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 P-1 approximation to the linear Boltzmann transport equation at intensity modulation frequencies greater than several hundred MHz. The P-1 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 P-1 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 P-1. 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 P-1 approximation equation and the SDE. This analysis shows that the P-1 approximation provides a more accurate description of the data over this range of modulation frequencies. Some researchers have claimed that the P-1 approximation predicts that a light pulse should propagate with an average speed of c/root 3 in a thick turbid medium. However, an examination of the Green's function that we obtained from the frequency-domain P-1 approximation model indicates that a photon density wave phase velocity of c/root 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 P-1 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/root 3.