Homogenization of Harmonic Maxwell Equations with Allowance for Interfacial Surface Currents: Layered Structure

  • Y. Amirat
  • V. V. ShelukhinEmail author


The Maxwell equations for a composite two-component laminated material with a periodic structure in the field of a time-harmonic source acting along the layers are considered. Two-scale homogenization of the equations is performed with allowance for complex conductivity of interfacial layers and their thickness. The boundary-value problem for systems of differential equations with boundary conditions is reduced to a problem in a weakly variational formulation. Unique solvability of the problem is established. The case of low frequencies of interfacial currents of different intensities with allowance for the frequency-dependent wave length and skin layer length is analyzed. Macro-equations are derived, and effective material constants are determined, such as the dielectric permittivity, magnetic permeability, and electrical conductivities. Conditions at which the effective parameters depend on interfacial currents are described. It is found that the effective dielectric permittivity can be negative at specially chosen parameters of interfacial layers if it is determined on the basis of the effective wave number.


Maxwell equation interfacial currents homogenization two-scale convergence 


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  1. 1.
    Y. Amirat and V. V. Shelukhin, “Homogenization of Time Harmonic Maxwell Equations: The Effect of Interfacial Currents,” Math. Methods Appl. Sci. 40 (8), 3140–3162 (2017).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    C. Tournassat, Y. Chapron, P. Leroy, et al., “Comparison of Molecular Dynamics Simulations with Triple Layer and Modified Gouy-Chapman Models in a 0.1 M NaCl-Montmorillonite System,” J. Colloid Interface Sci. 339, 533–541 (2009).ADSCrossRefGoogle Scholar
  3. 3.
    Yu. V. Bludov, A. Ferreira, N. M. R. Peres, and M. I. Vasilevskiy, “A Primer on Surface Plasmon-Polaritons in Grafene,” Int. J. Modern Phys. B 27 (10), 1341001 (2013).ADSCrossRefzbMATHGoogle Scholar
  4. 4.
    G. Kristensson, “Homogenization of Corrugated Interfaces in Electromagnetics,” Progr. Electromag. Res. 55, 1–31 (2005).CrossRefGoogle Scholar
  5. 5.
    V. V. Shelukhin and S. A. Terentev, “Frequency Dispersion of Dielectric Permittivity and Electric Conductivity of Rocks via Two-Scale Homogenization of the Maxwell Equations,” Progr. Electromag. Res. B 14, 175–202 (2009).CrossRefGoogle Scholar
  6. 6.
    V. V. Shelukhin and S. A. Terent’ev, “Homogenization of Maxwell Equations and Maxwell-Wagner Dispersion,” Dokl. Akad. Nauk, 424 (3), 402–406 (2009).MathSciNetzbMATHGoogle Scholar
  7. 7.
    N. Wellander, “Homogenization of the Maxwell Equations: Case 2. Nonlinear Conductivity,” Appl. Math. 47 (3), 255–283 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    A. N. Lagarkov, V. N. Kisel, and V. N. Semenenko, “Wide-Angle Absorption by the use of a Metamaterial Plate,” Progr. Electromag. Res. Lett. 1, 35–44 (2008).CrossRefGoogle Scholar
  9. 9.
    Y. Amirat and V. V. Shelukhin, “Homogenization of Time Harmonic Maxwell Equations and the Frequency Dispersion Effect,” J. Math. Pures Appl. 95, 420–443 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    L. D. Landau and E. M. Lifshits, Course of Theoretical Physics, Vol. 8: Electrodynamics of Continuous Media (Fizmatlit, Moscow, 2005; Pergamon, New York, 1984).Google Scholar
  11. 11.
    I. C. Bourg and G. Sposito, “Molecular Dynamics Simulations of the Electrical Double Layer on Smectite Surfaces Contacting Concentrated Mixed Electrolyte (NaCl-CaCl2) Solutions,” J. Colloid Interface Sci. 360, 701–715 (2011).ADSCrossRefGoogle Scholar
  12. 12.
    G. Nguetseng, “A General Convergence Result for a Functional Related to the Theory of Homogenization,” SIAM J. Math. Anal. 20 (3), 608–623 (1989).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    K. Sarabandi and F. T. Ulaby, “Techniques for Measuring the Dielectric Constant of Thin Materials,” IEEE Trans. Instrum. Measur. 4, 631–636 (1988).CrossRefGoogle Scholar
  14. 14.
    A. H. Boughriet, C. Legrand, and A. Chapoton, “Non-Iterative Stable Transmission/Reflection Method for Low-Loss Material Complex Permittivity Determination,” IEEE Trans. Microwave Theory Methods 45 (1), 52–57 (1997).ADSCrossRefGoogle Scholar
  15. 15.
    W. Brown, Jr., Dielectrics (West Berlin, 1956).Google Scholar
  16. 16.
    D. R. Smith, W. J. Padilla, D. C. Vier, et al., “Composite Medium with Simultaneously Negative Permeability and Permittivity,” Phys. Rev. Lett. 4 (18), 4184–4187 (2000).ADSCrossRefGoogle Scholar

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© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.University of AuvergneClermont-FerrandFrance
  2. 2.Lavrentyev Institute of Hydrodynamics, Siberian BranchRussian Academy of SciencesNovosibirskRussia
  3. 3.Novosibirsk State UniversityNovosibirskRussia

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