Exact Analytical Solution of the One-Dimensional Time-Dependent Radiative Transfer Equation with Linear Scattering

Abstract

The radiative transfer equation (RTE) is a cornerstone for describing the propagation of electromagnetic radiation in a medium, with applications spanning atmospheric science, astrophysics, remote sensing, and biomedical optics. Despite its importance, an exact analytical solution to the RTE has remained elusive, necessitating the use of numerical approximations such as Monte Carlo, discrete ordinate, and spherical harmonics methods. In this paper, we present an exact solution to the one-dimensional time-dependent RTE. We delve into the moments of the photon distribution, providing a clear view of the transition to the diffusion regime. This analysis offers a deeper understanding of light propagation in the medium. Furthermore, we demonstrate that the one-dimensional RTE is equivalent to the Klein-Gordon equation with an imaginary mass term determined by the inverse reduced scattering length. Contrary to naive expectations of superluminal solutions, we find that our solution is strictly causal under appropriate boundary conditions, determined by the light transport problem. We validate the found solution using Monte Carlo simulations and benchmark the performance of the latter. Our analysis reveals that even for highly forward scattering, dozens of random light scatterings are required for an accurate estimate, underscoring the complexity of the problem. Moreover, we propose a method for faster convergence by adjusting the parameters of Monte Carlo sampling. We show that a Monte Carlo method sampling photon scatterings with input parameters $(μ_s,g)$, where $μ_s$ is the inverse scattering length and $g$ is the scattering anisotropy parameter, is equivalent to that with $(μ_s(1-g)/2,-1)$. This equivalence leads to a significantly faster convergence to the exact solution, offering a substantial improvement of the Monte Carlo method for the one-dimensional RTE.

Figures (5)

• Figure 1: Comparison of the exact solution $L_+(l)$, where $l=ct$, (solid line) with the Monte Carlo estimates using $N_\gamma=10^7$ and $n_\text{scat}=3,10,20$, assuming $\mu_a=0.05 m^{-1}$, $\mu_s=0.1 m^{-1}$, $x=10\rm{m}$ and $g=0.9$ (upper plot) and $g=-0.9$ (bottom plot).
• Figure 2: Comparison of the exact solution $L_+(l)$, where $l=ct$, (solid line) with the improved Monte Carlo estimates (histograms) using $N_\gamma=10^7$ and assuming $g=0.9$ (upper, $n_\text{scat}=2$) and $g=-0.9$ (bottom, $n_\text{scat}=2,6,12$), $\mu_a=0.05 m^{-1}$, $\mu_s=0.1 m^{-1}$ and $x=10\rm{m}$.
• Figure 3: The ratio $n_\text{scat}^\text{orig}/n_\text{scat}^\text{fast}$ as a function of the asymmetry parameter $g$ for several values of $k$ and $x$, assuming $\varepsilon=10^{-3}$.
• Figure 4: Comparison between the analytic expansion of the exact solution $L_+(l)$, where $l=ct$, and the numerical result of Monte-Carlo for $x = 10m$ with $g = 0, \mu_a = 0.05m^{-1}, \mu_s = 0.1m^{-1}$.
• Figure 5: Comparison between the analytic expansion of the exact solution $L_+(l)$, where $l=ct$, and the numerical result of improved Monte-Carlo for $x = 10m$ with $g = 0, \mu_a = 0.05m^{-1}, \mu_s = 0.1m^{-1}$.