Analytical solutions for the mean-square displacement derived from transport theory

In this paper, exact analytical solutions are derived for the second-order moment and the mean-square displacement based on the transport theory fluence that is caused by an arbitrary anisotropic point source, which is located at the origin of a three-dimensional coordinate system. In particular, the derivations are carried out as a function of the number of scattering events. The resulting formulas in the steady-state and time domain depend, apart from the scattering coefficient and partly the absorption coefficient, only on the anisotropy factor of the considered rotationally invariant scattering phase function. We additionally present the second-order moment and the mean-square displacement for the fluence of fluorescence light in the steady-state domain.

Figure 1

MSD ${\langle r^2\rangle }_n$ of the steady-state fluence ${\mathrm{\Phi }}_n\left(\mathbf{x}\right)$ as function of the scattering order.

Figure 2

Second-order moment $X_n\left(t\right)$ of the time-dependent fluence ${\mathrm{\Phi }}_n(\mathbf{x},t)$ for two different scattering orders.

Figure 3

MSD ${\langle r^2\left(t\right)\rangle }_n$ of the time-dependent fluence ${\mathrm{\Phi }}_n(\mathbf{x},t)$ for two different scattering orders.