Experiments on strongly scattering colloidal suspensions of core-shell TiO2@silica nanoparticles reported that, near the expected transition to Anderson localization of light, transport depends strongly on the angle of incidence: the optical conductance falls as the incidence angle θ\theta grows, absorption appears concentrated near the input border, and the coherent backscattering cone loses its interference enhancement precisely where boundary internal reflection is strongest. The 2018 study that reported these effects raised, in its discussion, conclusions, and supplementary material, six unresolved issues that define the problem: separating absorption from scattering in the angle-dependent conductance; establishing a criterion for the critical regime of the localization transition when the local transport parameter klTkl_T is spatially inhomogeneous and localized and extended modes coexist; testing, via polarization-resolved backscattering cones, whether nonlinear processes break the interference of time-reversed paths; developing internal-reflection corrections to the transport mean free path valid beyond the diffusive regime; assessing retrapping of photons emitted from superficial localized states; and characterizing how scatterer correlations and inhomogeneity sustain the critical regime and its angular dependence.

Background

When light propagates through a disordered medium, repeated elastic scattering randomizes its propagation direction; transport is characterized by the transport mean free path lTl_T, the distance over which the direction becomes uncorrelated. Anderson localization is the interference-driven breakdown of such diffusive transport: waves returning to a previously visited scatterer interfere constructively with themselves, and diffusion halts. The Ioffe-Regel criterion klT∼1kl_T\sim1, where k=2π/λk=2\pi/\lambda is the optical wavenumber, marks the regime where localization is expected in simple scalar models of three-dimensional (3D) media, but no sharp, material-independent threshold for the phase transition to localization is known for real optical systems. A closely related interference hallmark is the coherent backscattering (CBS) cone: the interference of time-reversed multiple-scattering paths doubles the backscattered intensity in the exact backward direction (enhancement factor 2 in the weak-scattering limit) with a cone width of order 1/(klT)1/(kl_T), and the cone's enhancement and line shape are standard diagnostics of how close a medium is to localization. A further boundary effect is central to the present problem: because a scatterer suspension has a larger effective refractive index than the surrounding medium, photons reaching the sample boundary from the inside at angles beyond the critical angle are totally internally reflected. In the diffusive regime this internal reflection renormalizes the apparent transport mean free path and modifies boundary conditions, as treated by the formalisms of Lagendijk, Vreeker, and de Vries (1989) and Zhu, Pine, and Weitz (1991); whether such corrections remain valid in the localization regime, where transport is no longer diffusive, is unknown.

The observational status of 3D localization of light is contested. Time-of-flight measurements on TiO2 powders reported the critical regime near localization (Störzer et al., 2006), and frequency-domain measurements claimed a direct determination of the transition (Sperling et al., 2013), but these claims were challenged on the grounds that weak inelastic scattering (absorption or fluorescence) can mimic localization signatures (Scheffold and Wiersma, 2013), and theory for point scatterers found no localization in 3D ensembles (Skipetrov and Sokolov, 2014). Against this backdrop, a Brazilian group developed a colloidal testbed: rutile TiO2 cores coated with silica shells, suspended in ethanol. The shell provides colloidal stability via electrostatic (ζ\zeta-potential-mediated) repulsion and suppresses near-field dipole coupling between the high-index cores; earlier work on this system reported localization signatures (Jimenez-Villar et al., Nanoscale 2016).

The source study (Jimenez-Villar et al., Photonics Research 6, 929, 2018) measured, as a function of the incidence angle θ\theta, the total transmission, the propagation intensity profiles, the absorption near the input border, and the CBS cone of this suspension. It reported that the optical conductance decreases and the absorption near the input border increases as θ\theta grows; that the specular reflection of incoming photons is lower than the effective internal reflection experienced by coherently backscattered photons leaving in the opposite direction (an asymmetry the authors interpreted as nonreciprocal propagation); and that the CBS enhancement factor falls toward 1 as the internal reflection approaches 100%. The physical motivation is that in the diffusive regime transport is independent of incidence angle, whereas near the mobility edge (the diffusion-localization phase boundary) the density of localized states near the boundary is predicted to be enhanced by internal reflection, so that 'superficial' localized states become sensitive to boundary reflectivity and a strong θ\theta-dependence of transport is a candidate signature of the critical regime — the coexistence regime of localized and extended modes. The study's discussion, conclusions, and supplementary material raise unresolved issues that a verification of this candidate signature must address.

Problem Statement

Establish a verified, quantitative account of the incidence-angle dependence of light transport at the phase transition to Anderson localization in the TiO2@silica colloidal system and equivalent strongly scattering dielectric colloidal media. In the diffusive regime transport is independent of incidence angle; the source study's central claim is that near the mobility edge it is not. Confirming, correcting, or refuting that claim requires addressing the following open issues raised in the source study's discussion, conclusions, and supplementary material. (1) Absorption artifact or genuine transport: determine whether the reported angle-dependent asymptotic relative conductance G(∞;θ)G(\infty;\theta) — the source's θ=0\theta=0-normalized ratio of asymptotic slopes of 1/T(d;θ)1/T(d;\theta), extracted from total-transmission and propagation measurements, which rises with incidence angle while the optical conductance falls — survives a quantitative separation of absorption (concentrated near the input border, strongest at the largest incidence angles) from scattering. (2) Criterion for the critical regime: establish a criterion delimiting the critical regime of the localization transition for media in which repulsive interparticle interactions and size polydispersity produce a spatially inhomogeneous local klTkl_T, so that localized and extended modes coexist and a single averaged klTkl_T cannot characterize the regime. (3) Boundary-sensitive interference: test, by a polarization-resolved study of the CBS cone as a function of incidence angle, whether nonlinear processes (such as a dynamical refractive-index increase during the photon residence time in a localized state) break the interference of time-reversed paths, as suggested by the observed enhancement factor tending to 1 as internal reflection approaches 100%. (4) Boundary theory beyond diffusion: develop the correction of the transport mean free path lT0l_{T0} (the bare transport mean free path before any internal-reflection correction) for internal reflection beyond the diffusive-regime formalisms of Lagendijk et al. (1989) and Zhu et al. (1991), i.e., a boundary theory valid in the localization regime. (5) Superficial-state retrapping: assess whether photons emitted by nonlinear effects from superficial localized states — the near-boundary localized modes whose density is enhanced by internal reflection, as introduced in Background — can be retrapped, i.e. reabsorbed into other superficial states, increasing the effective mode density and quality factor inferred from the near-surface absorption and emission measurements. (6) Role of inhomogeneity: characterize how the inhomogeneous scatterer distribution arising from ζ\zeta-potential repulsion and polydispersity sustains the critical regime and shapes the angular dependence of transport. These components support one objective, namely to decide whether incidence-angle-resolved transport is a trustworthy experimental signature of the localization transition in dielectric colloids and to supply the measurement standards and boundary theory that decision requires. The task is resolved by deciding that umbrella question; a resolution of any one of (1)-(6) to a stated standard of evidence counts as a partial result, whereas a negative answer to (3) or (5) alone does not resolve it. The nonlinear conjectures in (3) and (5) originate in the source study's own speculation and admit negative answers.

References

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