# 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] Lingyu Li, Zhang Chen. Asymptotic behavior of non-autonomous random Ginzburg-Landau equation driven by colored noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3303-3333. doi: 10.3934/dcdsb.2020233 [2] Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311 [3] Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, 2021, 14 (2) : 199-209. doi: 10.3934/krm.2021002 [4] Fioralba Cakoni, Pu-Zhao Kow, Jenn-Nan Wang. The interior transmission eigenvalue problem for elastic waves in media with obstacles. Inverse Problems & Imaging, 2021, 15 (3) : 445-474. doi: 10.3934/ipi.2020075 [5] Tomáš Roubíček. An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 867-893. doi: 10.3934/dcdss.2017044 [6] Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic & Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034 [7] 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, 2021, 15 (3) : 555-597. doi: 10.3934/ipi.2021005 [8] Yves Capdeboscq, Shaun Chen Yang Ong. Quantitative jacobian determinant bounds for the conductivity equation in high contrast composite media. Discrete & Continuous Dynamical Systems - B, 2020, 25 (10) : 3857-3887. doi: 10.3934/dcdsb.2020228 [9] Fumihiko Nakamura. Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2457-2473. doi: 10.3934/dcdsb.2018055 [10] Adrian Viorel, Cristian D. Alecsa, Titus O. Pinţa. Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3319-3341. doi: 10.3934/dcds.2020407 [11] José A. Carrillo, Bertram Düring, Lisa Maria Kreusser, Carola-Bibiane Schönlieb. Equilibria of an anisotropic nonlocal interaction equation: Analysis and numerics. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3985-4012. doi: 10.3934/dcds.2021025 [12] Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258 [13] Huan Zhang, Jun Zhou. Asymptotic behaviors of solutions to a sixth-order Boussinesq equation with logarithmic nonlinearity. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021034 [14] Xuping Zhang. Pullback random attractors for fractional stochastic $p$-Laplacian equation with delay and multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021107 [15] Andrew Comech, Elena Kopylova. Orbital stability and spectral properties of solitary waves of Klein–Gordon equation with concentrated nonlinearity. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021063 [16] Yohei Yamazaki. Center stable manifolds around line solitary waves of the Zakharov–Kuznetsov equation with critical speed. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3579-3614. doi: 10.3934/dcds.2021008 [17] Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023 [18] Xiaozhong Yang, Xinlong Liu. Numerical analysis of two new finite difference methods for time-fractional telegraph equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3921-3942. doi: 10.3934/dcdsb.2020269 [19] Dmitry Treschev. Travelling waves in FPU lattices. Discrete & Continuous Dynamical Systems, 2004, 11 (4) : 867-880. doi: 10.3934/dcds.2004.11.867 [20] Scott Schmieding, Rodrigo Treviño. Random substitution tilings and deviation phenomena. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3869-3902. doi: 10.3934/dcds.2021020

2019 Impact Factor: 1.27