# American Institute of Mathematical Sciences

February  2021, 26(2): 1171-1195. doi: 10.3934/dcdsb.2020158

## Enhanced Backscattering of a partially coherent field from an anisotropic random lossy medium

 1 CMAP, CNRS, Ecole polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau Cedex, France 2 Department of Mathematics, University of California, Irvine CA 92697, USA

* Corresponding author: Knut Sølna

Received  September 2019 Revised  February 2020 Published  February 2021 Early access  May 2020

Fund Project: This research is supported by AFOSR grant FA9550-18-1-0217, NSF grant 1616954

The weak localization or enhanced backscattering phenomenon has received a lot of attention in the literature. The enhanced backscattering cone refers to the situation that the wave backscattered by a random medium exhibits an enhanced intensity in a narrow cone around the incoming wave direction. This phenomenon can be analyzed by a formal path integral approach. Here a mathematical derivation of this result is given based on a system of equations that describes the second-order moments of the reflected wave. This system derives from a multiscale stochastic analysis of the wave field in the situation with high-frequency waves and propagation through a lossy medium with fine scale random microstructure. The theory identifies a duality relation between the spreading of the wave and the enhanced backscattering cone. It shows how the cone, its regularity and width relate to the statistical structure of the random medium. We discuss how this information in particular can be used to estimate the internal structure of the random medium based on observations of the reflected wave.

Citation: Josselin Garnier, Knut Sølna. Enhanced Backscattering of a partially coherent field from an anisotropic random lossy medium. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1171-1195. doi: 10.3934/dcdsb.2020158
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##### References:
Physical interpretation of the scattering of a plane wave by a random medium. The output wave in direction $A$ is the superposition of many different scattering paths. One of these paths is plotted as well as the reversed path. The phase difference between the two outgoing waves is $k e = k d \sin A$
The backscattering enhancement cone in Eq. (81) (normalized by $\pi^2 P_{\rm tot}$). Here we use the Matérn covariance function (55). In the left plot $p = .6$, while in the right plot $p = .9$ so that the medium fluctuations are smoother in the right plot. In the plots the narrowest cones with largest peak values correspond to the largest $\beta$ values
Notations used in the paper
 $c_o$ background speed of propagation of the medium $\sigma_o$ background attenuation of the medium $\ell_z$ longitudinal correlation radius of the random medium $\ell_x$ transverse correlation radius of the random medium $\sigma$ standard deviation of the random medium $\omega$ (angular) frequency of the source $r_0$ radius of the source $\rho_0$ correlation radius of the source ${{{\boldsymbol k}}}_0$ transverse wavevector of the source $\lambda_o = \frac{2\pi c_o}{\omega}$ wavelength $L_{\rm att} = \frac{c_o}{2\sigma_o}$ attenuation length $\zeta_L= \frac{L}{L_{\rm att}}$ relative propagation distance $K_z= \frac{2\omega \ell_z}{c_o}$ relative wavenumber $\alpha= \frac{c_o^2}{2\sigma_o \omega \ell_x^2}$ strength of diffraction $\beta= \frac{ \omega^2 \sigma^2 \ell_z}{8 c_o \sigma_o}$ strength of forward scattering $\overline{D}_o$ cross spectral density central value (see Eq. (26)) $P_{\rm tot}$ mean reflected power (see Eq. (31))
 $c_o$ background speed of propagation of the medium $\sigma_o$ background attenuation of the medium $\ell_z$ longitudinal correlation radius of the random medium $\ell_x$ transverse correlation radius of the random medium $\sigma$ standard deviation of the random medium $\omega$ (angular) frequency of the source $r_0$ radius of the source $\rho_0$ correlation radius of the source ${{{\boldsymbol k}}}_0$ transverse wavevector of the source $\lambda_o = \frac{2\pi c_o}{\omega}$ wavelength $L_{\rm att} = \frac{c_o}{2\sigma_o}$ attenuation length $\zeta_L= \frac{L}{L_{\rm att}}$ relative propagation distance $K_z= \frac{2\omega \ell_z}{c_o}$ relative wavenumber $\alpha= \frac{c_o^2}{2\sigma_o \omega \ell_x^2}$ strength of diffraction $\beta= \frac{ \omega^2 \sigma^2 \ell_z}{8 c_o \sigma_o}$ strength of forward scattering $\overline{D}_o$ cross spectral density central value (see Eq. (26)) $P_{\rm tot}$ mean reflected power (see Eq. (31))
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