the classical waves may follow a different scaling behavior from that of electrons. For electrons, the effect of WL due to interference of recurrent scattering paths is limited within a spherical volume because of the presence of a finite dephas-ing length, L

_{Φ}. For classical waves, L

_{Φ} can be considered as infinite and it is the sample geometry that determines the contributions of the WL effects. Furthermore, the WL effects in both cubic and slab geometry have been found to be smaller than those in spherical geometry. As a result, the averaged static diffusion constant D(L) scales like ln(L)/L in cubic and slab geometry and the corresponding transmission follows

∝ ln L/L^{2}. These are in con-trast to the behaviors of D(L) ∝ 1/L and ∝ 1/L^{2} obtained previously for electrons or spherical samples. Furthermore, all of the static and dynamic transport quantities studied in slab geometry are found to follow the scaling behavior of D(L). We have also considered the position-dependent WL effects by using a plausible form of position-dependent diffusion constant D(z). The same scaling behavior is found, i.e., ∝ ln L/L^{2}.

Finally, we have studied the effects of WL in thin disordered slabs with internal reflections. It has been found in optical measurements that the appar-ent diffusion constant in thin samples decreases from its bulk value when the sample size is smaller than eight times of the mean free path. This anomalous diffusion behavior has puzzled researchers for many years. In this thesis, when we consider both the internal reflections and WL effects in the Bethe-Salpeter equation, our calculations indicate that the anomalous behavior found in pre-vious experiments can be achieved when the scattering strength is sufficiently strong by considering the renormalization of the extrapolation length due to WL effects.

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