Invisibility Regions and Regular Metamaterials

  • George DassiosEmail author
Part of the Springer Optimization and Its Applications book series (SOIA, volume 91)


For more than half a century people are trying to develop effective methods of remote sensing. RADAR and SONAR systems are the most advertised techniques toward this end, and the contemporary level of sophistication of both of these modalities is really amazing. In fact, so many things have been achieved with these identification methods that the inverse question has been naturally raised: Is it possible to isolate a region of space where nothing can be detected via scattering techniques? Much to our surprise the answer to this question is “yes” and the way to achieve it is knowing as “cloaking.” This is possible through the construction of a material, called “metamaterial” surrounding the cloaked region, which has particular preassigned properties. Cloaking has a history of less than a decade and almost all realistic cloaking regions share the shape of a sphere. However, spherically cloaked regions demand metamaterials with singular conductivity tensors, a consequence of the highly focusing effects of the spherical system as it collapses down to its center. We will demonstrate an ellipsoidal cloaking region, which, as a consequence of the fact that the ellipsoidal system springs from its characteristic focal ellipse, the necessary metamaterial that creates the invisibility region is regular throughout, leaving this way its realization at the level of engineering construction.


Electric Impedance Tomography Spherical Geometry Spherical System Material Tensor Reference Ellipsoid 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    N. Bleistein and J.K. Cohen. Nonuniqueness in the inverse source problem in acoustics and electromagnetics. Journal of Mathematical Physics, 18:194–201, 1977.CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    H. Chen and C.T. Chan. Acoustic cloaking in three dimensions using acoustic metamaterials. Applied Physics Letters, 91, 183518:1–3, 2007.Google Scholar
  3. 3.
    H. Chen and G. Uhlmann. Cloaking a sensor for three-dimensional Maxwell’s equations:transformation optics approach. Optics Express, 19:20518–20530, 2011.CrossRefGoogle Scholar
  4. 4.
    H. Chen, B-I. Wu, B. Zhang, and J.A. Kong. Electromagnetic wave interactions with a metamaterial cloak. Physical Review Letters, 99, 063903:1–4, 2007.Google Scholar
  5. 5.
    A.S. Cummer, B-I. Popa, D. Schurig, D.R. Smith, J. Pendry, M. Rahm, and A. Starr. Scattering theory derivation of a 3D acoustic cloaking shell. Physical Review Letters, PRL100,024301:1–4, 2008.Google Scholar
  6. 6.
    A.S. Cummer and D. Schurig. On path to acoustic cloaking. New Journal of Physics, 9(45):1–8, 2007.Google Scholar
  7. 7.
    G. Dassios. Ellipsoidal Harmonics. Theory and Applications. Cambridge University Press, Cambridge, 2012.Google Scholar
  8. 8.
    A.J. Devaney and E. Wolf. Radiating and nonradiating classical current distributions and the fields they generate. Physical Review D, 8:1044–1047, 1973.CrossRefGoogle Scholar
  9. 9.
    F.G. Friedlander. An inverse problem for radiation fields. Proceedings of the London Mathematical Society, 27:551–576, 1973.CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann. Full-wave invisibility of active devices at all frequencies. Communications in Mathematical Physics, 275:749–789, 2007.CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann. Cloaking devises, electromagnetic wormholes, and transformation optics. SIAM Review, 51:3–33, 2009.CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann. Invisibility and inverse problems. Bulletin of the Americal Mathematical Society, 46:55–97, 2009.CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann. Cloaking a sensor via transformation optics. Physical Review E, 83, 016603:1–6, 2011.Google Scholar
  14. 14.
    A. Greenleaf, M. Lassas, and G. Uhlmann. Anisotropic conductivities that cannot detected in EIT. Physilogical Measurement (special issue on Impedance Tomography), 24:413–420, 2003.Google Scholar
  15. 15.
    A. Greenleaf, M. Lassas, and G. Uhlmann. On nonuniqueness for the Calderón’s inverse problem. Mathematical Research Letters, 10:685–693, 2003.CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    A. Greenleaf, M. Lassas, and G. Uhlmann. The Calderón problem for conormal potentials, I: global uniqueness and reconstraction. Communications in Pure and Applied Mathematics, 56:328–352, 2003.CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    H. Helmholtz. Ueber einige Gesetze der Vertheilung elektrischer Ströme in körperlichen Leitern mit Anwendung auf die thierisch-elektrischen Versuche;. Annalen der Physik und Chemie, 89:211–233 and 353–377, 1853.Google Scholar
  18. 18.
    E.W. Hobson. The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press, U.K., Cambridge, first edition, 1931.Google Scholar
  19. 19.
    M. Kerker. Invisible bodies. Journal of the Optical Society of America, 65:376–379, 1975.CrossRefGoogle Scholar
  20. 20.
    U. Leonhardt. Optical conformal mapping. Science, 312:1777–1780, 2006.CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    A. Norris. Acoustic cloaking theory. Proceedings of the Royal Society A, 464:2411–2434, 2008.CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    J.B. Pendry, D. Schurig, and D.R. Smith. Controlling electromagnetic fields. Science, 312:1780–1782, 2006.CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    R. Weder. A rigorous analysis of high-order electromagnetic invisibility cloaks. Journal of Physics A: Mathematical and Theoretical, 41, 065207:1–21, 2008.Google Scholar
  24. 24.
    E.T. Whittaker and G.N. Watson. A Course of Modern Analysis. Cambridge University Press, 3rd edition, 1920.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of Chemical Engineering, Division of Applied MathematicsUniversity of Patras and ICE-HT/FORTHPatrasGreece

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