volume | PIER Journals

Abstract

Dealing with the project of metamaterials scientists often have to design circuit elements at a subwavelength (or ``microscopic'') scale. At that scale, they use the set of Maxwell's equations in free-space, and neither permittivity ε nor permeability μ are formally defined. However, the objective is to use the unit cells in order to build a bulk material with some desired ``macroscopic'' properties. At that scale the set of Maxwell's equations in matter is adopted. To pass from one approach to the other is not obvious. In this paper we analyse the classic definitions of polarization P and magnetization M, highlighting their limits. Then we propose a definition for P and M fully consistent with Maxwell's equations at any scale.

1. Maxwell, J. C., A Treatise on Electricity and Magnetism, Vol. 1, Clarendon Press, 1881.

2. Engheta, N. and R. W. Ziolkowski, "Metamaterials: Physics and Engineering Explorations," John Wiley & Sons, 2006.

3. Pendry, J. B., "Negative refraction," Contemporary Physics, Vol. 45, No. 3, 191-202, 2004.
doi:10.1080/00107510410001667434

4. Pendry, J. B. and D. R. Smith, "Reversing light with negative refraction," Physics Today, Vol. 57, 37-43, 2004.
doi:10.1063/1.1784272

5. Scheller, M., C. Jansen, and M. Koch, Applications of Effective Medium Theories in the Terahertz Regime, INTECH Open Access Publisher, 2010.

6. Herczynski, A., "Bound charges and currents," American Journal of Physics, Vol. 81, No. 3, 202-205, 2013.
doi:10.1119/1.4773441

7. Bobbio, S. and E. Gatti, Elettromagnetismo Ottica, Bollati Boringhieri, 1991.

8. Griffiths, D. J., Introduction to Electrodynamics, Prentice-Hall, 1999.

9. Landau, L. D., J. Bell, M. Kearsley, L. Pitaevskii, E. Lifshitz, and J. Sykes, Electrodynamics of Continuous Media, Vol. 8, Elsevier, 1984.

10. Dendy, R. O., Plasma Dynamics, Clarendon Press Oxford, 1990.

11. Drude, P., "Zur elektronentheorie der metalle," Annalen der Physik, Vol. 306, 566-613, 1900.
doi:10.1002/andp.19003060312

12. Vanderlinde, J., Classical Electromagnetic Theory, Springer Science & Business Media, 2006.

13. Gonano, C. A., "Estensione in ND di prodotto vettore e rotore e loro applicazioni,", Master's Thesis, Politecnico di Milano, Dec. 2011.

14. Sihvola, A. H., "Metamaterials and depolarization factors," Progress In Electromagnetics Research, Vol. 51, 65-82, 2005.
doi:10.2528/PIER04021001

15. Mallet, P., C.-A. Guerin, and A. Sentenac, "Maxwell-Garnett mixing rule in the presence of multiple scattering: Derivation and accuracy," Physical Review B, Vol. 72, No. 1, 014205, 2005.
doi:10.1103/PhysRevB.72.014205

16. Cheng, D. K. and J.-A. Kong, "Covariant descriptions of bianisotropic media," Proceedings of the IEEE, Vol. 56, No. 3, 248-251, 1968.
doi:10.1109/PROC.1968.6268

17. Paul, O. and M. Rahm, "Covariant description of transformation optics in nonlinear media," Optics Express, Vol. 20, No. 8, 8982-8997, 2012.
doi:10.1364/OE.20.008982

18. Pastorino, M., M. Raffetto, and A. Randazzo, "Electromagnetic inverse scattering of axially moving cylindrical targets," IEEE Transactions on Geoscience and Remote Sensing, Vol. 53, No. 3, 1452-1462, Mar. 2015.
doi:10.1109/TGRS.2014.2342933

19. Fernandes, P., M. Ottonello, and M. Raffetto, "Regularity of time-harmonic electromagnetic fields in the interior of bianisotropic materials and metamaterials," IMA Journal of Applied Mathematics, hxs039, 2012.

20. Landau, L. and E. Lifshitz, The Classical Theory of Fields, Vol. 2, Course of Theoretical Physics Series, Butterworth-Heinemann, 1980.

21. Gonano, C. A. and R. E. Zich, "Magnetic monopoles and Maxwell’s equations in N dimensions," 2013 International Conference on Electromagnetics in Advanced Applications (ICEAA), 1544-1547, Sep. 2013.
doi:10.1109/ICEAA.2013.6632510

22. Gonano, C. A. and R. E. Zich, "Cross product in N Dimensions - the doublewedge product,", arXiv:1408.5799, 2014.

23. Einstein, A., "Zur elektrodynamik bewegter körper," Annalen Der Physik, Vol. 322, No. 10, 891-921, 1905.
doi:10.1002/andp.19053221004

24. Janssen, M. and J. J. Stachel, "The optics and electrodynamics of moving bodies,", 2004.

25. Yin, X., H. Zhang, S.-J. Sun, Z. Zhao, and Y.-L. Hu, "Analysis of propagation and polarization characteristics of electromagnetic waves through nonuniform magnetized plasma slab using propagator matrix method," Progress In Electromagnetics Research, Vol. 137, 159-186, 2013.
doi:10.2528/PIER13010410

26. Xu, X. B. and L. Zeng, "Ferromagnetic cylinders in Earth’s magnetic field: A two-dimensional model of magnetization of submarine," Progress In Electromagnetics Research, Vol. 19, 319-335, 1998.
doi:10.2528/PIER97120100

Dr. Carlo Andrea Gonano

Dr. Riccardo Enrico Zich

Prof. Marco Mussetta