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

摘要

The predictions of the frequency-domain standard diffusion equation (SDE) model for light propagation in an infinite turbid medium diverge from the more complete P1 approximation to the linear Boltzmann transport equation at intensity modulation frequencies greater than several hundred MHz. The P1 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 P1 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 P1. 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 P1 aproximation equation and the SDE. This analysis shows that the P1 approximation provides a more accurate description of the data over this range of modulation frequencies. Some researchers have claimed that the P1 approximation predicts that a light pulse should propagate with an average speed of c/ sqrt 3 in a thick turbid medium. However, an examination of the Green's function that we obtained from the frequency-domain P1 approximation model indicates that a photon density wave phase velocity of c/ sqrt 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 P1 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/ sqrt 3.