# 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 and Imaging, 2013, 7 (2) : 445-470. doi: 10.3934/ipi.2013.7.445
##### References:
 [1] M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables," 55 of National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C., 1964. [2] H. Ammari, E. Iakovleva, D. Lesselier and G. Perrusson, Music-type electromagnetic imaging of a collection of small three-dimensional inclusions, SIAM J. Sci. Comput., 29 (2007), 674-709. doi: 10.1137/050640655. [3] H. Ammari, E. Iakovleva and S. Moskow, Recovery of small inhomogeneities from the scattering amplitude at a fixed frequency, SIAM J. Math. Anal., 34 (2003), 882-900. doi: 10.1137/S0036141001392785. [4] X. Antoine, B. Pinçon, K. Ramdani and B. Thierry, Far field modeling of electromagnetic time reversal and application to selective focusing on small scatterers, SIAM J. Appl. Math., 69 (2008), 830-844. doi: 10.1137/080715779. [5] T. Arens, A. Lechleiter and D. R. Luke, Music for extended scatterers as an instance of the factorization method, SIAM J. Appl. Math., 70 (2009), 1283-1304. doi: 10.1137/080737836. [6] C. Ben Amar, N. Gmati, C. Hazard and K. Ramdani, Numerical simulation of acoustic time reversal mirrors, SIAM J. Appl. Math., 67 (2007), 777-791. doi: 10.1137/060654542. [7] L. Borcea, G. Papanicolaou and F. G. Vasquez, Edge illumination and imaging of extended reflectors, SIAM J. Imaging Sciences, 1 (2008), 75-114. doi: 10.1137/07069290X. [8] L. Borcea, G. Papanicolaou and F. G. Vasquez, Edge illumination and imaging of extended reflectors, SIAM J. Imaging Sciences, 1 (2008), 75-114. doi: 10.1137/07069290X. [9] D. H. Chambers and J. G. Berryman, Target characterization using decomposition of the time-reversal operator: Electromagnetic scattering from small ellipsoids, Inverse Problems, 22 (2006), 2145-2163. doi: 10.1088/0266-5611/22/6/014. [10] Q. Chen, H. Haddar, A. Lechleiter and P. Monk, A sampling method for inverse scattering in the time domain, Inverse Problems, 26 (2010), 085001. doi: 10.1088/0266-5611/26/8/085001. [11] M. Cheney, The linear sampling method and the {MUSIC algorithm}, Inverse Problems, 17 (2001), 591-595. doi: 10.1088/0266-5611/17/4/301. [12] D. Colton, H. Haddar and M. Piana, The linear sampling method in inverse electromagnetic scattering theory, Inverse Problems, 19 (2003), S105-S137. doi: 10.1088/0266-5611/19/6/057. [13] D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393. doi: 10.1088/0266-5611/12/4/003. [14] D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," Springer-Verlag, Berlin, second edition, 1998. [15] A. Devaney, E. Marengo and F. Gruber, Time-reversal-based imaging and inverse scattering of multiply scattering point targets, J. Acoust. Soc. Amer., 118 (2005), 3129-3138. doi: 10.1121/1.2042987. [16] A. Fannjiang, On time reversal mirrors, Inverse Problems, 25 (2009), 095010. doi: 10.1088/0266-5611/25/9/095010. [17] M. Fink, Acoustic time-reversal mirrors, in "Imaging of Complex Media with Acoustic and Seismic Waves" (eds. E. H. Zarantonello and Author 2), Springer, (2002), 17-43. [18] M. Fink, D. Cassereau, A. Derode, C. Prada, P. Roux, M. Tanter and J.-L. Thomas, Time-reversed acoustics, Rep. Prog. Phys., 63 (2000), 1933-1995. [19] M. Fink and C. Prada, Acoustic time-reversal mirrors, Inverse Problems, 17 (2001), 1761-1773. [20] N. A. Gumerov and R. Duraiswami, Computation scattering from n spheres using multipole reexpansion, J. Acoust. Soc. Amer., 112 (2002), 2688-2701. doi: 10.1121/1.1517253. [21] C. Hazard and K. Ramdani, Selective acoustic focusing using time-harmonic reversal mirrors, SIAM J. Appl. Math., 64 (2004), 1057-1076. doi: 10.1137/S0036139903428732. [22] S. Hou, K. Solna and H. Zhao, Imaging of location and geometry for extended targets using the response matrix, J. Comput. Phys., 199 (2004), 317-338. doi: 10.1016/j.jcp.2004.02.010. [23] E. Iakovleva and D. Lesselier, Multistatic response matrix of spherical scatterers and the back-propagation of singular fields, IEEE Trans. Antenna. Prop., 56 (2008), 825-833. doi: 10.1109/TAP.2008.916913. [24] A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems, 14 (1998), 1489-1512. doi: 10.1088/0266-5611/14/6/009. [25] A. Kirsch, Factorization of the far-field operator for the inhomogeneous medium case and an application in inverse scattering theory, Inverse Problems, 15 (1999), 413-429. doi: 10.1088/0266-5611/15/2/005. [26] A. Kirsch, New characterizations of solutions in inverse scattering theory, Appl. Anal., 76 (2000), 319-350. doi: 10.1080/00036810008840888. [27] A. Kirsch, The MUSIC algorithm and the factorization method in inverse scattering theory for inhomogeneous media, Inverse Problems, 18 (2002), 1025-1040. doi: 10.1088/0266-5611/18/4/306. [28] R. Kress, Minimizing the condition number of boundary integral operators in acoustic and electromagnetic scattering, Quart. J. Mech. Appl. Math., 38 (1985), 323-341. doi: 10.1093/qjmam/38.2.323. [29] G. Micolau, "Etude Théorique et Numérique de la Méthode de la Décomposition de L'opérateur de Retournement Temporel (D.O.R.T.) en Diffraction ÉlectromagnÉtique," Ph.D thesis, Université d'Aix-Marseille, 2001. [30] B. Pinçon and K. Ramdani, Selective focusing on small scatterers in acoustic waveguides using time reversal mirrors, Inverse Problems, 23 (2007), 1-25. doi: 10.1088/0266-5611/23/1/001. [31] C. Prada, S. Manneville, D. Spoliansky and M. Fink, Decomposition of the time reversal operator: Detection and selective focusing on two scatterers, J. Acoust. Soc. Am., 99 (1996), 2067-2076. doi: 10.1121/1.415393. [32] E. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals," Princeton University Press, Princeton, NJ, 1993. [33] B. Thierry, "Analyse et Simulations Numériques du Retournement Temporel et de la Diffraction Multiple," Ph.D thesis, Nancy Université, 2011. [34] H. Tortel, G. Micolau and M. Saillard, Decomposition of the time reversal operator for electromagnetic scattering, J. Electromagn. Waves Appl., 13 (1999), 687-719. doi: 10.1163/156939399X01113. [35] R. Wong, "Asymptotic Approximations of Integrals," 34 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. doi: 10.1137/1.9780898719260. [36] A. Zaanen, "Linear Analysis. Measure and Integral, Banach and Hilbert Space, Linear Integral Equations," Interscience Publishers Inc., New York, 1953.

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##### References:
 [1] M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables," 55 of National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C., 1964. [2] H. Ammari, E. Iakovleva, D. Lesselier and G. Perrusson, Music-type electromagnetic imaging of a collection of small three-dimensional inclusions, SIAM J. Sci. Comput., 29 (2007), 674-709. doi: 10.1137/050640655. [3] H. Ammari, E. Iakovleva and S. Moskow, Recovery of small inhomogeneities from the scattering amplitude at a fixed frequency, SIAM J. Math. Anal., 34 (2003), 882-900. doi: 10.1137/S0036141001392785. [4] X. Antoine, B. Pinçon, K. Ramdani and B. Thierry, Far field modeling of electromagnetic time reversal and application to selective focusing on small scatterers, SIAM J. Appl. Math., 69 (2008), 830-844. doi: 10.1137/080715779. [5] T. Arens, A. Lechleiter and D. R. Luke, Music for extended scatterers as an instance of the factorization method, SIAM J. Appl. Math., 70 (2009), 1283-1304. doi: 10.1137/080737836. [6] C. Ben Amar, N. Gmati, C. Hazard and K. Ramdani, Numerical simulation of acoustic time reversal mirrors, SIAM J. Appl. Math., 67 (2007), 777-791. doi: 10.1137/060654542. [7] L. Borcea, G. Papanicolaou and F. G. Vasquez, Edge illumination and imaging of extended reflectors, SIAM J. Imaging Sciences, 1 (2008), 75-114. doi: 10.1137/07069290X. [8] L. Borcea, G. Papanicolaou and F. G. Vasquez, Edge illumination and imaging of extended reflectors, SIAM J. Imaging Sciences, 1 (2008), 75-114. doi: 10.1137/07069290X. [9] D. H. Chambers and J. G. Berryman, Target characterization using decomposition of the time-reversal operator: Electromagnetic scattering from small ellipsoids, Inverse Problems, 22 (2006), 2145-2163. doi: 10.1088/0266-5611/22/6/014. [10] Q. Chen, H. Haddar, A. Lechleiter and P. Monk, A sampling method for inverse scattering in the time domain, Inverse Problems, 26 (2010), 085001. doi: 10.1088/0266-5611/26/8/085001. [11] M. Cheney, The linear sampling method and the {MUSIC algorithm}, Inverse Problems, 17 (2001), 591-595. doi: 10.1088/0266-5611/17/4/301. [12] D. Colton, H. Haddar and M. Piana, The linear sampling method in inverse electromagnetic scattering theory, Inverse Problems, 19 (2003), S105-S137. doi: 10.1088/0266-5611/19/6/057. [13] D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393. doi: 10.1088/0266-5611/12/4/003. [14] D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," Springer-Verlag, Berlin, second edition, 1998. [15] A. Devaney, E. Marengo and F. Gruber, Time-reversal-based imaging and inverse scattering of multiply scattering point targets, J. Acoust. Soc. Amer., 118 (2005), 3129-3138. doi: 10.1121/1.2042987. [16] A. Fannjiang, On time reversal mirrors, Inverse Problems, 25 (2009), 095010. doi: 10.1088/0266-5611/25/9/095010. [17] M. Fink, Acoustic time-reversal mirrors, in "Imaging of Complex Media with Acoustic and Seismic Waves" (eds. E. H. Zarantonello and Author 2), Springer, (2002), 17-43. [18] M. Fink, D. Cassereau, A. Derode, C. Prada, P. Roux, M. Tanter and J.-L. Thomas, Time-reversed acoustics, Rep. Prog. Phys., 63 (2000), 1933-1995. [19] M. Fink and C. Prada, Acoustic time-reversal mirrors, Inverse Problems, 17 (2001), 1761-1773. [20] N. A. Gumerov and R. Duraiswami, Computation scattering from n spheres using multipole reexpansion, J. Acoust. Soc. Amer., 112 (2002), 2688-2701. doi: 10.1121/1.1517253. [21] C. Hazard and K. Ramdani, Selective acoustic focusing using time-harmonic reversal mirrors, SIAM J. Appl. Math., 64 (2004), 1057-1076. doi: 10.1137/S0036139903428732. [22] S. Hou, K. Solna and H. Zhao, Imaging of location and geometry for extended targets using the response matrix, J. Comput. Phys., 199 (2004), 317-338. doi: 10.1016/j.jcp.2004.02.010. [23] E. Iakovleva and D. Lesselier, Multistatic response matrix of spherical scatterers and the back-propagation of singular fields, IEEE Trans. Antenna. Prop., 56 (2008), 825-833. doi: 10.1109/TAP.2008.916913. [24] A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems, 14 (1998), 1489-1512. doi: 10.1088/0266-5611/14/6/009. [25] A. Kirsch, Factorization of the far-field operator for the inhomogeneous medium case and an application in inverse scattering theory, Inverse Problems, 15 (1999), 413-429. doi: 10.1088/0266-5611/15/2/005. [26] A. Kirsch, New characterizations of solutions in inverse scattering theory, Appl. Anal., 76 (2000), 319-350. doi: 10.1080/00036810008840888. [27] A. Kirsch, The MUSIC algorithm and the factorization method in inverse scattering theory for inhomogeneous media, Inverse Problems, 18 (2002), 1025-1040. doi: 10.1088/0266-5611/18/4/306. [28] R. Kress, Minimizing the condition number of boundary integral operators in acoustic and electromagnetic scattering, Quart. J. Mech. Appl. Math., 38 (1985), 323-341. doi: 10.1093/qjmam/38.2.323. [29] G. Micolau, "Etude Théorique et Numérique de la Méthode de la Décomposition de L'opérateur de Retournement Temporel (D.O.R.T.) en Diffraction ÉlectromagnÉtique," Ph.D thesis, Université d'Aix-Marseille, 2001. [30] B. Pinçon and K. Ramdani, Selective focusing on small scatterers in acoustic waveguides using time reversal mirrors, Inverse Problems, 23 (2007), 1-25. doi: 10.1088/0266-5611/23/1/001. [31] C. Prada, S. Manneville, D. Spoliansky and M. Fink, Decomposition of the time reversal operator: Detection and selective focusing on two scatterers, J. Acoust. Soc. Am., 99 (1996), 2067-2076. doi: 10.1121/1.415393. [32] E. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals," Princeton University Press, Princeton, NJ, 1993. [33] B. Thierry, "Analyse et Simulations Numériques du Retournement Temporel et de la Diffraction Multiple," Ph.D thesis, Nancy Université, 2011. [34] H. Tortel, G. Micolau and M. Saillard, Decomposition of the time reversal operator for electromagnetic scattering, J. Electromagn. Waves Appl., 13 (1999), 687-719. doi: 10.1163/156939399X01113. [35] R. Wong, "Asymptotic Approximations of Integrals," 34 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. doi: 10.1137/1.9780898719260. [36] A. Zaanen, "Linear Analysis. Measure and Integral, Banach and Hilbert Space, Linear Integral Equations," Interscience Publishers Inc., New York, 1953.
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