volume | PIER Journals

Abstract

A key structure in so-called metamaterial mediums is the elementary split-ring resonator. We consider in this paper the low-frequency electromagnetic scattering by a split-ring particle modelled as a perfectly conducting wire ring, furnished with a narrow gap, and derive analytical solutions for the electric and magnetic dipole moments for different kinds of incidence and polarisation in the quasi-static approximation. Through a vectorial homogenisation process, the expressions discovered for the dipole moments and the related polarisability dyadics are linked with the macroscopic constitutive equations for the medium. We further show that the condition for resonance of a medium consisting of simple split-rings cannot be achieved by means of the given quasi-static terms without violating the underlying assumptions of homogenisation. Nevertheless, the results are applicable for sparse medium of rings, and we derive numerical guidelines for the applicability with some examples of the effect of the considered split-ring medium on electromagnetic wave propagation.

1. Engheta, N. and R. W. Ziolkowski (eds.), Electromagnetic Metamaterials: Physics and Engineering Explorations, Wiley, 2006.

2. Pendry, J. B., A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Microw. Theory and Techniques, Vol. 47, 2075-2084, 1999.
doi:10.1109/22.798002 Google Scholar

3. Merlin, R., "Metamaterials and the Landau-Lifshitz permeability argument: Large permittivity begets high-frequency magnetism," Proc. Nat. Acad. Sci., Vol. 106, No. 6, 1693-1698, Feb. 10, 2009.
doi:10.1073/pnas.0808478106 Google Scholar

4. Simovski, C. R. and B. Sauviac, "Toward creating isotropic microwave composites with negative refraction," Radio Science, Vol. 39, 1-18, 2004, RS2014. Google Scholar

5. Lewellyn Smith, S. G. and A. M. J. Davis, "The split ring resonator," Proc. R. Soc. A, Vol. 466, 3117-3134, 2010.
doi:10.1098/rspa.2010.0047 Google Scholar

6. Movchan, A. B. and S. Guenneau, "Split-ring resonators and localized modes," Phys. Rev. B, Vol. 70, 125116, 2004.
doi:10.1103/PhysRevB.70.125116 Google Scholar

7. Popa, B.-I. and S. A. Cummer, "Compact dielectric particles as a building block for low-loss magnetic metamaterials," Phys. Rev. Lett., Vol. 1000, 207401, 2008.
doi:10.1103/PhysRevLett.100.207401 Google Scholar

8. Jones, D. S., "Low frequency electromagnetic radiation," J. Inst. Maths Applics, Vol. 23, 421-447, 1979.
doi:10.1093/imamat/23.4.421 Google Scholar

9. Stevenson, A. F., "Solution of electromagnetic scattering problems as power series in the ratio (dimension of scatterer)/wavelength," J. Appl. Phys., Vol. 24, 1134-1142, 1953.
doi:10.1063/1.1721461 Google Scholar

10. Kleinman, R. E. and T. B. A. Senior, "Rayleigh scattering," Low and High Frequency Asymptotics, 1-70, V. K. Varadan and V. V. Varadan (eds.), North-Holland, 1986. Google Scholar

11. Arvas, E., R. F. Harrington, and J. R. Mautz, "Radiation and scattering from electrically small conducting bodies of arbitrary shape," IEEE Trans. Antennas Propag., Vol. 34, No. 1, 66-77, Jan. 1986.
doi:10.1109/TAP.1986.1143716 Google Scholar

12. Van Bladel, J., Electromagnetic Fields, 2nd Ed., IEEE Press & Wiley Interscience, 2007.
doi:10.1002/047012458X

13. Jackson, J. D., Classical Electrodynamics, 3rd Ed., Wiley, 1999.

14. Hadad, Y. and B. Z. Steinberg, "Electrodynamic synergy of micro-properties and macro-structure in particle arrays," 2010 URSI Electromagnetic Theory Symposium, 839-842, 2010. Google Scholar

15. Belov, P. A. and C. R. Simovski, "On homogenization of electromagnetic crystals formed by uniaxial resonant scatterers," Phys. Rev. E, Vol. 72, 026615, 2005.
doi:10.1103/PhysRevE.72.026615 Google Scholar

16. Silveirinha, M. G., "Generalized Lorentz-Lorenz formulas for microstructured materials," Phys. Rev. B, Vol. 76, 245117, 2007.
doi:10.1103/PhysRevB.76.245117 Google Scholar

17. Bossavit, A., "Homogenization of split-ring arrays, seen as the exploitation of translational symmetry," Metamaterials and Plasmonics: Fundamentals, Modelling, Applications, 77-90, Springer Netherlands, 2008.

18. Holt, A. and G. Karlström, "Inclusion of the quadrupole moment when describing polarization. The effect of the dipole-quadrupole polarizability," J. Comput. Chem., Vol. 29, 2033-2038, 2008.
doi:10.1002/jcc.20976 Google Scholar

19. Rayleigh, L., "On the influence of obstacles arranged in rectangular order upon the properties of the medium," Phil. Mag., Vol. 34, 481-502, 1892. Google Scholar

20. McPhedran, R. C. and D. R. McKenzie, "The conductivity of lattices of spheres I. The simple cubic lattice," Proc. R. Soc. Lond. A, Vol. 359, 45-63, 1978.
doi:10.1098/rspa.1978.0031 Google Scholar

21. Qi, J., H. Kettunen, H. Wallén, and A. Sihvola, "Quasi-dynamic homogenization of geometrically simple dielectric composites," ACES Journal, Vol. 25, No. 12, 1036-1045, 2010. Google Scholar

22. Sjöberg, D., "Simple wave solutions for the Maxwell equations in bianisotropic nonlinear media, with application to oblique incidence," Wave Motion, Vol. 32, No. 3, 217-232, 2000.
doi:10.1016/S0165-2125(00)00039-1 Google Scholar

23. Gustafsson, M., C. Sohl, C. Larsson, and D. Sjöberg, "Physical bounds on the all-spectrum transmission through periodic arrays," EPL Europhysics Letters, Vol. 87, No. 3, 34002, 2009.
doi:10.1209/0295-5075/87/34002 Google Scholar

24. Sjöberg, D., M. Gustafsson, and C. Larsson, "Physical bounds on the all-spectrum transmission through periodic arrays: Oblique incidence," EPL Europhysics Letters, Vol. 92, 34009, 2010.
doi:10.1209/0295-5075/92/34009 Google Scholar

25. Sjöberg, D., "Variational principles for the static electric and magnetic polarizabilities of anisotropic media with perfect electric conductor inclusions," J. Phys. A: Math. Theor., Vol. 42, 335403, 2009.
doi:10.1088/1751-8113/42/33/335403 Google Scholar

26. Jones, D. S., "Scattering by inhomogeneous dielectric particles," Q. J. Mech. Appl. Math., Vol. 38, 135, 1985.
doi:10.1093/qjmam/38.1.135 Google Scholar

Dr. Johan Sten

Dr. Daniel Sjöberg