Generalized diffusion theory for radiative transfer in fully anisotropic scattering media

Abstract

A generalized anisotropic-diffusion framework is developed for transport problem in media described by a tensorial scattering coefficient and a scalar Henyey--Greenstein asymmetry factor. In this regime the classical similarity relation between scattering and transport parameters fails, and each principal diffusion coefficient depends on all components of the microscopic scattering rate. Explicit expressions are derived for the direction-averaged mean free path, the diagonal elements of the diffusion tensor, and boundary condition lengths via rapidly convergent spherical-harmonics expansions, along with open-source implementations. The resulting predictions are validated against anisotropic Monte Carlo simulations, showing excellent agreement across broad ranges of structural anisotropy and phase-function asymmetry factors. The theory provides a compact, general route connecting microscopic anisotropic scattering to macroscopic diffusion coefficients and boundary conditions in bounded geometries.

Figures (6)

• Figure 1: Schematic illustration of the time evolution of the diffuse intensity pattern at the surface of an anisotropic slab following excitation by a point-like pulsed source. Direction-dependent scattering coefficients ($\mu_{\text{s}, x}$, $\mu_{\text{s}, y}$, $\mu_{\text{s}, z}$) lead to anisotropic spreading of light, producing elongated diffusion profiles at increasing time delays. The coordinate system shown defines the reference frame used throughout this work. Adapted from pini2024experimental.
• Figure 2: Ratios between the diffusion tensor elements and their corresponding simplistic counterparts for different values of $\mu_{\text{s}, y}/\mu_{\text{s}, x}$ and $\mu_{\text{s}, z}/\mu_{\text{s}, x}$, with $\mu_{\text{s}, x}$ fixed. This highlights the systematic bias introduced when anisotropy is ignored in diffusion-based parameter estimation.
• Figure 3: Representation of the angular distribution of the radiance $P(\vu{s})$ in the diffusive limit for different anisotropic scattering configurations.
• Figure 4: (a) Ratios between the diffusion tensor elements and their corresponding simplistic approximations as a function of the asymmetry factor $g$. Solid colored curves denote the analytical predictions, while black dashed curves show anisotropic Monte Carlo results. Shaded areas indicate $2\sigma$ confidence intervals. Results are shown for $4\mu_{\text{s}, x}=2\mu_{\text{s}, y}=\mu_{\text{s}, z}$, no free parameters are used in the analytical curves. The gray curve represents the isotropic case. (b) Corresponding ratios for the equivalent isotropic source position $z_0$. The light blue curve representing the previously proposed approximation alerstam2014anisotropic$z_0=\ell_z^\ast$ is shown for comparison.
• Figure 5: (a) Steady-state transmittance $T(x, y)$ from a 10-thick slab with $\mu_{\text{s}, x} = \qty{2.5}{\per\milli\meter}$, $\mu_{\text{s}, y} = \qty{7.5}{\per\milli\meter}$, $\mu_{\text{s}, z} = \qty{10}{\per\milli\meter}$, $n = 1.3$, $\mu_\text{a}=\qty{2e-6}{\per\milli\meter}$, and $g=0$. Data are plotted as log-scale contour lines marking consecutive decades. (b, c, d) Time-resolved transmittance at different locations $(x_0, y_0)$ on the slab's output surface, identified by the markers in panel (a). The prediction from anisotropic theory (dashed) is compared against the results of Monte Carlo simulations with e12 trajectories.