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Chapter 9:
Wave Interference and Modes in Random Media
Mass and energy are transported via waves. These waves are quantum mechanical for classical particles, such as electrons, and classical for quantum mechanical particles, such as photons. For particles that do not mutually interact, transport through disordered media reduces to the study of wave scattering by inhomogeneities in the phase velocity of a medium. Examples of disturbances within uniform or periodic media are atomic dislocations in resistors, molecules in the atmosphere, or dielectric fluctuations in composite media. When the wave is multiply scattered, but scattering is sufficiently weak that the wave returns only rarely to a coherence volume within the sample through which it has passed, average transport may be described by particle diffusion. However, the wave nature is still strongly exhibited in fluctuations on a wavelength scale and in the statistics of transport. Though details of the scattering process depend on the type of waves and the specific environment, many essential characteristics of wave transport on length scales greater than both the wavelength λ and the transport mean free path ℓ are strikingly similar. Thus, for example, in the limit of particle diffusion, electronic conductance follows Ohm's law, and optical transmission through clouds and white paint correspondingly falls inversely with thickness. In this review, we consider features of multiply scattered waves in samples in which the wave is temporally coherent throughout the sample. This is readily achieved for classical wave scattering from static dielectric structures, but is obtained only for electrons interacting with restless atoms at ultralow temperatures in mesoscopic samples intermediate in size between the microscopic atomic scale and the macroscopic scale. The superposition of randomly scattered waves in static disordered systems produces a random spatial pattern of field or intensity referred to as the speckle pattern because of its grainy appearance, as shown schematically in Fig. 9.1. The speckle pattern at different frequencies provides a complex fingerprint of the interaction of a wave with the sample. However, in general, it is not possible to infer the internal structure of a body, even from a complete set of such patterns for all incident wave vectors. Indeed, even the forward problem of calculating the speckle pattern from a given structure for 3D systems cannot be solved at present, except for samples with dimensions considerably larger than the wavelength scale. Nonetheless, essential elements of a description of wave transport can be inferred from the statistics of the speckle pattern of radiation scattered from ensembles of random samples. Here, we examine the statistics of random transmission variables within a random ensemble of sample configurations, as well as the statistics of the evolution of the speckle pattern as a whole.
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