Vol. 14

Latest Volume
All Volumes
All Issues

Boundary Effects in the Electromagnetic Response of a Metamaterial in the Case of Normal Incidence

By Aaron D. Scher and Edward F. Kuester
Progress In Electromagnetics Research B, Vol. 14, 341-381, 2009


In this paper we investigate boundary effects and other consequences of spatial dispersion by analyzing in detail the response of a metamaterial half-space to a monochromatic plane wave normally incident from free-space. The metamaterial is composed of an orthorhombic lattice of identical particles, each of which exhibits both an electric and magnetic response. Rather than relying on the conventional boundary conditions and the Clausius-Mossotti equations, we use instead the point-dipole interaction model and an expansion of polarization in eigenmodes to determine the structure's dispersion relation and electromagnetic response. Using the nearestneighbor approximation, we show how truncating the crystal lattice excites an "ordinary" mode and two "extraordinary" modes that are necessary to satisfy the boundary conditions at the interface. For most cases, the extraordinary modes are evanescent, and thus form a thin transition layer at the surface. However, under certain conditions, typically near particle resonances, either one or both of these modes can be propagating.


Aaron D. Scher and Edward F. Kuester, "Boundary Effects in the Electromagnetic Response of a Metamaterial in the Case of Normal Incidence," Progress In Electromagnetics Research B, Vol. 14, 341-381, 2009.


    1. Smith, D. R., D. C. Vier, T. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Physical Review E, Vol. 71, 036617(1)-036617(11), Mar. 2005.

    2. Mahan, G. D. and G. Obermair, "Polaritons at surfaces," Physical Review, Vol. 183, 834-841, 1969.

    3. Philpott, M. R., "Reflection of light by a semi-infinite dielectric," Journal of Chemical Physics, Vol. 60, 1410-1419, 1974.

    4. Philpott, M. R., "Polaritons in a spatially dispersive dielectric half space," Physical Review B, Vol. 14, 3471-3487, 1976.

    5. Philpott, M. R., "Effect of spatial dispersion on the S-polarized optical properties of a slab dielectric," Journal of Chemical Physics, Vol. 60, 2520-2529, 1974.

    6. Mead, C. A., "Exactly soluble model for crystal with spatialdispersion," Physical Review B, Vol. 15, 519-532, 1977.

    7. Mead, C. A., "Formally closed solution for a crystal with spatialdispersion," Physical Review B, Vol. 17, 4644-4651, 1978.

    8. Belov, P. A. and C. R. Simovski, "Boundary conditions for interfaces of electromagnetic crystals and the generalized Ewald-Oseen extinction principle," Physical Review B, Vol. 73, 045102(1)-045102(14), Jan. 2006.

    9. Pekar, S. I., Crystal Optics and Additional Light Waves, Benjamin/Cummings Pub. Co., Menlo Park, CA, 1983.

    10. Tretyakov, S. A. and A. J. Viitanen, "Plane waves in regular arrays of dipole scatterers and effective-medium modeling," Journal of the Optical Society of America A-Optics Image Science and Vision , Vol. 17, 1791-1797, Oct. 2000.

    11. Yatsenko, V. and S. I. Maslovski, "Electromagnetic interaction of parallel arrays of dipole scatterers," Progress In Electromagnetics, Vol. 25, 285-307, 2000.

    12. Belov, P. A. and C. R. Simovski, "Oblique propagation of electromagnetic waves in regular 3D lattices of scatterers (dipole approximation)," Proc. SPIE, Vol. 4073, 266-276, 2000.

    13. Belov, P. A. and C. R. Simovski, "Homogenization of electromagnetic crystals formed by uniaxial resonant scatterers," Physical Review E, Vol. 72, 026615(1)-026615(15), Aug. 2005.

    14. Collin, R. E., "Field Theory of Guided Waves," IEEE Press , 2nd edition, New York, 1991.

    15. Simovski, C. R. and S. L. He, "Frequency range and explicit expressions for negative permittivity and permeability for an isotropic medium formed by a lattice of perfectly conducting Omega particles ," Physics Letters A, Vol. 311, 254-263, May 12, 2003 .

    16. Simovski, C. R. and S. A. Tretyakov, "Local constitutive parameters of metamaterials from an effective-medium perspective," Physical Review B, Vol. 75, 195111(1)-195111(10), May 2007.

    17. Simovski, C. R., "Analytical modelling of double-negative composites," Metamaterials, Vol. 2, 169-185, 2008.

    18. Simovski, C. R., "Bloch material parameters of magneto-dielectric metamaterials and the concept of Bloch lattices," Metamaterials, Vol. 1, 62-80, 2007.

    19. Berman, D. H., "An extinction theorem for electromagnetic waves in a point dipole model," American Journal of Physics, Vol. 71, 917-924, Sep. 2003.

    20. Wang, J. F., S. B. Qu, H. Ma, Y. M. Yang, and X. Wu, "Wideangle polarization independent planar left-handed metamaterials based on dielectric resonators," Progress in Electromagnetics Research B, Vol. 12, 243-248, 2009.

    21. Abramowitz, M. and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, Mineola, NY, 1972.

    22. Scher, A. D., Boundary effects in the electromagnetic response of a metamaterial using the point-dipole interaction model, Ph.D. Dissertation, University of Colorado at Boulder, 2008.

    23. Scher, A. D. and E. F. Kuester, "Extracting the bulk effective parameters of a metamaterial via the scattering from a single planar array of particles," Metamaterials, Vol. 3, No. 1, 44-45, 2009.

    24. Sipe, J. E. and J. V. Kranendonk, "Macroscopic electromagnetic theory of resonant dielectrics," Physical Review A, Vol. 9, 1806-1822, 1974.

    25. Jackson, J. D., Classical Electrodynamics, 3rd Ed., Wiley, New York, 1999.

    26. Kempa, K., R. Ruppin, and J. B. Pendry, "Electromagnetic response of a point-dipole crystal," Physical Review B, Vol. 72, 205103(1)-205103(6), Nov. 2005.

    27. Brillouin, L., Wave Propagation in Periodic Structures; Electric Filters and Crystal Lattices, 2nd Ed., Dover Publications, New York, 1953.

    28. Kar, N. and A. Bagchi, "Local-field effect near the surface of dipolar lattices," Solid State Communications, Vol. 33, 645-648, 1980.

    29. Mochan, W. L. and R. G. Barrera, "Surface local field-effect," Journal de Physique Colloque C5, Vol. 45, 207-212, 1984.

    30. Poppe, G. P. M. and C. M. J. Wijers, "Exact solution of the optical-response of thick slabs in the discrete dipole approach," Physica B, Vol. 167, 221-237, Dec. 1990.

    31. Gadomskii, O. N. and S. V. Sukhov, "Microscopic theory of a transition layer on the ideal surface of semiinfinite dielectric media and the near-field effect," Optics and Spectroscopy, Vol. 89, 261-267.

    32. Drude, P., The Theory of Optics, Longmans, Green, and Co., New York, 1902.

    33. Mochan, W. L. and R. G. Barrera, "Intrinsic surface-induced optical anisotropies of cubic crystals: Local field effect," Physical Review Letters, Vol. 55, 1192-1195, 1185.

    34. Idemen, M., "Straightforward derivation of boundary-conditions on sheet simulating an anisotropic thin-layer," Electronics Letters, Vol. 24, 663-665, May 26, 1988.

    35. Idemen, M., "Universal boundary relations of the electromagneticfield," Journal of the Physical Society of Japan, Vol. 59, 71-80, Jan. 199.

    36. Nicolson, A. M. and G. F. Ross, "Measurement of intrinsic properties of materials by time-domain techniques," IEEE Transactions on Instrumentation and Measurement, Vol. 19, 377-382, 1970.

    37. Smith, D. R., S. Schultz, P. Markos, and C. M. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Physical Review B, Vol. 65, 195104(1)-195104(5), May 15, 2002.

    38. Clercx, H. J. H. and G. Bossis, "Electrostatic interactions in slabs of polarizable particles," Journal of Chemical Physics, Vol. 98, 8284-8293, 2004.

    39. Koschny, T., P. Markos, D. R. Smith, and C. M. Soukoulis, "Resonant and antiresonant frequency dependence of the effective parameters of metamaterials," Physical Review E, Vol. 68, 065602(1)-065602(4), Dec. 2003.

    40. Chen, X. D., T. M. Grzegorczyk, B. I. Wu, J. Pacheco, and J. A. Kong, "Robust method to retrieve the constitutive effective parameters of metamaterials," Physical Review E, Vol. 70, 016608(1)-016608(7), 2004.