Skip to main content
SpringerLink
Log in

Negative index materials and their applications: Recent mathematics progress

  • Published:
Chinese Annals of Mathematics, Series B Aims and scope Submit manuscript

Abstract

Negative index materials are artificial structures whose refractive index has negative value over some frequency range. These materials were first investigated theoretically by Veselago in 1946 and were confirmed experimentally by Shelby, Smith, and Schultz in 2001. Mathematically, the study of negative index materials faces two difficulties. Firstly, the equations describing the phenomenon have sign changing coefficients, hence the ellipticity and the compactness are lost in general. Secondly, the localized resonance, i.e., the field explodes in some regions and remains bounded in some others as the loss goes to 0, might appear. In this survey, the author discusses recent mathematics progress in understanding properties of negative index materials and their applications. The topics are reflecting complementary media, superlensing and cloaking by using complementary media, cloaking a source via anomalous localized resonance, the limiting absorption principle and the well-posedness of the Helmholtz equation with sign changing coefficients.

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. Agmon, S., Douglis, A. and Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math., 17, 1964, 35–92.

    Article  MathSciNet  MATH  Google Scholar 

  2. Ammari, H., Ciraolo, G., Kang, H., et. al., Anomalous localized resonance using a folded geometry in three dimensions, Proc. R. Soc. Lond. Ser. A, 469, 2013, 20130048.

    Article  MATH  Google Scholar 

  3. Ammari, H., Ciraolo, G., Kang, H., et. al., Spectral theory of a Neumann-Poincaré-type operator and analysis of cloaking due to anomalous localized resonance, Arch. Rational Mech. Anal., 218, 2013, 667–692.

    Article  MATH  Google Scholar 

  4. Bethuel, F., Brezis, H. and Helein, F., Ginzburg Landau vortices, Progress in Nonlinear Differential Equations and Their Applications., 13, Birkhäuser, Boston, 1994.

    MATH  Google Scholar 

  5. Bonnet-Ben Dhia, A. S., Chesnel, L. and Ciarlet, P., T-coercivity for scalar interface problems between dielectrics and metamaterials, ESAIM Math. Model. Numer. Anal., 46, 2012, 1363–1387.

    Article  MathSciNet  MATH  Google Scholar 

  6. Bonnet-Ben Dhia, A. S., Chesnel, L. and Ciarlet, P., T-coercivity for the Maxwell problem with signchanging coefficients, Comm. Partial Differential Equations, 39, 2014, 1007–1031.

    Article  MathSciNet  MATH  Google Scholar 

  7. Bonnet-Ben Dhia, A. S., Ciarlet, P. and Zwölf, C. M., Two-and three-field formulations for wave transmission between media with opposite sign dielectric constants, J. Comput. Appl. Math., 204, 2007, 408–417.

    Article  MathSciNet  MATH  Google Scholar 

  8. Bonnet-Ben Dhia, A. S., Ciarlet, P. and 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, 2008, 1605–1631.

    Article  MathSciNet  MATH  Google Scholar 

  9. Bonnet-Ben Dhia, A. S., Ciarlet, P. and Zwölf, C. M., Time harmonic wave diffraction problems in materials with sign-shifting coefficients, J. Comput. Appl. Math., 234, 2010, 1912–1919.

    Article  MathSciNet  MATH  Google Scholar 

  10. Bouchitté, G. and Felbacq, D., Homogenization near resonances and artificial magnetism from dielectrics, C. R. Math. Acad. Sci. Paris, 339, 2004, 377–382.

    Article  MathSciNet  MATH  Google Scholar 

  11. Bouchitté, G. and Schweizer, B., Cloaking of small objects by anomalous localized resonance, Quart. J. Mech. Appl. Math., 63, 2010, 437–463.

    Article  MathSciNet  MATH  Google Scholar 

  12. Bouchitté, G. and Schweizer, B., Homogenization of Maxwell’s equations in a split ring geometry, Multiscale Model. Simul., 8, 2010, 717–750.

    Article  MathSciNet  MATH  Google Scholar 

  13. Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universititex, Springer-Verlag, New York, 2010.

    Book  Google Scholar 

  14. Bruno, O. P. and Lintner, S., Superlens-cloaking of small dielectric bodies in the quasistatic regime, J. Appl. Phys., 102, 2007, 12452.

    Article  Google Scholar 

  15. Costabel, M. and Stephan, E., A direct boundary integral equation method for transmission problems, J. Math. Anal. Appl., 106, 1985, 367–413.

    Article  MathSciNet  MATH  Google Scholar 

  16. Guenneau, S. and Zolla, F., Homogenization of 3d finite chiral photonic crystals, Physica B: Condens. Matter, 394, 2007, 145–147.

    Article  MATH  Google Scholar 

  17. Kettunen, H., Lassas, M. and Ola, P., On absence and existence of the anomalous localized resonance without the quasi-static approximation. http://arxiv.org/abs/1406.6224

  18. Kohn, R. V., Lu, J., Schweizer, B. and Weinstein, M. I., A variational perspective on cloaking by anomalous localized resonance, Comm. Math. Phys., 328, 2014, 1–27.

    Article  MathSciNet  MATH  Google Scholar 

  19. Kohn, R. V., Onofrei, D., Vogelius, M. S. and Weinstein, M. I., Cloaking via change of variables for the Helmholtz equation, Comm. Pure Appl. Math., 63, 2010, 973–1016.

    MathSciNet  MATH  Google Scholar 

  20. Kohn, R. V. and Shipman, S. P., Magnetism and homogenization of microresonators, Multiscale Model. Simul., 7, 2008, 62–92.

    Article  MathSciNet  MATH  Google Scholar 

  21. Lai, Y., Chen, H., Zhang, Z. and Chan, C. T., Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell, Phys. Rev. Lett., 102, 2009, 093901.

    Article  Google Scholar 

  22. Milton, G. W., Nicorovici, N. A., McPhedran, R. C. and 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, 2005, 3999–4034.

    Article  MathSciNet  MATH  Google Scholar 

  23. Milton, G. W. and Nicorovici, N. A. P., On the cloaking effects associated with anomalous localized resonance, Proc. R. Soc. Lond. Ser. A, 462, 2006, 3027–3059.

    Article  MathSciNet  MATH  Google Scholar 

  24. Milton, G. W., Nicorovici, N. P., McPhedran, R. C., et. al., Solutions in folded geometries, and associated cloaking due to anomalous resonance, New J. Phys., 10, 2008, 115021.

    Google Scholar 

  25. Nguyen, H.-M., Cloaking via change of variables for the Helmholtz Equation in the whole space, Comm. Pure Appl. Math., 63, 2010, 1505–1524.

    Article  MathSciNet  MATH  Google Scholar 

  26. Nguyen, H.-M., Approximate cloaking for the Helmholtz equation via transformation optics and consequences for perfect cloaking, Comm. Pure Appl. Math., 65, 2012, 155–186.

    Article  MathSciNet  MATH  Google Scholar 

  27. Nguyen, H.-M., Asymptotic behavior of solutions to the Helmholtz equations with sign changing coefficients, Trans. Amer. Math. Soc., 367, 2015, 6581–6595.

    Article  MathSciNet  MATH  Google Scholar 

  28. Nguyen, H.-M., Superlensing using complementary media, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32, 2015, 471–484.

    Article  MathSciNet  MATH  Google Scholar 

  29. 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, 2015, 41–46.

    Article  MathSciNet  MATH  Google Scholar 

  30. Nguyen, H.-M., Cloaking via anomalous localized resonance for doubly complementary media in the quasistatic regime, J. Eur. Math. Soc., 17, 2015, 1327–1365.

    Article  MathSciNet  MATH  Google Scholar 

  31. Nguyen, H.-M., Cloaking using complementary media in the quasistatic regime, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33, 2016, 1509–1518.

    Article  MathSciNet  MATH  Google Scholar 

  32. Nguyen, H.-M., Limiting absorption principle and well-posedness for the Helmholtz equation with sign changing coefficients, J. Math. Pures Appl., 106, 2016, 342–374.

    Article  MathSciNet  MATH  Google Scholar 

  33. Nguyen, H.-M., Reflecting complementary and superlensing using complementary media for electromagnetic waves, submitted.

  34. Nguyen, H.-M., Cloaking an arbitrary object via anomalous localized resonance: the cloak is independent of the object, submitted. https://arxiv.org/abs/1607.06492

  35. Nguyen, H.-M., Cloaking via anomalous localized resonance for doubly complementary media in the finite frequency regime, submitted. https://arxiv.org/abs/1511.08053

  36. Nguyen, H.-M., Cloaking using complementary for electromagnetic waves, submitted. https://arxiv.org/abs/1701.02339

  37. Nguyen, H.-M. and Nguyen, H. L., Complete resonance and localized resonance in plasmonic structures, ESAIM: Math. Model. Numer. Anal., 49, 2015, 741–754.

    Article  MathSciNet  MATH  Google Scholar 

  38. Nguyen, H.-M. and Nguyen, H. L., Cloaking using complementary media for the Helmholtz equation and a three spheres inequality for second order elliptic equations, Trans. Amer. Math. Soc., Ser. B, 2, 2015, 93–112.

    Article  MathSciNet  MATH  Google Scholar 

  39. Nguyen, H.-M. and Vogelius, M. S., Full range scattering estimates and their application to cloaking, Arch. Rational Mech. Anal., 203, 2012, 769–807.

    Article  MathSciNet  MATH  Google Scholar 

  40. Nguyen, H.-M. and Vogelius, M. S., Approximate cloaking for the wave equation via change of variables, SIAM J. Math. Anal., 44, 2012, 1894–1924.

    Article  MathSciNet  MATH  Google Scholar 

  41. Nguyen, H.-M. and Vogelius, M. S., Approximate cloaking for the full wave equation: A study of the Lorentz model, J. Math. Pures Appl., 106, 2016, 797–836.

    Article  MathSciNet  MATH  Google Scholar 

  42. Nicorovici, N. A., McPhedran, R. C. and Milton, G. M., Optical and dielectric properties of partially resonant composites, Phys. Rev. B, 49, 1994, 8479–8482.

    Article  Google Scholar 

  43. Ola, P., Remarks on a transmission problem, J. Math. Anal. Appl., 196, 1995, 639–658.

    Article  MathSciNet  MATH  Google Scholar 

  44. Pendry, J. B., Negative refraction makes a perfect lens, Phys. Rev. Lett., 85, 2000, 3966–3969.

    Article  Google Scholar 

  45. Pendry, J. B., Perfect cylindrical lenses, Optics Express, 1, 2003, 755–760.

    Article  Google Scholar 

  46. Pendry, J. B., Holden, A. J., Robbins, D. J. and Stewart, W. J., Magnetism from conductors and enhanced nonlinear phenomena, IEEE Trans. Microw. Theory Tech., 47, 1999, 2075–2084.

    Article  Google Scholar 

  47. Ramakrishna, S. A. and Pendry, J. B., Spherical perfect lens: Solutions of Maxwell’s equations for spherical geometry, Phys. Rev. B, 69, 2004, 115115.

    Article  Google Scholar 

  48. Shelby, R. A., Smith, D. R. and Schultz, S., Experimental Verification of a Negative Index of Refraction, Science, 292, 2001, 77–79.

    Article  Google Scholar 

  49. Veselago, V. G., The electrodynamics of substances with simultaneously negative values of ε and µ, Usp. Fiz. Nauk, 92, 1964, 517–526.

    Article  Google Scholar 

Download references

Acknowledgments

The author thanks Professor Haïm Brezis deeply for his encouragement, his guidance, and his support from his Ph.D studies. His joint work with Bethuel and Hélein on the Ginzburg-Landau equation in [4] inspired the author to introduce the removing localized singularity technique to handle the localized resonance which appears naturally in the study of negative index materials. The author also thanks Professor Graeme Milton for useful discussions on the subject.

Author information

Authors and Affiliations

  1. EPFL SB MATHAA CAMA, Station 8, CH-1015, Lausanne, Switzerland

    Hoai-Minh Nguyen

Authors
  1. Hoai-Minh Nguyen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hoai-Minh Nguyen.

Additional information

Dedicated to Haïm Brezis for his 70th birthday with esteem

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nguyen, HM. Negative index materials and their applications: Recent mathematics progress. Chin. Ann. Math. Ser. B 38, 601–628 (2017). https://doi.org/10.1007/s11401-017-1086-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11401-017-1086-5

Keywords

2000 MR Subject Classification

Search

Navigation