Advertisement

Negative Index Materials: Some Mathematical Perspectives

  • Hoai-Minh Nguyen
Article
  • 23 Downloads

Abstract

Negative index materials are artificial structures whose refractive index has a negative value over some frequency range. These materials were postulated and investigated theoretically by Veselago in 1964 and were confirmed experimentally by Shelby, Smith, and Schultz in 2001. New fabrication techniques now allow for the construction of negative index materials at scales that are interesting for applications, which has made them a very active topic of investigation. In this paper, we report various mathematical results on the properties of negative index materials and their applications. The topics discussed herein include superlensing using complementary media, cloaking using complementary media, cloaking an object via anomalous localized resonance, and the well-posedness and the finite speed propagation in media consisting of dispersive metamaterials. Some of the results have been refined and have simpler proofs than the original ones.

Keywords

Superlensing Cloaking Finite speed propagation Complementary media Negative index metamaterials. 

Mathematics Subject Classification (2010)

35B34 35B35 35J05 35Q60. 

Notes

Acknowledgements

This paper is an extended version of the lecture given by the author at VIASM annual meeting in 2017 at Vietnam Institute for Advanced Study in Mathematics. The author warmly thanks the institute for the hospitality.

References

  1. 1.
    Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II. Comm. Pure Appl. Math. 17, 35–92 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abdulle, A., Huber, M.E., Lemaire, S.: An optimization-based numerical method for diffusion problems with sign-changing coefficients. C. R. Math. Acad. Sci. Paris 355, 472–478 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    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)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bethuel, F., Brezis, H., Helein, F.: Ginzburg Landau Vortices. Progress in Nonlinear Differential Equations and Their Applications, vol. 13. Birkhäuser, Boston (1994)zbMATHGoogle Scholar
  5. 5.
    Bonnet-Ben Dhia, A.S., Chesnel, L., Ciarlet, P.: T-coercivity for scalar interface problems between dielectrics and metamaterials. ESAIM Math. Model. Numer. Anal. 46, 1363–1387 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bonnet-Ben Dhia, A.S., Ciarlet, P., Zwölf, C. M.: A new compactness result for electromagnetic waves. Application to the transmission problem between dielectrics and metamaterials. Math. Models Methods Appl. Sci. 18, 1605–1631 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bonnetier, E., Nguyen, H. -M.: Superlensing using hyperbolic metamaterials: the scalar case. J. Éc. polytech. Math. 4, 973–1003 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bouchitté, G., Felbacq, D.: Homogenization near resonances and artificial magnetism from dielectrics. C. R. Math. Acad. Sci. Paris 339, 377–382 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bouchitté, G., Schweizer, B.: Cloaking of small objects by anomalous localized resonance. Quart. J. Mech. Appl. Math. 63, 437–463 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chen, Y., Lipton, R.: Resonance and double negative behavior in metamaterials. Arch. Ration. Mech. Anal. 209, 835–868 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Costabel, M., Stephan, E.: A direct boundary integral equation method for transmission problems. J. Math. Anal. Appl. 106, 367–413 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Evans, L.C.: Partial Differential Equations Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (1998)Google Scholar
  13. 13.
    Gralak, B., Tip, A.: Macroscopic Maxwell’s equations and negative index materials. J. Math. Phys. 51, 052902 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Guenneau, S., Zolla, F.: Homogenization of 3D finite chiral photonic crystals. Phys. B: Condens. Matter 394, 145–147 (2007)CrossRefzbMATHGoogle Scholar
  15. 15.
    Jackson, J.D.: Classical Electrodynamics. Wiley, NY (1999)zbMATHGoogle Scholar
  16. 16.
    Cassier, M., Hazard, C., Joly, P.: Spectral theory for Maxwell’s equations at the interface of a metamaterial. Part I: generalized Fourier transform. Comm. Partial Diff. Equat. 42(11), 1707–1748 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kohn, R.V., Lu, J., Schweizer, B., Weinstein, M.I.: A variational perspective on cloaking by anomalous localized resonance. Comm. Math. Phys. 328, 1–27 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kohn, R.V., Shipman, S.P.: Magnetism and homogenization of microresonators. Multiscale Model Simul. 7, 62–92 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lai, Y., Chen, H., Zhang, Z.Q., 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
  20. 20.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Milton, G.W., Nicorovici, N.A.P.: On the cloaking effects associated with anomalous localized resonance. Proc. R. Soc. Lond. Ser. A 462, 3027–3059 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Nguyen, H. -M.: Asymptotic behavior of solutions to the Helmholtz equations with sign changing coefficients. Trans. Am. Math. Soc. 367, 6581–6595 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Nguyen, H. -M.: Superlensing using complementary media. Ann. Inst. H. Poincaré Anal. Non Linéaire 32, 471–484 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Nguyen, H. -M.: Cloaking via anomalous localized resonance for doubly complementary media in the quasistatic regime. J. Eur. Math. Soc. (JEMS) 17, 1327–1365 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Nguyen, H. -M.: Cloaking using complementary media in the quasistatic regime. Ann. Inst. H. Poincaré Anal. Non Linéaire 33, 1509–1518 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Nguyen, H. -M.: Negative index materials and their applications: recent mathematics progress. Chin. Ann. Math. Ser. B 38, 601–628 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Nguyen, H.-M.: Superlensing using complementary media and reflecting complementary media for electromagnetic waves. Adv. Nonlinear Anal. to appear,  https://doi.org/10.1515/anona-2017-0146
  30. 30.
    Nguyen, H.-M.: Cloaking via anomalous localized resonance for doubly complementary media in the finite frequency regime. J. Anal. Math. to appear, arXiv:https://arxiv.org/abs/1511.08053
  31. 31.
    Nguyen, H.-M.: Cloaking using complementary media for electromagnetic waves. ESAIM Control Optim. Calc. Var., to appear,  https://doi.org/10.1051/cocv/2017078
  32. 32.
    Nguyen, H. -M., Nguyen, H.L.: Complete resonance and localized resonance in plasmonic structures. ESAIM: Math. Model. Numer. Anal. 49, 741–754 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    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. Ser. B 2, 93–112 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Nguyen, H.-M., Vinoles, V.: Electromagnetic wave propagation in dispersive metamaterials. submitted, arXiv:https://arxiv.org/abs/1710.08648
  35. 35.
    Nguyen, H. -M., Nguyen, L.: Generalized impedance boundary conditions for scattering by strongly absorbing obstacles for the full wave equation: the scalar case. Math. Models Methods Appl. Sci. 25, 1927–1960 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Nguyen, H. -M., Vogelius, M.S.: Approximate cloaking for the full wave equation via change of variables: the Drude-Lorentz model. J. Math. Pures Appl. 106, 797–836 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Nicorovici, N.A., McPhedran, R.C., Milton, G.M.: Optical and dielectric properties of partially resonant composites. Phys. Rev. B 49, 8479–8482 (1994)CrossRefGoogle Scholar
  38. 38.
    Ola, P.: Remarks on a transmission problem. J. Math. Anal. Appl. 16, 639–658 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Pendry, J.B.: Negative refraction makes a perfect lens. Phys. Rev. Lett. 85, 3966–3969 (2000)CrossRefGoogle Scholar
  40. 40.
    Pendry, J.B.: Perfect cylindrical lenses. Opt. Express 1, 755–760 (2003)CrossRefGoogle Scholar
  41. 41.
    Protter, M.H.: Unique continuation for elliptic equations. Trans. Am. Math. Soc. 95, 81–91 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Ramakrishna, S.A., Pendry, J.B.: Focusing light using negative refraction. J. Phys. Condens. Matter 15, 6345–6364 (2003)CrossRefGoogle Scholar
  43. 43.
    Ramakrishna, S.A., Pendry, J.B.: Spherical perfect lens: solutions of Maxwell’s equations for spherical geometry. Phys. Rev. B 69, 115115 (2004)CrossRefGoogle Scholar
  44. 44.
    Shelby, R.A., Smith, D.R., Schultz, S.: Experimental Verification of a Negative Index of Refraction. Science 292, 77–79 (2001)CrossRefGoogle Scholar
  45. 45.
    Veselago, V.G.: The electrodynamics of substances with simultaneously negative values of 𝜖 and μ. Usp. Fiz. Nauk 92, 517–526 (1964)CrossRefGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of MathematicsEPFL SB CAMALausanneSwitzerland

Personalised recommendations