Light propagation through a finite slab of a disordered medium is commonly modeled with incoherent building blocks: single-particle Mie scattering parameters, the independent-scattering approximation, and ray-tracing Monte Carlo. For thin films of 'Mie glasses' (monodisperse TiO2 spheres of radius 225 nm at 2.5% filling fraction in a mesoporous titania matrix), Miranda-Muñoz and co-workers (Adv. Opt. Mater. 8, 1901196, 2020) showed this incoherent framework reproduces measured spectral and angular response without any phase-correlation effects, but also obtained two different transport mean free paths at the same wavelength: lt=15l_t = 15 µm from single-particle Mie parameters versus an effective lt=10l_t = 10 µm estimated from Monte-Carlo total-transmittance spectra of slabs spanning the few-ltl_t regime. The open problem is twofold: (i) establish quantitative threshold conditions — in refractive-index contrast, filling fraction, strength of structural correlations between scatterers, and spectral proximity to single-particle Mie resonances — beyond which coherent (phase-correlated) multiple-scattering effects such as dependent scattering, near-field coupling, recurrent scattering loops, and ultimately localization must be included for accurate prediction of transport in finite slabs; and (ii) determine the origin of, and a reconciliation procedure for, the discrepancy between single-scatterer-derived and slab-scale transport metrics in finite geometries.

Background

When light traverses a disordered medium, its propagation is characterized by the scattering mean free path lscl_{sc} (mean distance between scattering events) and the transport mean free path lt=lsc/(1−⟨cos⁡θ⟩)l_t = l_{sc}/(1-\langle\cos\theta\rangle), where ⟨cos⁡θ⟩\langle\cos\theta\rangle is the scattering-angle asymmetry factor; ltl_t sets the depth beyond which the propagation direction is randomized. Mie theory is the exact classical solution for scattering by a sphere, and the independent-scattering approximation (ISA) assumes each scattering event is governed by the single-particle cross section and phase function, so the ensemble response follows from radiative transfer, commonly sampled by ray-tracing Monte Carlo. This incoherent picture neglects phase-correlation (coherent) effects: dependent scattering, in which the ensemble cross sections deviate from the sum of single-particle ones once inter-particle distances become comparable to the wavelength, entering through the medium's structure factor (the Fourier transform of the pair-correlation function of scatterer positions, which encodes their spatial correlations); near-field coupling between close scatterers; recurrent scattering loops (wave paths that return to a scatterer already visited); interference corrections; and Anderson localization (the interference-induced halt of wave propagation). For slabs of thickness LL comparable to ltl_t — a 'finite-size' regime between ballistic transmission and fully developed diffusion — the number ss of scattering events per photon is small and the angular and spectral response depends on the distribution over ss, which only a photon-resolved model captures. Miranda-Muñoz, Esteso, Jiménez-Solano, Lozano, and Míguez (Adv. Opt. Mater. 8, 1901196, 2020) studied such a system experimentally and by Monte Carlo: 'Mie glasses' of monodisperse crystalline TiO2 spheres (radius 225 ± 20 nm, filling fraction 2.5%) randomly dispersed in a nonabsorbing mesoporous anatase matrix. They found the measured spectral and angular response of films with LL from below ltl_t up to 4lt4l_t is reproduced without phase-correlation effects, and introduced per-photon representations in which emerging photons are binned by the number ss of undergone scattering events. Two facts mark the limits of this success. First, at λ = 600 nm the transport mean free path from single-particle Mie parameters (lt=15l_t = 15 µm, from lsc=3.6l_{sc} = 3.6 µm) differs by roughly a third from the value (lt=10l_t = 10 µm) independently estimated from their Monte-Carlo total-transmittance spectra of slabs of diverse thicknesses — the authors note both are 'of the same order' but do not explain the difference, which may involve boundary, sub-diffusive (corrections to diffusion theory that appear when LL is not large compared to ltl_t), and fitting-convention effects in the few-ltl_t regime. Second, their conclusion that coherent effects are negligible was established only for low filling fraction, moderate refractive-index contrast, wavelengths away from strong single-particle resonances, and statistically uncorrelated scatterer positions. The broader context shows these limits are active research fronts: theory and simulations establish that short-range structural correlations modify transport through structure-factor corrections (Vynck, Pierrat, and Carminati, Phys. Rev. A 94, 033851, 2016); dependent-scattering corrections measurably alter photon transport in dense polydisperse suspensions at volume fractions up to 20% (Fujii et al., Opt. Express 28, 22962, 2020) and in spatially correlated particulate media (Li and Bala Chandran, Int. J. Heat Mass Transfer 182, 121951, 2022); radiative-transfer-plus-coherent-backscattering codes — coherent backscattering being the interference-induced enhancement of intensity scattered exactly backwards — have been extended to dense discrete random media (Markkanen and Penttilä, J. Quant. Spectrosc. Radiat. Transfer 310, 108733, 2023); reflection-matrix measurements on dense TiO2 powders reveal subdiffusive early-time transport from recurrent scattering loops and near-field coupling (Cobus, Maret, and Aubry, Phys. Rev. B 106, 014208, 2022); and brute-force finite-difference time-domain simulations (direct grid integration of Maxwell's equations) demonstrate three-dimensional Anderson localization in random metallic sphere packings while finding its absence for dielectric spheres at refractive-index contrast up to 10 (Yamilov et al., Nat. Phys. 19, 1308, 2023), with full-wave studies of dense irregular dielectric particles reporting localization-like slowdown at high contrast (Grynko et al., arXiv:2312.14393) and a clean microwave-scale experimental observation in metal aggregates appearing in 2026 (Goïcoechea et al., arXiv:2606.04897). What no later work supplies is a general quantitative answer, for dielectric particle ensembles of the Mie-glass type, to when the incoherent framework breaks down and how its transport parameters should be consistently defined and extracted in finite slabs.

Problem Statement

For optically disordered dielectric media consisting of randomly dispersed Mie scatterers in a finite slab, determine the quantitative conditions — expressed through refractive-index contrast between scatterer and matrix, scatterer filling fraction, the strength and range of structural correlations between scatterer positions, and spectral detuning from single-particle Mie resonances — beyond which the incoherent description of light transport (single-particle Mie parameters combined with the independent-scattering approximation and ray-tracing Monte Carlo, with all phase-correlation effects neglected) fails to reproduce, within a tolerance that the analysis states and justifies, the observable transport properties of the slab, namely its total and angular transmittance and reflectance spectra and the distribution of the number of scattering events per photon, so that coherent multiple-scattering effects (dependent scattering via the structure factor, near-field inter-particle coupling, recurrent scattering loops, interference corrections, and localization) must be included. In parallel, establish the physical origin of, and a reconciliation procedure for, the discrepancy between the transport mean free path derived from single-scatterer Mie parameters (lt=lsc/(1−⟨cos⁡θ⟩)l_t = l_{sc}/(1-\langle\cos\theta\rangle)) and the effective transport mean free path extracted from slab-scale total-transmittance data in the few-ltl_t regime — including the specific 15 µm versus 10 µm case at λ = 600 nm in the TiO2 Mie-glass system of Miranda-Muñoz et al. — identifying how candidate mechanisms such as boundary-reflection, sub-diffusive, and fitting-convention effects may enter, and giving the range of slab thicknesses over which a single effective ltl_t is meaningful. Two further directions — extending the event-resolved (ss-resolved) description to absorbing media, anisotropic or non-spherical scatterers, and angularly correlated or spatially structured disorder, and translating the simulation-based design mappings into fabrication tolerances — are companion extensions of the same modeling framework; they are named here as outside the problem's core, whose completion boundary is set by the two questions above.

References

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