# 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  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
##### 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] Patrick W. Dondl, Martin Jesenko. Threshold phenomenon for homogenized fronts in random elastic media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 353-372. doi: 10.3934/dcdss.2020329 [2] Fioralba Cakoni, Pu-Zhao Kow, Jenn-Nan Wang. The interior transmission eigenvalue problem for elastic waves in media with obstacles. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020075 [3] Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021002 [4] Alexander Dabrowski, Ahcene Ghandriche, Mourad Sini. Mathematical analysis of the acoustic imaging modality using bubbles as contrast agents at nearly resonating frequencies. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021005 [5] Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323 [6] Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256 [7] Adrian Viorel, Cristian D. Alecsa, Titus O. Pinţa. Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020407 [8] Yohei Yamazaki. Center stable manifolds around line solitary waves of the Zakharov–Kuznetsov equation with critical speed. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021008 [9] Feifei Cheng, Ji Li. Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 967-985. doi: 10.3934/dcds.2020305 [10] Tetsuya Ishiwata, Young Chol Yang. Numerical and mathematical analysis of blow-up problems for a stochastic differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 909-918. doi: 10.3934/dcdss.2020391 [11] Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020352 [12] Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020284 [13] Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119 [14] Eduard Marušić-Paloka, Igor Pažanin. Homogenization and singular perturbation in porous media. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020279 [15] Timothy Chumley, Renato Feres. Entropy production in random billiards. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1319-1346. doi: 10.3934/dcds.2020319 [16] Jakub Kantner, Michal Beneš. Mathematical model of signal propagation in excitable media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 935-951. doi: 10.3934/dcdss.2020382 [17] Weihong Guo, Yifei Lou, Jing Qin, Ming Yan. IPI special issue on "mathematical/statistical approaches in data science" in the Inverse Problem and Imaging. Inverse Problems & Imaging, 2021, 15 (1) : I-I. doi: 10.3934/ipi.2021007 [18] Wei-Chieh Chen, Bogdan Kazmierczak. Traveling waves in quadratic autocatalytic systems with complexing agent. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020364 [19] Olivier Pironneau, Alexei Lozinski, Alain Perronnet, Frédéric Hecht. Numerical zoom for multiscale problems with an application to flows through porous media. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 265-280. doi: 10.3934/dcds.2009.23.265 [20] George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003

2019 Impact Factor: 1.27

## Tools

Article outline

Figures and Tables