Communications in Mathematical Physics

, Volume 317, Issue 1, pp 253–266 | Cite as

Enhancement of Near Cloaking Using Generalized Polarization Tensors Vanishing Structures. Part I: The Conductivity Problem

  • Habib Ammari
  • Hyeonbae Kang
  • Hyundae Lee
  • Mikyoung Lim


The aim of this paper is to provide an original method of constructing very effective near-cloaking structures for the conductivity problem. These new structures are such that their first Generalized Polarization Tensors (GPT) vanish. We show that this in particular significantly enhances the cloaking effect. We then present some numerical examples of Generalized Polarization Tensors vanishing structures.


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  1. 1.
    Alú A., Engheta N.: Achieving transparency with plasmonic and metamaterial coatings. Phys. Rev. E 72, 016623 (2005)ADSCrossRefGoogle Scholar
  2. 2.
    Alú A., Engheta N.: Cloaking and transparency for collections of particles with metamaterial and plasmonic covers. Optics Express 15, 7578–7590 (2007)ADSCrossRefGoogle Scholar
  3. 3.
    Ammari, H., Boulier, T., Garnier, J., Jing, W., Kang, H., Wang, H.: Target identification using dictionary matching of generalized polarization tensors. Submitted to Found. Comp. Math. available at [math.oc], 2012
  4. 4.
    Ammari, H., Ciraolo, G., Kang, H., Lee, H., Milton, G.: Spectral theory of a Neumann-Poincaré-type operator and analysis of cloaking due to anomalous localized resonance. Submitted, available at [math.AP], 2012
  5. 5.
    Ammari, H., Deng, Y., Kang, H., Lee, H.: Reconstruction of inhomogeneous conductivities via generalized polarization tensors. SubmittedGoogle Scholar
  6. 6.
    Ammari H., Garnier J., Jugnon V., Kang H., Lee H., Lim M.: Enhancement of near-cloaking. Part III: Numerical simulations, statistical stability, and related questions. Contemp. Math. 577, 1–24 (2012)CrossRefGoogle Scholar
  7. 7.
    Ammari, H., Kang, H.: Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory, Applied Mathematical Sciences, Vol. 162, New York: Springer-Verlag, 2007Google Scholar
  8. 8.
    Ammari, H., Kang, H., Lee, H., Lim, M.: Enhancement of near-cloaking. Part II: The Helmholtz equation. Commun. Math. Phys., 2012. doi:10.1007/s00220-012-1620-y
  9. 9.
    Astala K., Lassas M., Päivärinta L.: Calderon’s inverse problem for anisotropic conductivity in the plane. Comm. Part. Diff. Equat. 30, 207–224 (2005)MATHCrossRefGoogle Scholar
  10. 10.
    Astala K., Päivärinta L.: Calderon’s inverse conductivity problem in the plane. Ann. Math. 163, 265–299 (2006)MATHCrossRefGoogle Scholar
  11. 11.
    Bryan K., Leise T.: Impedance Imaging, inverse problems, and Harry Potter’s Cloak. SIAM Rev. 52, 359–377 (2010)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Greenleaf A., Kurylev Y., Lassas M., Uhlmann G.: Cloaking devices, electromagnetic wormholes, and transformation optics. SIAM Rev. 51, 3–33 (2009)MathSciNetADSMATHCrossRefGoogle Scholar
  13. 13.
    Greenleaf A., Lassas M., Uhlmann G.: On nonuniqueness for Calderon’s inverse problem. Math. Res. Lett. 10, 685–693 (2003)MathSciNetMATHGoogle Scholar
  14. 14.
    Nguyen H.M.: Cloaking via change of variables for the Helmholtz equation in the whole space. Comm. Pure Appl. Math. 63, 1505–1524 (2010)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Kohn R.V., Onofrei D., Vogelius M.S., Weinstein M.I.: Cloaking via change of variables for the Helmholtz equation. Comm. Pure Appl. Math. 63, 973–1016 (2010)MathSciNetMATHGoogle Scholar
  16. 16.
    Kohn R.V., Shen H., Vogelius M.S., Weinstein M.I.: Cloaking via change of variables in electric impedance tomography. Inverse Problems 24, 015016 (2008)MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    Kohn R., Vogelius M.: Determining conductivity by boundary measurements. Comm. Pure and Appl. Math. 37, 289–298 (1984)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Leonhardt U.: Optical conforming mapping. Science 312(5781), 1777–1780 (2006)MathSciNetADSMATHCrossRefGoogle Scholar
  19. 19.
    Leonhardt U., Tyc T.: Broadband invisibility by non-euclidean cloaking. Science 323, 110–111 (2009)ADSCrossRefGoogle Scholar
  20. 20.
    Liu H.: Virtual reshaping and invisibility in obstacle scattering. Inverse Problems 25, 044006 (2009)CrossRefGoogle Scholar
  21. 21.
    Milton, G.W.: The Theory of Composites. Cambridge Monographs on Applied and Computational Mathematics, Cambridge: Cambridge University Press, 2001Google Scholar
  22. 22.
    Milton G.W., Nicorovici N.A.: On the cloaking effects associated with anomalous localized resonance. Proc. R. Soc. A 462, 3027–3059 (2006)MathSciNetADSMATHCrossRefGoogle Scholar
  23. 23.
    Milton G.W., Nicorovici N.A., McPhedran R.C., Podolskiy V.A.: A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance. Proc. R. Soc. A 461, 3999–4034 (2005)MathSciNetADSMATHCrossRefGoogle Scholar
  24. 24.
    Nachman A.: Global uniqueness for a two-dimensional inverse boundary value problem. Ann. Math. 142, 71–96 (1996)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Pendry J.B., Schurig D., Smith D.R.: Controlling electromagnetic fields. Science 312, 1780–1782 (2006)MathSciNetADSMATHCrossRefGoogle Scholar
  26. 26.
    Sylvester J., Uhlmann G.: A global uniqueness theorem for an inverse boundary value problem. Ann. Math. 125, 153–169 (1987)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Habib Ammari
    • 1
  • Hyeonbae Kang
    • 2
  • Hyundae Lee
    • 2
  • Mikyoung Lim
    • 3
  1. 1.Department of Mathematics and ApplicationsEcole Normale SupérieureParisFrance
  2. 2.Department of MathematicsInha UniversityIncheonKorea
  3. 3.Department of Mathematical SciencesKorean Advanced Institute of Science and TechnologyDaejeonKorea

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