Modeling the Doppler spectrum of waves backscattered from an expanding cloud for anisotropic phase functions

Abstract

We study the Doppler spectrum of a collimated beam of light backscattered from a cloud of moving particles. The problem we address is attracting attention in the context of the application of photonic Doppler velocimetry (PDV) to probe ejecta from shock-loaded metal samples. The Doppler spectrum is calculated on the basis of numerically solving the transport equation for the field correlation function. We transform the original transport equation into a system of Milne-like equations, which are then integrated with the discrete-ordinate code. The calculations are carried out for a plane cloud of relatively large metal particles (comparable to or larger than the wavelength) moving away from the free surface bounding the cloud. The effect of anisotropic single scattering on the Doppler spectrum is analyzed depending on the cloud's optical thickness and albedo under conditions characteristic of the experiment (finite field of view of the PDV probe, wave reflection from the cloud-bounding surface). A sharp asymmetric peak in the spectrum at the Doppler shift corresponding to the free-surface velocity is shown to be caused by the snake waves and should be observed up to the ejecta cloud thickness of the order of a few transport mean free paths. We demonstrate that the difference in amplitude between the Doppler spectrum calculated with the exact phase function and that obtained in the transport approximation proves to be fairly small for most realistic values of the ejecta cloud parameters. A comparison with available Monte Carlo simulation data is also presented.