In this work an analysis of transient wave propagation in forward scattering random media is presented. The analysis is based on evaluation of the two-frequency mutual coherence function, which is an important quantity in itself since it provides a measure of the coherence bandwidth. The coherence function is calculated by using the path integral technique; specifically, by resorting to a cumulant expansion of the path integral. In contrast to the formulas available in the literature, the solution obtained is not limited by the strength of disorder and applies equally well to both dispersive and nondispersive media, with arbitrary spectra of inhomogeneities. For the regime of weak scattering (or relatively short propagation distances) the first cumulant gives an excellent approximation coinciding with the results obtained earlier in a particular case of the Kolmogorov turbulence by solving the corresponding differential equation numerically. In the regime of strong scattering (long distances), which to our knowledge has not been covered previously, our solution demonstrates a different type of scaling dependence. It is shown that, even for power spectra with fractal behavior in a wide range of spatial frequencies, the coherence function is very sensitive to fine details of the spectrum at both small and large spatial scales. Using the cumulant expansion, the temporal moments of the pulsed wave propagating in a random medium are also considered. It is found that the temporal moments of the pulse are determined exactly by accounting for a corresponding number of the cumulants. In particular, the average time delay of the pulse is determined by the first cumulant, and the pulse width is obtained by accounting for the first two cumulants. Although the consideration of the problem is based on the model of a continuous medium, the results are also applicable to wave propagation in media containing discrete particles scattering predominantly in the forward direction.
An original approach to the description of classical wave localization in weakly scattering random media is developed. The approach accounts explicitly for the correlation properties of the disorder, and is based on the idea of spectral filtering. According to this idea, the Fourier space (power spectrum) of the scattering potential is divided into two different domains. The first one is related to the global (Bragg) resonances and consists of spectral components lying within a limiting sphere of the Ewald construction. These resonances, arising in the momentum space as a result of a self-averaging, determine the dynamic behavior of the wave in a typical realization. The second domain, consisting of the components lying outside the limiting sphere, is responsible for the effect of local (stochastic) resonances observed in the configuration space. Combining a perturbative path-integral technique with the idea of spectral filtering allows one to eliminate the contribution of local resonances, and to distinguish between possible stochastic and dynamical localization of waves in a given system with arbitrary correlated disorder. In the one-dimensional (1D) case, the result, obtained for the localization length by using such an indirect procedure, coincides exactly with that predicted by a rigorous theory. In higher dimensions, the results, being in agreement with general conclusions of the scaling theory of localization, add important details to the common picture. In particular, the effect of the high-frequency localization length saturation is predicted for 2D systems. Some possible links with the problem of wave transport in periodic or near-periodic systems (photonic crystals) are also discussed.
An original approach is developed for the description of spectral coherence and time-domain transport of wave fields scattered in random media. This approach accounts explicitly for the correlation properties of the disorder and is universal with respect to the dimensionality of the system. Specifically, a two-frequency mutual coherence function is evaluated by using a procedure of embedding the initial Helmholtz equation into an auxiliary problem of a directed wave propagating in a higher-dimensional space. The resulting Schrödinger-like equation is solved perturbatively by means of a cumulant path integral technique. Mean intensity profiles and temporal moments of a narrowband wave packet scattered in a random medium are calculated by using the Fourier transformation of the coherence function. The theory describes the ballistic to diffusive transition in wave transport, and is consistent with experimental results. Since the coherence function is expressed via an arbitrary form power spectrum, the results obtained open a new avenue for studying wave transport in anisotropic and/or fractally correlated systems.
A theoretical model is proposed to describe narrowband pulse dynamics in two-dimensional systems with arbitrary correlated disorder. In anisotropic systems with elongated cigarlike inhomogeneities, fast propagation is predicted in the direction across the structure where the wave is exponentially localized and tunneling of evanescent modes plays a dominant role in typical realizations. Along the structure, where the wave is channeled as in a waveguide, the motion of the wave energy is relatively slow. Numerical simulations performed for ultra-wide-band pulses show that even at the initial stage of wave evolution, the radiation diffuses predominantly in the direction along the major axis of the correlation ellipse. Spectral analysis of the results relates the long tail of the wave observed in the transverse direction to a number of frequency domain "lucky shots" associated with the long-living resonant modes localized inside the sample.
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