Skip to main content
SpringerLink
Log in

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

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alú A., Engheta N.: Achieving transparency with plasmonic and metamaterial coatings. Phys. Rev. E 72, 016623 (2005)

    Article  ADS  Google Scholar 

  2. Alú A., Engheta N.: Cloaking and transparency for collections of particles with metamaterial and plasmonic covers. Optics Express 15, 7578–7590 (2007)

    Article  ADS  Google Scholar 

  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 http://arxiv.org/abs/1204.3035 [math.oc], 2012

  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 http://arxiv.org/abs/1109.0979v2 [math.AP], 2012

  5. Ammari, H., Deng, Y., Kang, H., Lee, H.: Reconstruction of inhomogeneous conductivities via generalized polarization tensors. Submitted

  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)

    Article  Google Scholar 

  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, 2007

  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. Astala K., Lassas M., Päivärinta L.: Calderon’s inverse problem for anisotropic conductivity in the plane. Comm. Part. Diff. Equat. 30, 207–224 (2005)

    Article  MATH  Google Scholar 

  10. Astala K., Päivärinta L.: Calderon’s inverse conductivity problem in the plane. Ann. Math. 163, 265–299 (2006)

    Article  MATH  Google Scholar 

  11. Bryan K., Leise T.: Impedance Imaging, inverse problems, and Harry Potter’s Cloak. SIAM Rev. 52, 359–377 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Greenleaf A., Kurylev Y., Lassas M., Uhlmann G.: Cloaking devices, electromagnetic wormholes, and transformation optics. SIAM Rev. 51, 3–33 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. Greenleaf A., Lassas M., Uhlmann G.: On nonuniqueness for Calderon’s inverse problem. Math. Res. Lett. 10, 685–693 (2003)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  ADS  Google Scholar 

  17. Kohn R., Vogelius M.: Determining conductivity by boundary measurements. Comm. Pure and Appl. Math. 37, 289–298 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  18. Leonhardt U.: Optical conforming mapping. Science 312(5781), 1777–1780 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. Leonhardt U., Tyc T.: Broadband invisibility by non-euclidean cloaking. Science 323, 110–111 (2009)

    Article  ADS  Google Scholar 

  20. Liu H.: Virtual reshaping and invisibility in obstacle scattering. Inverse Problems 25, 044006 (2009)

    Article  Google Scholar 

  21. Milton, G.W.: The Theory of Composites. Cambridge Monographs on Applied and Computational Mathematics, Cambridge: Cambridge University Press, 2001

  22. Milton G.W., Nicorovici N.A.: On the cloaking effects associated with anomalous localized resonance. Proc. R. Soc. A 462, 3027–3059 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. Nachman A.: Global uniqueness for a two-dimensional inverse boundary value problem. Ann. Math. 142, 71–96 (1996)

    Article  MathSciNet  Google Scholar 

  25. Pendry J.B., Schurig D., Smith D.R.: Controlling electromagnetic fields. Science 312, 1780–1782 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. Sylvester J., Uhlmann G.: A global uniqueness theorem for an inverse boundary value problem. Ann. Math. 125, 153–169 (1987)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Applications, Ecole Normale Supérieure, 45 Rue d’Ulm, 75005, Paris, France

    Habib Ammari

  2. Department of Mathematics, Inha University, Incheon, 402-751, Korea

    Hyeonbae Kang & Hyundae Lee

  3. Department of Mathematical Sciences, Korean Advanced Institute of Science and Technology, Daejeon, 305-701, Korea

    Mikyoung Lim

Authors
  1. Habib Ammari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hyeonbae Kang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hyundae Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Mikyoung Lim
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Habib Ammari.

Additional information

Communicated by S. Zelditch

This work was supported by National Institute for Mathematical Sciences (2010 Thematic Program, TP1003), ERC Advanced Grant Project MULTIMOD–267184, Korea Research Foundation through grant KRF-2008-220-C00002, and NRF grants No. 2009-0085987, 2010-0017532, and 2010-0004091, and grants from Inha University.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ammari, H., Kang, H., Lee, H. et al. Enhancement of Near Cloaking Using Generalized Polarization Tensors Vanishing Structures. Part I: The Conductivity Problem. Commun. Math. Phys. 317, 253–266 (2013). https://doi.org/10.1007/s00220-012-1615-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-012-1615-8

Keywords

Search

Navigation