Correlations in coherent multiple scattering

We present details of calculations and a comprehensive review on the various correlation functions for coherent electronic or classical wave transmission through a disordered scattering medium, in the diffusive limit λ {left double angle bracket} l^* {left double angle bracket} L_φ, where λ is the wavelength, l^* is the transport mean free path, and L_φ is the phase coherence length. The connection between mesoscopic conductance fluctuations in disordered metals and speckle pattern fluctuations for classical waves is reviewed. The short-range correlation function C⁽¹⁾ is shown to be responsible for the conventional speckle pattern fluctuations and the so called "memory effect". The long-range correlation function C⁽²⁾ is shown to dominate in the correlations and fluctuations of the total transmission coefficient. The "infinite range" background correlation function C⁽³⁾ is shown to be related to the universal conductance fluctuations in disordered mesoscopic conductors. We also discuss many generalized correlation functions such as those for frequency correlations, spatial correlations, correlations in Fabry-Perot interferometer devices, and in reflection geometries. A special kind of correlation function, corresponding to the sensitivity of speckle pattern intensities to the motion of impurities, will also be reviewed. Finally, the equivalence between the angular correlation functions and the auto-correlation functions in "diffusing wave spectroscopy" will be discussed.