Gigahertz photon density waves in a turbid medium: Theory and experiments.

The predictions of the frequency-domain standard diffusion equation (SDE) model for light propagation in an infinite turbid medium diverge from the more complete [Formula Presented] approximation to the linear Boltzmann transport equation at intensity modulation frequencies greater than several hundred MHz. The [Formula Presented] approximation is based on keeping only the terms l=0 and l=1 in the expansion of the angular photon density in spherical harmonics, and the nomenclature [Formula Presented] approximation is used since the spherical harmonics of order l=1 can be written in terms of the first order Legendre polynomial, which is traditionally represented by the symbol [Formula Presented]. Frequency-domain data acquired in a quasi-infinite turbid medium at modulation frequencies ranging from 0.38 to 3.2 GHz using a superheterodyning microwave detection system were analyzed using expressions derived from both the [Formula Presented] aproximation equation and the SDE. This analysis shows that the [Formula Presented] approximation provides a more accurate description of the data over this range of modulation frequencies. Some researchers have claimed that the [Formula Presented] approximation predicts that a light pulse should propagate with an average speed of c/ √3 in a thick turbid medium. However, an examination of the Green’s function that we obtained from the frequency-domain [Formula Presented] approximation model indicates that a photon density wave phase velocity of c/ √3 is only asymptotically approached in a regime where the light intensity modulation frequency aproaches infinity. The Fourier transform of this frequency-domain result shows that in the time domain, the [Formula Presented] approximation predicts that only the leading edge of the pulse (i.e., the photons arriving at the detector at the earliest time) approaches a speed of c/√3. © 1996 The American Physical Society