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The invisibility via anomalous localized resonance of a source for electromagnetic waves

  • Hoai-Minh NguyenEmail author
Research
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Abstract

We investigate the invisibility via anomalous localized resonance of a general source in anisotropic media for electromagnetic waves. To this end, we first introduce the concept of doubly complementary media in the electromagnetic setting. These are media consisting of negative-index metamaterials in a shell and positive-index materials in its complement for which the shell is complementary to a part of the core and a part of the exterior of the core–shell structure. We then provide criteria for establishing the invisibility of a source in these media. We show that (i) a source is invisible if the power is blown up; (ii) a source is invisible if it is sufficiently close to the plasmonic shell structure, and it is visible if it is far from this plasmonic structure; (iii) if the plasmonic structure is complementary to an annulus of constant isotropic medium, there is a critical length that characterizes the cloaking phenomena, as first observed in the two-dimensional acoustic quasistatic setting by Milton and Nicorovici.

Notes

References

  1. 1.
    Alessandrini, G., Rondi, L., Rosset, E., Vessella, S.: The stability for the Cauchy problem for elliptic equations. Inverse Probl. 25, 123004 (2009)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alonso, A., Valli, A.: Some remarks on the characterization of the space of tangential traces of \(H(\text{ rot }; \Omega )\) and the construction of an extension operator. Manuscr. Math. 89, 159–178 (1996)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Alu, A., Engheta, N.: Achieving transparency with plasmonic and metamaterial coatings. Phys. Rev. E 95, 106623 (2005)Google Scholar
  4. 4.
    Ammari, H., Ciraolo, G., Kang, H., Lee, H., Milton, G.W.: Anomalous localized resonance using a folded geometry in three dimensions. Proc. R. Soc. Lond. Ser. A 469, 20130048 (2013)CrossRefGoogle Scholar
  5. 5.
    Ammari, H., Ciraolo, G., Kang, H., Lee, H., Milton, G.W.: Spectral theory of a Neumann-Poincaré-type operator and analysis of cloaking due to anomalous localized resonance. Arch. Rational Mech. Anal. 218, 667–692 (2013)CrossRefGoogle Scholar
  6. 6.
    Ando, K., Kang, H., Liu, H.: Plasmon resonance with finite frequencies: a validation of the quasi-static approximation for diametrically small inclusions. SIAM J. Appl. Math. 76, 731–749 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ball, J., Capdeboscq, Y., Tsering-Xiao, B.: On uniqueness for time harmonic anisotropic Maxwell’s equations with piecewise regular coefficients. Math. Models Methods Appl. Sci. 22, 1250036 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bouchitté, G., Bourel, C., Felbacq, D.: Homogenization near resonances and artificial magnetism in three dimensional dielectric metamaterials. Arch. Ration. Mech. Anal. 225, 1233–1277 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bouchitté, G., Schweizer, B.: Cloaking of small objects by anomalous localized resonance. Q. J. Mech. Appl. Math. 63, 437–463 (2010)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Buffa, A., Costabel, M., Sheen, D.: On traces for \(H(\text{ curl },\Omega )\) in Lipschitz domains. J. Math. Anal. Appl. 276, 845–867 (2002)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory. Applied Mathematical Sciences, vol. 98, 2nd edn. Springer, Berlin (1998)CrossRefGoogle Scholar
  12. 12.
    Hadamard, J.: Sur les fonction entières. Bull. Soc. Math. France 24, 94–96 (1896)zbMATHGoogle Scholar
  13. 13.
    Kettunen, H., Lassas, M., Ola, P.: On absence and existence of the anomalous localized resonance without the quasi-static approximation. SIAM J. Appl. Math. 78, 609–628 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kirsch, A., Hettlich, F.: The Mathematical Theory of Time-Harmonic Maxwell’s Equations, Expansion, Integral, and Variational Methods. Springer, Berlin (2015)CrossRefGoogle Scholar
  15. 15.
    Kohn, R.V., Lu, J., Schweizer, B., Weinstein, M.I.: A variational perspective on cloaking by anomalous localized resonance. Commun. Math. Phys. 328, 1–27 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lai, Y., Chen, H., Zhang, Z., Chan, C.T.: Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell. Phys. Rev. Lett. 102, 093901 (2009)CrossRefGoogle Scholar
  17. 17.
    McPhedran, R.C., Nicorovici, N.A., Botten, L.C., Milton, G.W.: Cloaking by plasmonic resonance among systems of particles: cooperation or combat. C. R. Phys. 10, 391–399 (2009)CrossRefGoogle Scholar
  18. 18.
    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. Lond. Ser. A 461, 3999–4034 (2005)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Milton, G.W., Nicorovici, N.A.: On the cloaking effects associated with anomalous localized resonance. Proc. R. Soc. Lond. Ser. A 462, 3027–3059 (2006)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Nguyen, H.-M.: Asymptotic behavior of solutions to the Helmholtz equations with sign changing coefficients. Trans. Am. Math. Soc. 367, 6581–6595 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Nguyen, H.-M.: Superlensing using complementary media. Ann. Inst. H. Poincaré Anal. Non Linéaire 32, 471–484 (2015)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Nguyen, H.-M.: Cloaking via anomalous localized resonance. A connection between the localized resonance and the blow up of the power for doubly complementary media. C. R. Math. Acad. Sci. Paris 353, 41–46 (2015)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Nguyen, H.-M.: Cloaking via anomalous localized resonance for doubly complementary media in the quasi static regime. J. Eur. Math. Soc. (JEMS) 17, 1327–1365 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Nguyen, H.-M.: Cloaking using complementary media in the quasistatic regime. Ann. Inst. H. Poincaré Anal. Non Linéaire 33, 1509–1518 (2016)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Nguyen, H.-M.: Limiting absorption principle and well-posedness for the Helmholtz equation with sign changing coefficients. J. Math. Pures Appl. 106, 342–374 (2016)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Nguyen, H.-M.: Cloaking an arbitrary object via anomalous localized resonance: the cloak is independent of the object. SIAM J. Math. Anal. 49, 3208–3232 (2017)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Nguyen, H.-M.: Superlensing using complementary media and reflecting complementary media for electromagnetic waves. Adv. Nonlinear Anal. 7, 449–467 (2018)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Nguyen, H.-M.: Negative index materials: some mathematical perspectives. Acta Math. Vietnam. 44, 325–349 (2019)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Nguyen, H.-M.: Cloaking using complementary for electromagnetic waves. ESAIM Control Optim. Calc. Var. 25, 29 (2019)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Nguyen, H.-M.: Cloaking via anomalous localized resonance for doubly complementary media in the finite frequency regime. J. Anal. Math. 138, 157–184 (2019)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Nguyen, H.-M., Nguyen, H.L.: Complete resonance and localized resonance in plasmonic structures. ESAIM Math. Model. Numer. Anal. 49, 741–754 (2015)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Nguyen, H.-M., Nguyen, H.L.: Cloaking using complementary media for the Helmholtz equation and a three spheres inequality for second order elliptic equations. Trans. Am. Math. Soc. B 2, 93–112 (2015)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Nguyen, T., Wang, J.-N.: Quantitative uniqueness estimate for the Maxwell system with Lipschitz anisotropic media. Proc. Am. Math. Soc. 140, 595–605 (2012)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Nicorovici, N.A., McPhedran, R.C., Milton, G.W.: Optical and dielectric properties of partially resonant composites. Phys. Rev. B 49, 8479–8482 (1994)CrossRefGoogle Scholar
  35. 35.
    Shelby, R.A., Smith, D.R., Schultz, S.: Experimental verification of a negative index of refraction. Science 292, 77–79 (2001)CrossRefGoogle Scholar
  36. 36.
    Veselago, V.G.: The electrodynamics of substances with simultaneously negative values of \(\varepsilon \) and \(\mu \). Usp. Fiz. Nauk 92, 517–526 (1964)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsEPFL SB CAMALausanneSwitzerland

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