Frequency-Domain Homogenization of Maxwell Equations in Complex Media

Ouail Ouchetto, Brahim Essakhi


This paper is devoted to the homogenization of the Maxwell equations with periodically oscillating coefficients in the bianisotropic media which represents the most general linear media. In the first time, the limiting homogeneous constitutive law is rigorously justified in the frequency domain and is found from the solution of a local problem on the unit cell. The homogenization process is based on the two-scale convergence conception. In the second time, the implementation of the homogeneous
constitutive law by using the finite element method and the introduction of the boundary conditions in the discrete problem are introduced. Finally, the numerical results associated of the perforated chiral media are presented.

Full Text:



PLUM, E., ZHOU, J., DONG, J., et al. Metamaterial with negative index due to chirality. Physical Review B, 2009, vol. 79, no 3, p. 035407.

SIMOVSKI, Constantin R. et TRETYAKOV, Sergei A. Local constitutive parameters of metamaterials from an effective-medium perspective. Physical Review B, 2007, vol. 75, no 19, p. 195111.

KARAMANOS, Theodosios D., ASSIMONIS, Stylianos D., DIMITRIADIS, Alexandros I., et al. Effective parameter extraction of 3D metamaterial arrays via first-principles homogenization theory. Photonics and Nanostructures-Fundamentals and Applications, 2014, vol. 12, no 4, p. 291-297.

PENDRY, John B., HOLDEN, A. J., ROBBINS, D. J., et al. Magnetism from conductors and enhanced nonlinear phenomena. Microwave Theory and Techniques, IEEE Transactions on, 1999, vol. 47, no 11, p. 2075-2084.

SMITH, David R. et PENDRY, John B. Homogenization of metamaterials by field averaging. J. Opt. Soc. Am. B, 2006, vol. 23, no 3, p. 391-403.

TSUKERMAN, Igor. Effective parameters of metamaterials: a rigorous homogenization theory via Whitney interpolation. J. Opt. Soc. Am. B, 2011, vol. 28, no 3, p. 577-586.

OUCHETTO, Ouail, OUCHETTO, Hassania, ZOUHDI, Said, et al. Homogenization of Maxwell’s Equations in Lossy Biperiodic Metamaterials. Antennas and Propagation, IEEE Transactions on, 2013, vol. 61, no 8, p. 4214-4219.

ISHIMARU, Akira, LEE, Seung-Woo, KUGA, Yasuo, et al. Generalized constitutive relations for metamaterials based on the quasi-static Lorentz theory. Antennas and Propagation, IEEE Transactions on, 2003, vol. 51, no 10, p. 2550-2557.

BELOV, Pavel A. et SIMOVSKI, Constantin R. Homogenization of electromagnetic crystals formed by uniaxial resonant scatterers. Physical Review E, 2005, vol. 72, no 2, p. 026615.

LAMB, W., WOOD, D. M., et ASHCROFT, N. W. Long-wavelength electromagnetic propagation in heterogeneous media. Physical review B, 1980, vol. 21, no 6, p. 2248.

ALÙ , Andrea. Restoring the physical meaning of metamaterial constitutive parameters. Physical Review B, 2011, vol. 83, no. 8 , p. 081102.

SILVEIRINHA, Mario G. Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters. Physical Review B, 2007, vol. 75, no 11, p. 115104.

NGUETSENG, Gabriel. A general convergence result for a functional related to the theory of homogenization.

SIAM Journal on Mathematical Analysis, 1989, vol. 20, no 3, p. 608-623.

NGUETSENG, Gabriel. Asymptotic analysis of a stiff variational problem arising in mechanics, SIAM Journal on Mathematical Analysis, 1990, vol. 21, p. 1394-1414.

ALLAIRE, Gr´egoire. Homogenization and two-scale convergence. SIAM Journal on Mathematical Analysis, 1992, vol. 23, no 6, p. 1482-1518.

BOURGEAT, Alain, MIKELIC, Andro, et WRIGHT, Steve. Stochastic two-scale convergence in the mean and applications. J. reine angew. Math, 1994, vol. 456, no 1, p. 19-51.

NEUSS-RADU, Maria. Some extensions of two-scale convergence. Comptes rendus de l’Académie des sciences. Série 1, Math´ematique, 1996, vol. 322, no 9, p. 899-904.

CIORANESCU, Doina et DONATO, Patrizia. An introduction to homogenization. Oxford : Oxford University Press, 1999.

AMIRAT, Youcef, HAMDACHE, Kamel, et ZIANI, Abdelhamid. Homogenization of degenerate waveequations with periodic coefficients. SIAM Journal on Mathematical Analysis, 1993, vol. 24, no 5, p. 1226-1253.

AVELLANEDA, M., HOU, Th Y., et PAPANICOLAOU, G. C. Finite difference approximations for partial differential equations with rapidly oscillating coefficients. ESAIM: Mathematical Modelling and Numerical Analysis-Modélisation Mathématique et Analyse Numérique, 1991, vol. 25, no 6, p. 693-710.

S. SHKOLLER, S. et HEGEMIER, G. Homogenization of plane wave composite using two-scale convergence. Int. J. Solids Structures, 1995, vol. 32, no 6/7, p.783-794.

WELLANDER, Niklas. Homogenization of some linear and nonlinear partial differential equations. 1998.

WELLANDER, Niklas. Homogenization of the Maxwell equations: Case I. Linear theory. Applications of

Mathematics, 2001, vol. 46, no 1, p. 29-51.

WELLANDER, Niklas. Homogenization of the Maxwell equations: Case II. Nonlinear theory. Applications of Mathematics, 2002, vol. 47, no 3, p. 255-283.

KRISTENSSON, Gerhard et WELLANDER, Niklas. Homogenization of the Maxwell equations at fixed frequency. SIAM Journal on Applied Mathematics, 2003, vol. 64, no 1, p. 170-195.

LUKKASSEN, Dag, NGUETSENG, Gabriel, et WALL, Peter. Two-scale convergence. Int. J. Pure Appl. Math, 2002, vol. 2, no 1, p. 35-86.

CIORANESCU, Doina, DAMLAMIAN, Alain, et GRISO, Georges. Periodic unfolding and homogenization. Comptes Rendus Mathematique, 2002, vol. 335, no 1, p. 99-104.

CIORANESCU, Doina, DAMLAMIAN, Alain, et GRISO, Georges. The periodic unfolding method in homogenization. SIAM Journal on Mathematical Analysis, 2008, vol. 40, no 4, p. 1585-1620.

NGUETSENG, Gabriel. Homogenization structures and applications. II. Z. Anal. Anwendungen, 2004, vol.23, p. 483-508.

WELLANDER, Niklas. The two-scale Fourier transform approach to homogenization; periodic homogenization in Fourier space. Asymptotic Analysis, 2009, vol. 62, no 1, p. 1-40.

BARBATIS, G. et STRATIS, I. G. Homogenization of Maxwell’s equations in dissipative bianisotropic media. Mathematical methods in the applied sciences, 2003, vol. 26, no 14, p. 1241-1253.

MIARA, B., ROHAN, E., GRISO, G., et al. Application of multi-scale modelling to some elastic, piezoelectric and electromagnetic composites. Mechanics of advanced materials and structures, 2007, vol. 14, no 1, p. 33-42.

BOSSAVIT, Alain, GRISO, Georges, et MIARA, Bernadette. Modelling of periodic electromagnetic structures bianisotropic materials with memory effects. Journal de mathématiques pures et appliquées, 2005, vol. 84, no 7, p. 819-850.

OUCHETTO, Ouail, QIU, Cheng-Wei, ZOUHDI, Saïd, et al. Homogenization of 3-d periodic bianisotropic metamaterials. Microwave Theory and Techniques, IEEE Transactions on, 2006, vol. 54, no 11, p. 3893-3898.

OUCHETTO, Ouail, ZOUHDI, Saïd, BOSSAVIT, Alain, et al. Modeling of 3-D periodic multiphase composites by homogenization. Microwave Theory and Techniques, IEEE Transactions on, 2006, vol. 54, no 6, p. 2615-2619.

OUCHETTO, O., ZOUHDI, S., BOSSAVIT, A., et al. Homogenization of structured electromagnetic materials and metamaterials. Journal of materials processing technology, 2007, vol. 181, no 1, p. 225-229.

OUCHETTO, O., ZOUHDI, S., BOSSAVIT, A., et al. Homogenization of 3d structured composites of complex shaped inclusions. In : Progress Electromagn. Res. Symp. 2005. p. 112.

OUCHETTO, O., ZOUHDI, S., RAZEK, A., et al. Effective constitutive parameters of structured chiral metamaterials. Microwave and optical technology letters, 2006, vol. 48, no 9, p. 1884-1886.

OUCHETTO, O., ZOUHDI, S., BOSSAVIT, A., et al. Effective constitutive parameters of periodic composites. In : Microwave Conference, 2005 European. IEEE, 2005. p. 2 pp.

LINDELL, Ismo V., SIHVOLA, A. H., TRETYAKOV, S. A., et al. Electromagnetic waves in chiral and bi-isotropic media, Norwood, MA: Artech House, 1994.

GDUVAUT, Georges et LIONS, Jacques Louis. Inequalities in mechanics and physics. Springer, 1976.

CIARLET P. The finite element method for elliptic problem, North-Holland, Amsterdam, 1978


  • There are currently no refbacks.

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.