Nonstationary Intensity Statistics in Diffusive Waves

It is a long-standing belief that, in the diffusion regime, the intensity statistics is always stationary and its probability distribution follows a negative exponential decay. Here, we demonstrate that, in fact, in reflection from strong disordered media, the intensity statistics changes through different stages of the diffusion. We present a statistical model that describes this nonstationary property and takes into account the evolving balance between recurrent scattering and near field coupling. The predictions are further verified by systematic experiments in the optical regime. This statistical nonstationary is akin to the nonequilibrium but steady-state diffusion of particulate systems.

Figure 1

Origin of PDF nonstationarity. (a) Strong recurrent scattering (green) at short paths leads to deviations from a negative exponential while near-field coupling (red) along longer paths tends to change it back. (b) The energy ratio $r(s)$ gauges the energetic contribution of correlated waves with respect to total energy. Because of the competition between creation of loops and energy leaking through near-field coupling, $r(s)$ depends on the trajectory length [blue curve, see Eq. (3)].

Figure 2

Path-resolved intensity fluctuations for different realizations $\alpha$ of a semi-infinite inhomogeneous medium with 1.5 refractive index consisting of 50% volume fraction of ${\mathrm{TiO}}_2$ particles with a diameter of $0.33\mu \mathrm{m}$. The intensity fluctuations corresponding to three different paths of length 80, 400, and $720\mu \mathrm{m}$ are shown on the right panels.

Figure 3

Path-length-resolved normalized variance (contrast) of intensity fluctuations in strong scattering media displaying subdiffusive behavior of propagation. The solid lines denote the evaluations based on mesoscopic wave transport theory in Eq. (3). The dots are the corresponding experimental results for media with different volume fractions and transport mean free paths as indicated. The shaded areas represent the standard deviation of five different datasets, each including 200 measurements.

Figure 4

Contour plot of the normalized variance (contrast) of intensity fluctuations for waves propagating along different path lengths through random media with varying volume fractions. The pink dotted line indicates the normalized variance of $C=0.96$ calculated based on a ballistic attenuation model (see text). The blue dotted line indicates the theoretical prediction for the same value $C=0.96$ using the model based on Eq. (3). The blue solid dots denote the experimental path lengths for which the same value of normalized variance is attained. The error bars designate the standard deviations from five different datasets, while each includes 200 measurements. The brown crosses indicate the path lengths for which the phase transitions are observed in the diffusion of light (see text).