# American Institute of Mathematical Sciences

May  2013, 7(2): 445-470. doi: 10.3934/ipi.2013.7.445

## Far field model for time reversal and application to selective focusing on small dielectric inhomogeneities

 1 Inria (CORIDA Team), Villers-lès-Nancy, F-54600, France 2 Mathematics Department, US Naval Academy, 572C Holloway Road, Annapolis, MD 21402-5002, United States 3 Université de Lorraine, IECL, UMR 7502, Vandoeuvre-les-Nancy, F-54506, France

Received  November 2011 Revised  February 2013 Published  May 2013

Based on the time-harmonic far field model for small dielectric inclusions in $3$D, we study the so-called DORT method (DORT is the French acronym for Diagonalization of the Time Reversal Operator''). The main observation is to relate the eigenfunctions of the time-reversal operator to the location of small scattering inclusions. For non penetrable sound-soft acoustic scatterers, this observation has been rigorously proved for $2$ and $3$ dimensions by Hazard and Ramdani in [21] for small scatterers. In this work, we consider the case of small dielectric inclusions with far field measurements. The main difference with the acoustic case is related to the magnetic permeability and the related polarization tensors. We show that in the regime $kd\rightarrow \infty$ ($k$ denotes here the wavenumber and $d$ the minimal distance between the scatterers), each inhomogeneity gives rise to -at most- 4 distinct eigenvalues (one due to the electric contrast and three to the magnetic one) while each corresponding eigenfunction generates an incident wave focusing selectively on one of the scatterers. The method has connections to the MUSIC algorithm known in Signal Processing and the Factorization Method of Kirsch.
Citation: Corinna Burkard, Aurelia Minut, Karim Ramdani. Far field model for time reversal and application to selective focusing on small dielectric inhomogeneities. Inverse Problems & Imaging, 2013, 7 (2) : 445-470. doi: 10.3934/ipi.2013.7.445
##### References:

show all references

##### References:
 [1] Kenrick Bingham, Yaroslav Kurylev, Matti Lassas, Samuli Siltanen. Iterative time-reversal control for inverse problems. Inverse Problems & Imaging, 2008, 2 (1) : 63-81. doi: 10.3934/ipi.2008.2.63 [2] Jingzhi Li, Jun Zou. A direct sampling method for inverse scattering using far-field data. Inverse Problems & Imaging, 2013, 7 (3) : 757-775. doi: 10.3934/ipi.2013.7.757 [3] Huai-An Diao, Peijun Li, Xiaokai Yuan. Inverse elastic surface scattering with far-field data. Inverse Problems & Imaging, 2019, 13 (4) : 721-744. doi: 10.3934/ipi.2019033 [4] Huey-Er Lin, Jian-Guo Liu, Wen-Qing Xu. Effects of small viscosity and far field boundary conditions for hyperbolic systems. Communications on Pure & Applied Analysis, 2004, 3 (2) : 267-290. doi: 10.3934/cpaa.2004.3.267 [5] Batoul Abdelaziz, Abdellatif El Badia, Ahmad El Hajj. Some remarks on the small electromagnetic inhomogeneities reconstruction problem. Inverse Problems & Imaging, 2017, 11 (6) : 1027-1046. doi: 10.3934/ipi.2017047 [6] Benoît Pausader, Walter A. Strauss. Analyticity of the nonlinear scattering operator. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 617-626. doi: 10.3934/dcds.2009.25.617 [7] Roland Griesmaier, Nuutti Hyvönen, Otto Seiskari. A note on analyticity properties of far field patterns. Inverse Problems & Imaging, 2013, 7 (2) : 491-498. doi: 10.3934/ipi.2013.7.491 [8] Rodrigo I. Brevis, Jaime H. Ortega, David Pardo. A source time reversal method for seismicity induced by mining. Inverse Problems & Imaging, 2017, 11 (1) : 25-45. doi: 10.3934/ipi.2017002 [9] Rodica Toader. Scattering in domains with many small obstacles. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 321-338. doi: 10.3934/dcds.1998.4.321 [10] Peter C. Gibson. On the measurement operator for scattering in layered media. Inverse Problems & Imaging, 2017, 11 (1) : 87-97. doi: 10.3934/ipi.2017005 [11] Anudeep Kumar Arora. Scattering of radial data in the focusing NLS and generalized Hartree equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6643-6668. doi: 10.3934/dcds.2019289 [12] Julien Chambarel, Christian Kharif, Olivier Kimmoun. Focusing wave group in shallow water in the presence of wind. Discrete & Continuous Dynamical Systems - B, 2010, 13 (4) : 773-782. doi: 10.3934/dcdsb.2010.13.773 [13] Zhigang Wang. Vanishing viscosity limit of the rotating shallow water equations with far field vacuum. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 311-328. doi: 10.3934/dcds.2018015 [14] Olha Ivanyshyn. Shape reconstruction of acoustic obstacles from the modulus of the far field pattern. Inverse Problems & Imaging, 2007, 1 (4) : 609-622. doi: 10.3934/ipi.2007.1.609 [15] Giovanni Alessandrini, Eva Sincich, Sergio Vessella. Stable determination of surface impedance on a rough obstacle by far field data. Inverse Problems & Imaging, 2013, 7 (2) : 341-351. doi: 10.3934/ipi.2013.7.341 [16] Qi Wang, Yanren Hou. Determining an obstacle by far-field data measured at a few spots. Inverse Problems & Imaging, 2015, 9 (2) : 591-600. doi: 10.3934/ipi.2015.9.591 [17] Albert Fannjiang, Knut Solna. Time reversal of parabolic waves and two-frequency Wigner distribution. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 783-802. doi: 10.3934/dcdsb.2006.6.783 [18] Kazufumi Ito, Karim Ramdani, Marius Tucsnak. A time reversal based algorithm for solving initial data inverse problems. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 641-652. doi: 10.3934/dcdss.2011.4.641 [19] Masaru Ikehata. The enclosure method for inverse obstacle scattering using a single electromagnetic wave in time domain. Inverse Problems & Imaging, 2016, 10 (1) : 131-163. doi: 10.3934/ipi.2016.10.131 [20] Bingbing Ding, Ingo Witt, Huicheng Yin. Blowup time and blowup mechanism of small data solutions to general 2-D quasilinear wave equations. Communications on Pure & Applied Analysis, 2017, 16 (3) : 719-744. doi: 10.3934/cpaa.2017035

2018 Impact Factor: 1.469

## Metrics

• HTML views (0)
• Cited by (3)

• on AIMS