Diffusive light transport in semitransparent media

Lorenzo Pattelli, Giacomo Mazzamuto, Diederik S. Wiersma, and Costanza Toninelli

Phys. Rev. A 94, 043846 – Published 24 October 2016

Abstract

It is common knowledge that diffusion theory cannot describe light propagation in semitransparent media, i.e., media with a low optical thickness. However, even in an optically thin slab, late-time transport will be eventually determined by a multiple scattering process whose characteristics are still largely unexplored. We numerically demonstrate that, even for an optical thickness as low as 1, after a short transient, propagation along the slab plane becomes diffusive. Nonetheless, we show that such a diffusion process is governed by modified statistical distributions which result from a highly nontrivial interplay with boundary conditions.

Received 22 September 2015

DOI:https://doi.org/10.1103/PhysRevA.94.043846

Physics Subject Headings (PhySH)

Diffusion Light propagation, transmission & absorption

Random & disordered media

Atomic, Molecular & Optical

Authors & Affiliations

Lorenzo Pattelli¹, Giacomo Mazzamuto^1,2, Diederik S. Wiersma^1,3, and Costanza Toninelli^1,2,4,*

¹European Laboratory for Non-linear Spectroscopy (LENS), 50019 Sesto Fiorentino, Florence, Italy
²Istituto Nazionale di Ottica (CNR-INO), Via Carrara 1, 50019 Sesto Fiorentino, Florence, Italy
³Department of Physics, Università di Firenze, 50019 Sesto Fiorentino, Florence, Italy
⁴QSTAR, Largo Enrico Fermi 2, 50125 Florence, Italy

^*toninelli@lens.unifi.it

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Issue

Vol. 94, Iss. 4 — October 2016

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Images

Figure 1
Sketch of the configuration used for Monte Carlo simulations. An infinite slab is illuminated by a pencil beam pulse $δ (t) δ (r)$ of energy packets performing a random walk inside the scattering material. A few representative trajectories and normalized transmitted intensities are shown in the case of an optically thin, index-matched slab ( $n_{in} = n_{out} = 1$ ) with thickness $L_{0} = l_{s} = 1 mm$ and scattering anisotropy $g = 0$ .
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Figure 2
(a) MSW expansion for three slab configurations with $l_{t} = L_{0}$ , $g = 0$ , and different refractive index contrasts, showing a perfectly linear growth after a short transient (shaded). (b) Dependence on the refractive index contrast $n$ and scattering anisotropy $g$ of the diffusion coefficient $D$ as inferred from a linear fit of the mean square width $w^{2} (t > 4 τ)$ for different samples with $OT = 1$ . Solid points represent the values retrieved from the linear fits shown in panel (a).
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Figure 3
Late-time modification of the step-length and scattering-angle distributions for an optically thin slab with $OT = 1$ , $g = 0$ , and $n = 1$ , 1.016, and 1.1. Panel (a) shows the probability distribution of step lengths between consecutive scattering events performed by those energy packets that are transmitted at $t = 90 ps$ . The retrieved distributions exhibit heavier tails than the nominal one (dashed line). Scattering angles become unevenly sampled at late times as well, as shown in panel (b).
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Figure 4
Time evolution of the ratio $〈 l^{2} 〉 / 2 〈 l 〉$ appearing in Eq. (2) and of $〈 cos θ 〉$ , as obtained by the simulations. Each point is obtained considering only the energy packets transmitted within the corresponding time bin. Dashed lines represent the nominal values for the two distributions.
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Figure 5
Time evolution of the step length distribution for $n = 1$ for energy packets transmitted at $t = 10$ , 20, 30, 40, 50, 60, 70, 80, and 90 ps. The gray curves and the blue curves show, respectively, the histogram of the step lengths drawn through the pseudorandom number generator and of the steps taken inside the sample. The two only differ for the last step of each trajectory.
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Figure 6
(a) Dependence on the refractive index contrast $n$ and scattering anisotropy $g$ of the decay time $τ$ as inferred from a single-exponential fit of the spatially integrated transmitted intensity for $OT = 1$ . Especially for lower $g$ factors, the ratio can clearly go above 1 for certain values of the refractive index contrast. The dependence of τ on OT for values highlighted as filled symbols is shown in panel (b). Dashed lines represent the DA prediction; solid lines serve as guides to the eye.
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