A diffusion model for light scattering in ejecta

TL;DR

This work derives a generalized diffusion equation from the radiative transfer framework to model light transport in ejecta produced by extreme shocks, accommodating the ejecta’s inhomogeneous density and moving scatterers. By employing a P1 angular expansion and separating ballistic and diffuse components, the authors obtain a diffusion equation with a transport diffusion constant and an effective, time- and position-dependent absorption that accounts for decorrelation due to scatterer motion. The model is tested against full RTE Monte Carlo simulations across multiple configurations, showing good agreement in thick, quasi-static-like ejecta and highlighting its computational efficiency relative to the RTE; however, it exhibits notable failures when velocity distributions are anisotropic or have fixed modulus, illustrating fundamental limits of the diffusion approximation. The diffusion framework provides a practical tool for rapid PDV spectrogram analysis in relevant regimes and offers directions for extending transport theories (e.g., delta-Eddington, Fokker-Planck) to better capture angular dispersion and dynamic effects in ejecta.

Abstract

We derive a diffusion equation for light scattering from ejecta produced by extreme shocks on metallic samples. This model is easier to handle than a more conventional model based on the Radiative Transfer Equation (RTE), and is a relevant tool to analyze spectrograms obtained from Photon Doppler Velocimetry (PDV) measurements in the deep multiple scattering regime. We also determine the limits of validity of the diffusive model compared to the RTE, based on a detailed analysis of various ejecta properties in configurations with increasing complexity.

Figures (6)

• Figure 1: Illustration of the microjetting mechanism in a typical shock ejecta experiment. Upon reaching the machined free surface, the shock wave first comes into contact with the inwardly directed grooves. Under right angle conditions, the shock wave is reflected and the inward grooves become outward microjets. Due to the velocity gap between the jet-heads and the free surface, the microjets are stretched until surface tension is no longer sufficient to hold matter together and fragmentation begins. This results in the creation of a particle cloud, i.e., an ejecta.
• Figure 2: Schematic representation of a typical shock-loaded experiment with a PDV setup. The probe illuminates the ejecta and the free surface with a highly collimated laser beam (typical numerical aperture of $\theta_p=4.2m rad$ and pupil size $\phi_p = 1.6m m$). The backscattered field is collected by the probe acting as the measuring arm and interferes with the reference arm at the detector. The beating signal is registered with a high bandwidth oscilloscope before being analyzed.
• Figure 3: Illustration of the two ballistic components existing in an ejecta illuminated by a plane-wave in the direction $\bm{u}_0=-\bm{u}_z$. The free surface is on the left.
• Figure 4: Translation invariant and infinite along $x$-axis and $y$-axis scattering slab of length $L$ along its longitudinal $z$-axis. This slab is illuminated from the right by a plane wave of intensity $I_0$ and frequency $\omega_0$ propagating in the direction $\bm{u}_0 = -\bm{u}_z$. The front of the particle cloud is moving at velocity $v_m$ and constitutes the right interface. The boundary condition for the diffuse energy density at this interface is given by the usual extrapolation length $z_{0,r}=(2/3)\ell_t(z_r)$. The left interface is the reflective free surface moving at velocity $v_s$. The position $z=0$ marks the initial position of the free surface and the particle cloud at $t=0$.
• Figure 5: Flux for different situations as a function of the correlation time $\tau$. The RTE calculations are in blue and the diffusion equation results are in red. (a) Total transmission for the statistical homogeneous medium with isotropic velocity distribution and open boundaries for both interfaces. (b) Same as (a) but for the flux collected by the probe in reflection. The computation using the modified effective absorption term in the diffusion equation is represented by a dotted line. (c) Same as (b) still with open boundaries (solid lines) or taking into account the free surface (dotted lines). (d) Same as (c) but with the free surface and a statistically homogeneous lognormal particle size distribution and an inhomogeneous particle number density. (e) Same as (d) but with an inhomogeneous isotropic velocity distribution. (f) Same as (e) but with an inhomogeneous anisotropic velocity distribution.