In this paper we analyze the electromagnetic response of a metamaterial slab in the case of normal incidence using the point-dipole interaction model and an expansion of polarization by eigenmodes. The problem is simplified by assuming that the lattice dimensions are smaller than a half wavelength and invoking the nearest neighbor approximation. In this manner, we find the structure supports three modes: an ordinary mode and two extraordinary modes. In the long-wavelength limit, the ordinary mode propagates with the same wave number as that predicted using the classic Clausius-Mossotti relations, while, for most cases, the two extraordinary modes are confined to thin surface transition layers near the boundaries of the slab. A systematic method is presented to find the scattering from the slab, and the results are confirmed by full wave simulation using Ansoft HFSS.
2. Maystre, D. and S. Enoch, "Perfect lenses made with left-handed materials: Alice's mirror?," Journal of the Optical Society of America A | Optics Image Science and Vision, Vol. 21, 122-131, 2004.
3. Larkin, I. A. and M. I. Stockman, "Imperfect perfect lens," Nano Letters, Vol. 5, 339-343, 2005.
4. Smith, D. R., D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, "Limitations on subdiffraction imaging with a negative refractive index slab," Applied Physics Letters, Vol. 82, 1506-1508, 2003.
5. French, O. E., K. I. Hopcraft, and E. Jakeman, "Perturbation on the perfect lens: The near-perfect lens," New Journal of Physics, Vol. 8, No. 271, 2006.
6. Merlin, R., "Analytical solution of the almost-perfect-lens problem," Applied Physics Letters, Vol. 84, 1290-1292, 2004.
7. Enoch, S., G. Tayeb, P. Sabouroux, N. Guerin, P. Vincent, and , "A metamaterial for directive emission," Physical Review Letters, Vol. 89, 213902-1-213902-4, 2002.
8. Alu, A., F. Bilotti, N. Engheta, and L. Vegni, "Metamaterial covers over a small aperture," IEEE Transactions on Antennas and Propagation, Vol. 54, 1632-1643, 2006.
9. Li, B., B. Wu, and C. H. Liang, "Study on high gain circular waveguide array antenna with metamaterial structure," Progress In Electromagnetics Research, Vol. 60, 207-219, 2006.
10. Saenz, E., K. Guven, E. Ozbay, I. Ederra, and R. Gonzalo, "Enhanced directed emission from metamaterial based radiation source," Applied Physics Letters, Vol. 92, 204103-1-204103-3, 2008.
11. Engheta, N., "An idea for thin subwavelength cavity resonators using metamaterials with negative permittivity and permeability," IEEE Antennas and Wireless Propagation Letters, Vol. 1, 10-13, 2002.
12. Caiazzo, M., S. Maci, and N. Engheta, "A metamaterial surface for compact cavity resonators," IEEE Antennas and Wireless Propagation Letters, Vol. 3, 261-264, 2004.
13. Holloway, C. L., D. C. Love, E. F. Kuester, A. Salandrino, and N. Engheta, "Sub-wavelength resonators: On the use of metafilms to overcome the lambda/2 size limit," IET Microw. Antennas Propag., Vol. 2, 120-129, 2008.
14. Bilotti, F., L. Nucci, and L. Vegni, "An SRR based microwave absorber," Microwave and Optical Technology Letters, Vol. 48, 2171-2175, 2006.
15. 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, 44-55, 2009.
16. Simovski, C. R. and S. A. Tretyakov, "Local constitutive parameters of metamaterials from an effective-medium perspective," Physical Review B, Vol. 75, No. 195111-1, 1-10, 2007.
17. Smith, D. R., S. Schultz, P. Markos, and C. M. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coe±cients," Physical Review B, Vol. 65, 195104-1-195104-5, 2002.
18. Smith, D. R., D. C. Vier, T. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Physical Review E, Vol. 71, No. 036617, 1-11, 2005.
19. Kar, N. and A. Bagchi, "Local-field effect near the surface of dipolar lattices," Solid State Communications, Vol. 33, 645-648, 1980.
20. 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, 1990.
21. Clercx, H. J. H. and G. Bossis, "Electrostatic interactions in slabs of polarizable particles," Journal of Chemical Physics, Vol. 98, 8284-8293, 1993.
22. Berman, D. H., "An extinction theorem for electromagnetic waves in a point dipole model," American Journal of Physics, Vol. 71, 917-924, 2003.
23. Sivukhin, D. V., "Molecular theory of the reflection and refraction of light," Zh. Eksp. Teor. Fiz., Vol. 18, 976-994, 1948.
24. Scher, A. D., Boundary effects in the electromagnetic metamaterial using the point-dipole interaction model, Ph.D. Dissertation, University of Colorado, Boulder, 2008.
25. Belov, P. A. and C. R. Simovski, "Homogenization of electromagnetic crystals formed by uniaxial resonant scatterers," Physical Review E, Vol. 72, No. 026615, 1-15, 2005.
26 . Silveirinha, M. G., "Generalized Lorentz-Lorenz formulas for microstructured materials," Physical Review B, Vol. 76, 245117-1-245117-9, 2007.
27. Silveirinha, M. G., "Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters," Physical Review B, Vol. 75, No. 115104, 1-15, 2007.
28. Ewald, P. P., On the Foundations of Crystal Optics, Air Force Cabridge Research Laboratories, 1970.
29. Mahan, G. D. and G. Obermair, "Polaritons at surfaces," Physical Review, Vol. 183, 834-841, 1969.
30. Philpott , M. R., "Reflection of light by a semi-in¯nite dielectric," Journal of Chemical Physics, Vol. 60, 1410-1419, 1974.
31. Philpott, M. R., "Polaritons in a spatially dispersive dielectric half space ," Physical Review B, Vol. 14, 3471-3487, 1976.
32. Philpott, M. R., "Effect of spatial dispersion of S-polarized optical properties of a slab dielectric," Journal of Chemical Physics, Vol. 60, 2520-2529, 1974.
33. Gadomsky, O. N. and K. V. Krutitsky, "Near-field effect in surface optics," Journal of the Optical Society of America B --- Optical Physics, Vol. 13, 1679-1690, 1996.
34. Krutitsky, K. V. and S. V. Suhov, "Near-field effect in classical optics of ultra-thin films," Journal of Physics B --- Atomic Molecular and Optical Physics, Vol. 30, 5341-5358, 1997.
35. Simovski, C. R., P. A. Belov, M. S. Kondratjev, and , "Electromagnetic interaction of chiral particles in three-dimensional arrays," Journal of Electromagnetic Waves and Applications,, Vol. 13, No. 2, 189-204, 1999.
36. 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.
37. 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, 2000.
38. Simovski, C. R., M. Popov, and S. L. He, "Dielectric properties of a thin film consisting of a few layers of molecules or particles," Physical Review B, Vol. 62, 13718-13730, 2000.
39. 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, 2000.
40. 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, 2003.
41. Simovski, C. R., "Bloch material parameters of magneto-dielectric metamaterials and the concept of Bloch lattices," Metamaterials, Vol. 1, 62-80, 2007.
42. Simovski, C. R., "Analytical modelling of double-negative composites," Metamaterials, Vol. 2, 169-185, 2008.
43. 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, No. 045102, 1-14, 2006.
44. Scher, A. D. and E. 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.
45. Shore, Shore and A. D. Yaghjian, "Electromagnetic waves on partially ¯nite periodic arrays of lossless or lossy penetrable spheres," IEICE Transactions on Communications, Vol. E91b, 1819-1824, 2008.
46. Sipe, J. E. and J. V. Kranendonk, "Macroscopic electromagnetic theory of resonant dielectrics," Physical Review A, Vol. 9, 1806-1822, 1976.