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

Corinna Burkard; Aurelia Minut; Karim Ramdani

doi:10.3934/ipi.2013.7.445

Article Contents

2013, Volume 7, Issue 2: 445-470. Doi: 10.3934/ipi.2013.7.445

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

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

Received: October 31, 2011

Revised: January 31, 2013

Published: April 2013

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Abstract

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.

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Mathematics Subject Classification: Primary: 74J20, 35P25; Secondary: 78M35, 35P15, 35B40.

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