Long-Range Correlations in the Speckle Patterns of Directed Waves

Abstract

We develop a general method for calculating statistical properties of the speckle pattern of coherent waves propagating in disordered media. In some aspects this method is similar to the BoltzmannLangevin approach for the calculation of classical fluctuations. We apply the method to the case where the incident wave experiences many small angle scattering events during propagation, but the total angle change remains small. In many aspects our results for this case are different from results previously known in the literature. The correlation function of the wave intensity at two points separated by a distance r, has a long range character. It decays as a power of r and changes sign. We also consider sensitivities of the speckles to changes of external parameters, such as the wave frequency and the incidence angle.PACS numbers: 72.15. Rn, 73.20.Fz, In this article we consider statistical properties of waves propagating trough a disordered medium and described by the stationary (scalar) wave equation,where n(r) is the index of refraction assumed to be a random Gausian function. This problem is relevant for a variety of important physical situations, ranging from electromagnetic waves propagating through the interstellar space or the atmosphere, to electron transport in disordered conductors. The wave density, I(r) = |ψ(r)| 2 , exhibits sample specific random fluctuations (speckles) due to interference of waves traveling along different paths. The statistical properties of speckles have been discussed in the past. The problem can be characterized by several characteristic lengths: the propagation distance, Z, the elastic mean free path, ℓ and the transport length ℓ tr ∼ ℓ/θ 0 , which is the typical distance for backscattering. Here θ 0 ∼ kξ is the typical scattering angle on the distance ℓ, and ξ is the correlation length of n(r

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