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

Abstract

The predictions of the frequency-domain standard diffusion equation (SDE) model for light propagation in an infinite turbid medium diverge from the more complete ${\mathit{P}}_{1}$ approximation to the linear Boltzmann transport equation at intensity modulation frequencies greater than several hundred MHz. The ${\mathit{P}}_{1}$ 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 ${\mathit{P}}_{1}$ 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 ${\mathit{P}}_{1}$. 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 ${\mathit{P}}_{1}$ aproximation equation and the SDE. This analysis shows that the ${\mathit{P}}_{1}$ approximation provides a more accurate description of the data over this range of modulation frequencies. Some researchers have claimed that the ${\mathit{P}}_{1}$ approximation predicts that a light pulse should propagate with an average speed of c/ \ensuremath{\surd}3 in a thick turbid medium. However, an examination of the Green's function that we obtained from the frequency-domain ${\mathit{P}}_{1}$ approximation model indicates that a photon density wave phase velocity of c/ \ensuremath{\surd}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 ${\mathit{P}}_{1}$ 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/\ensuremath{\surd}3. \textcopyright{} 1996 The American Physical Society.