Anisotropic diffusive transport: Connecting microscopic scattering and macroscopic transport properties

Erik Alerstam

Phys. Rev. E 89, 063202 – Published 16 June 2014; Erratum Phys. Rev. E 90, 029903 (2014)

Abstract

This work concerns the modeling of radiative transfer in anisotropic turbid media using diffusion theory. A theory for the relationship between microscopic scattering properties (i.e., an arbitrary differential scattering cross-section) and the macroscopic diffusion tensor, in the limit of independent scatterers, is presented. The theory is accompanied by a numerical method capable of performing the calculations. In addition, a boundary condition appropriate for modeling systems with anisotropic radiance is derived. It is shown that anisotropic diffusion theory, when based on these developments, indeed can describe radiative transfer in anisotropic turbid media. More specifically, it is reported that solutions to the anisotropic diffusion equation are in excellent agreement with Monte Carlo simulations, both in steady-state and time-domain. This stands in contrast to previous work on the topic, where inadequate boundary conditions and/or incorrect relations between microscopic scattering properties and the diffusion tensor have caused disagreement between simulations and diffusion theory. The present work thus falsify previous claims that anisotropic diffusion theory cannot describe anisotropic radiative transfer, and instead open for accurate quantitative diffusion-based modeling of anisotropic turbid materials.

Received 9 December 2013
Revised 13 February 2014

DOI:https://doi.org/10.1103/PhysRevE.89.063202

Erratum

Erratum: Anisotropic diffusive transport: Connecting microscopic scattering and macroscopic transport properties [Phys. Rev. E 89, 063202 (2014)]

Erik Alerstam

Phys. Rev. E 90, 029903 (2014)

Authors & Affiliations

Erik Alerstam^*

Division of Atomic Physics, Department of Physics, Lund University, P.O. Box 118, 221 00 Lund, Sweden

^*Present address: Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109, USA; erik.alerstam@jpl.nasa.gov

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Vol. 89, Iss. 6 — June 2014

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Images

Figure 1
A polar plot (arbitrary units) of $P (ϕ)$ , the azimuthal component of $P (\hat{s}) = P (θ) P (ϕ)$ , as a function of number of steps, $i$ . In this system $P (θ)$ uniformly distributed $\in [0, π]$ and is thus not shown. The anisotropy is caused by a direction-dependent single-scattering phase function (with the tensor elements $g_{x x} = 0.4$ , and $g_{y y} = g_{z z} = 0.8$ , as defined in Sec. 3), while the scattering coefficient is direction independent. Starting from an isotropic distribution (at $i = 0$ ), it takes many steps for the distribution to converge to the stationary distribution (full convergence not shown).
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Figure 2
Diffusion tensor elements calculated using random walk theory (RWT) [Eq. (21)] and the simplistic method [Eq. (12)] as a function of a single varying microscopic scattering property. In (a) $μ_{s, z z}$ is varied while $g_{x x} = g_{y y} = g_{z z} = 0.8$ , and $μ_{s, x x} = μ_{s, y y} = 100$ cm $^{- 1}$ are kept constant. In (b) $g_{z z}$ is varied, and $g_{x x} = g_{y y} = 0.8$ , and $μ_{s, x x} = μ_{s, y y} = μ_{s, z z} = 100$ cm $^{- 1}$ are kept constant. In both (a) and (b), the refractive index is $n = 1.0$ and $μ_{a} = 0$ . Due to symmetry, $D_{y y} = D_{x x}$ . It is clear that the simplistic method does not provide an accurate estimate of the diffusion tensor, unless the diffusion is isotropic.
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Figure 3
Steady-state transmission through 2-cm-thick, nonabsorbing slabs, as modeled by Monte Carlo and anisotropic diffusion theory. (a) and (b) show Case 1 (in linear and logarithmic scale, respectively), where the anisotropy is due to a direction-dependent scattering coefficient, $μ_{s, x x} = μ_{s, y y} = 100$ cm $^{- 1}$ , and $μ_{s, z z} = 50$ cm $^{- 1}$ , with $g_{x x} = g_{y y} = g_{z z} = 0.8$ , and $n = 1.5$ . The panels to the right (c) and (d) show Case 2 (in linear and logarithmic scale, respectively), where the single-scattering anisotropy factor is direction dependent with $g_{x x} = g_{y y} = 0.8$ , and $g_{z z} = 0.6$ , while $n = 1.0$ , and $μ_{s, x x} = μ_{s, y y} = μ_{s, z z} = 100$ cm $^{- 1}$ . Using RWT, anisotropic diffusion reproduce the Monte Carlo results within the accuracy of diffusion theory. However, using the simplistic method for deriving the diffusion tensor, and/or using the boundary condition for isotropic radiance, results in large deviations compared to Monte Carlo.
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Figure 4
(a) Isointensity curves for the steady-state transmission (units: [m $^{- 2}$ ]) through a 2-cm-thick slab with optical properties: $μ_{s, x x} = μ_{s, y y} = μ_{s, z z} = 100$ cm $^{- 1}$ , $g_{x x} = g_{y y} = 0.8$ , $g_{z z} = 0.4$ , $μ_{a} = 0$ , and $n = 1.5$ . Panels (b)–(d) show the time-resolved transmittance through the slab at the positions marked in (a). Each plot shows the results of a Monte Carlo simulation (random walkers collected from a $2 \times 2$ mm $^{2}$ area centered around the collection coordinates), as well as results from anisotropic diffusion theory. Diffusion theory with the RWT-derived diffusion tensor and extrapolation length show excellent agreement with Monte Carlo results, while using the simplistic diffusion tensor and/or the isotropic boundary condition results in large deviations.
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