Anomalous Radiative Transfer in Heterogeneous Media

Abstract

Monte Carlo (MC) simulations are the gold standard for describing various transport phenomena and have largely contributed to the understanding of these processes. However, while their implementation for classical transport governed by exponential step-length distributions is well-established, widely accepted approaches are still lacking for the more general class of anomalous transport phenomena. In this work, a set of rules for performing MC simulations in anomalous diffusion media is identified, which is also applicable in the case of finite-size geometries and/or heterogeneous inclusions. The results are presented in the context of radiative transfer, however their implications extend to all types of anomalous transport. The proposed set of rules exhibits full compatibility with the pathlength invariance property for random trajectories, and with the important radiometric concept of fluence. Additionally, it reveals the counter-intuitive possibility of introducing interfaces between independent subdomains with identical properties, which arise from the fact that non-exponential step-length distributions have a "memory" that can in principle be reset when traversing a boundary. These results have far-reaching consequences not just for the physical interpretation of the corrections required to handle these discontinuities, but also for their experimental verification, due to their expected effects on the observable pathlength distributions.

Keywords: Anomalous transport; Generalized Radiative Transfer Equation; Heterogeneous media; Monte Carlo simulations; Random walks trajectories.

Figure 1:

a) Propagation of a trajectory in an illustrative anomalous medium comprising two regions with different scattering coefficients ${\mu }_{\text{s}1}$ and ${\mu }_{\text{s}2}$ and refractive indices $n_1$ and $n_2$. The external environment has a refractive index $n_0$. Step lengths $s_{1,2}$ inside the scattering domain are extracted according to $p_{\text{u}}(\ell )$ distributions (black). Steps departing from a boundary (either at the injection point, or following refraction or reflection events) are extracted with $p_{\text{u}}(\ell )$ distributions (blue). b) Comparison between exponential $(k=0)$ and non-exponential $(k\ne 0)p_{\text{c}}$ distributions taken from the family of generalized Pareto distributions with equal mean step value $\langle \ell \rangle =1\text{mm}$. c) Comparison between the corresponding $p_{\text{u}}(\ell )$ derived as the survival functions of the respective $p_{\text{u}}(\ell )$. For the exponential case we have that $p_{\text{c}}(\ell )=p_{\text{u}}(\ell )$.

Figure 2:

Total $\langle L\rangle$ spent inside a non-absorbing sphere of radius $r=5\text{mm}$, surrounded by a non-scattering region of unit refractive index, illuminated by a Lambertian source, normalized to the IP prediction, ${\langle L\rangle }_{\text{IP}}$. For each value of ${\mu }_{\text{s}}$, three configurations are studied: a homogeneous case with a refractive index contrast with the external environment (panels a, d); a 4-layered sphere with alternating refractive indices $n_i$ (b, e); and a 4-layered sphere with alternating indices $n_i$ and scattering coefficients ${\mu }_{\text{s},i}$ (c, f). The index $i$ indicates the layers from the inside to the outside of the sphere. The gray dashed line indicates the IP prediction. Blue and red symbols refer to MC simulations obtained with and without the use of Equation (1) after interface events.

Figure 3:

a, c) Fluence rate $\Phi$ in each sub-volume of a 10-layered sphere of radius 5mm with refractive index discontinuities (values shown on top) and constant scattering ${\mu }_{\text{s}}=2{\text{mm}}^{-1}$. The gray dashed line represents the values predicted by Equation (12). b, d) Comparison between the fluence rate $\Phi$ and the average total pathlength $\langle L_i\rangle$ of all trajectories inside each layer.

Figure 4:

Comparison between total pathlength distributions obtained for a homogeneous non-absorbing sphere of radius $r$ and refractive index $n=1.4$ under isotropic incident radiance, in the case of (a,c) matched and (b,d) mismatched refractive index with the environment. For better visibility, curve pairs related to $2r{\mu }_{\text{s}}=1$ and 10 are rescaled by factors of ${10}^2$ and ${10}^4$, respectively.

Figure 5:

a) Probability density function $p(L)$ and b) normalized average pathlength $\langle L\rangle /{\langle L\rangle }_{\text{IP}}$ for five non-absorbing spheres with $r=5\text{mm}$ comprising an increasing number of identical independent layers with identical properties $k=0.7$, ${\mu }_{\text{s}}=1{\text{mm}}^{-1}$, $n=1.4$ and $n_0=1$. Results are shown only in the case where both $p_{\text{c}}$ and $p_{\text{u}}$ are used.