# American Institute of Mathematical Sciences

## 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  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, doi: 10.3934/dcdsb.2020158
##### References:

show all references

##### 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))
 [1] Josselin Garnier. Ghost imaging in the random paraxial regime. Inverse Problems & Imaging, 2016, 10 (2) : 409-432. doi: 10.3934/ipi.2016006 [2] Romina Gaburro, Clifford J Nolan. Enhanced imaging from multiply scattered waves. Inverse Problems & Imaging, 2008, 2 (2) : 225-250. doi: 10.3934/ipi.2008.2.225 [3] Stephen Coombes, Helmut Schmidt, Carlo R. Laing, Nils Svanstedt, John A. Wyller. Waves in random neural media. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2951-2970. doi: 10.3934/dcds.2012.32.2951 [4] Wenjia Jing, Olivier Pinaud. A backscattering model based on corrector theory of homogenization for the random Helmholtz equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5377-5407. doi: 10.3934/dcdsb.2019063 [5] Tzong-Yow Lee and Fred Torcaso. Wave propagation in a lattice KPP equation in random media. Electronic Research Announcements, 1997, 3: 121-125. [6] Guillaume Bal, Olivier Pinaud. Self-averaging of kinetic models for waves in random media. Kinetic & Related Models, 2008, 1 (1) : 85-100. doi: 10.3934/krm.2008.1.85 [7] Guillaume Bal. Homogenization in random media and effective medium theory for high frequency waves. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 473-492. doi: 10.3934/dcdsb.2007.8.473 [8] Michael Taylor. Random walks, random flows, and enhanced diffusivity in advection-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1261-1287. doi: 10.3934/dcdsb.2012.17.1261 [9] Andrew Homan. Multi-wave imaging in attenuating media. Inverse Problems & Imaging, 2013, 7 (4) : 1235-1250. doi: 10.3934/ipi.2013.7.1235 [10] Liliana Borcea, Dinh-Liem Nguyen. Imaging with electromagnetic waves in terminating waveguides. Inverse Problems & Imaging, 2016, 10 (4) : 915-941. doi: 10.3934/ipi.2016027 [11] Khaled El Dika. Asymptotic stability of solitary waves for the Benjamin-Bona-Mahony equation. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 583-622. doi: 10.3934/dcds.2005.13.583 [12] Seung-Yeal Ha, Shi Jin, Jinwook Jung. A local sensitivity analysis for the kinetic Kuramoto equation with random inputs. Networks & Heterogeneous Media, 2019, 14 (2) : 317-340. doi: 10.3934/nhm.2019013 [13] Aslihan Demirkaya, Panayotis G. Kevrekidis, Milena Stanislavova, Atanas Stefanov. Spectral stability analysis for standing waves of a perturbed Klein-Gordon equation. Conference Publications, 2015, 2015 (special) : 359-368. doi: 10.3934/proc.2015.0359 [14] Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311 [15] Patrick W. Dondl, Martin Jesenko. Threshold phenomenon for homogenized fronts in random elastic media. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020329 [16] Ennio Fedrizzi. High frequency analysis of imaging with noise blending. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 979-998. doi: 10.3934/dcdsb.2014.19.979 [17] Guillaume Bal, Lenya Ryzhik. Stability of time reversed waves in changing media. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 793-815. doi: 10.3934/dcds.2005.12.793 [18] Georgios Fotopoulos, Markus Harju, Valery Serov. Inverse fixed angle scattering and backscattering for a nonlinear Schrödinger equation in 2D. Inverse Problems & Imaging, 2013, 7 (1) : 183-197. doi: 10.3934/ipi.2013.7.183 [19] Lingyu Li, Zhang Chen. Asymptotic behavior of non-autonomous random Ginzburg-Landau equation driven by colored noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020233 [20] Guillaume Bal, Tomasz Komorowski, Lenya Ryzhik. Kinetic limits for waves in a random medium. Kinetic & Related Models, 2010, 3 (4) : 529-644. doi: 10.3934/krm.2010.3.529

2019 Impact Factor: 1.27

## Tools

Article outline

Figures and Tables