Derivation of the correlation diffusion equation with static background and analytical solutions.

A new correlation diffusion equation has been derived from a correlation transport equation allowing one to take into account the presence of moving scatterers and static background. Solutions for the reflectance from a semi-infinite medium have been obtained (point-like and ring detectors). The solutions have been tested by comparisons with "gold standard" Monte Carlo (MC) simulations. These formulas suitably describe the electric field autocorrelation function, for Brownian or random movement of the scatterers, even in the case where the probability for a photon to interact with a moving scatterer is very low. The proposed analytical models and the MC simulations show that the "classical" model, often used in diffuse correlation spectroscopy, underestimates the normalized field autocorrelation function for increasing correlation times.