We investigate the wave propagation properties in lossy structures with graded permittivity and permeability involving left-handed metamaterials. An exact analytic solution to Helmholtz' equation for a lossy case with both real and imaginary parts of permittivity and permeability profile, changing according to a hyperbolic tangent function along the direction of propagation, is obtained. It allows for different loss factors in RHM and LHM media. Thereafter, the corresponding numerical solution for the field intensity along the composite structure is obtained by means of a dispersive numerical model of lossy metamaterials that uses a transmission line matrix method based on Z-transforms. We present the expressions and graphical results for the field intensity along the composite structure and compare the analytic and numerical solutions, showing that there is an excellent agreement between them.
1. Veselago, V. G., "The electrodynamics of substances with simultaneously negative values of ε and μ," Sov. Phys. Uspekhi, Vol. 10, No. 4, 509-514, 1968. doi:10.1070/PU1968v010n04ABEH003699
2. Pendry, J. B., A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low frequency plasmons in thin wire structures," J. Phys. Condens. Mat., Vol. 10, No. 22, 4785-4809, 1998. doi:10.1088/0953-8984/10/22/007
3. Pendry, J. B., A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Transactions on Microwave Theory and Techniques, Vol. 47, No. 11, 2075-2084, 1999. doi:10.1109/22.798002
4. Falcone, F., T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, F. Martiacute, and M. Sorolla, "Babinet principle applied to the design of metasurfaces and metamaterials," Phys. Rev. Lett., Vol. 93, 197401, 2004. doi:10.1103/PhysRevLett.93.197401
5. Dolling, G., C. Enkrich, M. Wegener, J. F. Zhou, C. M. Soukoulis, and S. Linden, "Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials," Opt. Lett., Vol. 30, 3198-3200, 2005. doi:10.1364/OL.30.003198
6. Zhang, S., W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, "Experimental demonstration of near-infrared negative-index metamaterials," Phys. Rev. Lett., Vol. 95, 1-4, 2005.
7. Kafesaki, M., I. Tsiapa, N. Katsarakis, T. Koschny, C. M. Soukoulis, and E. N. Economou, "Left-handed metamaterials: The fish-net structure and its variations," Phys. Rev. B, Vol. 75, 235114, 2007. doi:10.1103/PhysRevB.75.235114
8. Valentine, J., S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature, Vol. 455, 376-379, 2008. doi:10.1038/nature07247
9. Shelby, R. A., D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science, Vol. 292, 77-79, 2001. doi:10.1126/science.1058847
10. Xiao, S., U. K. Chettiar, A. V. Kildishev, V. P. Drachev, and , "Yellow-light negative-index metamaterials," Opt. Lett., Vol. 34, 3478-3480, 2009. doi:10.1364/OL.34.003478
11. Cai, W. and V. Shalaev, Optical Metamaterials: Fundamentals and Applications, Springer, Dordrecht, 2009.
12. Ramakrishna, S. A. and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Materials, SPIE Press Bellingham, WA & CRC Press, Taylor & Francis Group, Boca Raton, FL, 2009.
14. Fang, N., H. Lee, C. Sun, and X. Zhang, "Subdiffraction-limited optical imaging with a silver superlens," Science, Vol. 308, 534-537, 2005. doi:10.1126/science.1108759
15. Engheta, N., "An idea for thin, subwavelength cavity resonators using metamaterials with negative permittivity and permeability," IEEE Anten. Wirel. Propag. Lett., Vol. 1, 10-13, 2002. doi:10.1109/LAWP.2002.802576
16. Zhu, W., I. Rukhlenko, and M. Premaratne, "Linear transfor-mation optics for plasmonics," Journal of the Optical Society of America B: Optical Physics, Vol. 29, No. 10, 2659-2664, 2012. doi:10.1364/JOSAB.29.002659
17. Novitsky, A. V., S. V. Zhukovsky, L. M. Barkovsky, and A. V. Lavrinenko, "Field approach in the transformation optics concept," Progress In Electromagnetics Research, Vol. 129, 485-515, 2012.
18. Chen, X., "Implicit boundary conditions in transformation-optics cloaking for electromagneticwaves," Progress In Electromagnetics Research, Vol. 121, 521-534, 2011. doi:10.2528/PIER11101010
19. Zhu, W., I. D. Rukhlenko, and M. Premaratne, "Manipulating energy flow in variable-gap plasmonic waveguides," Opt. Lett., Vol. 37, No. 24, 5151-5153, 2012. doi:10.1364/OL.37.005151
21. Pendry, J. B., D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science, Vol. 312, No. 5781, 1780-1782, 2006. doi:10.1126/science.1125907
22. Ergin, T., N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, "Three-dimensional invisibility cloak at optical wavelengths," Science, Vol. 328, No. 5976, 337-339, 2010. doi:10.1126/science.1186351
23. Jacob, Z., L. V. Alekseyev, and E. Narimanov, "Optical hyperlens: Far-field imaging beyond the diffraction limit," Opt. Express, Vol. 14, No. 8, 8247-8256, 2006. doi:10.1364/OE.14.008247
24. Cai, W., U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics, Vol. 1, 224-227, 2007. doi:10.1038/nphoton.2007.28
25. Fung, T. H., L. L. Leung, J. J. Xiao, and K. W. Yu, "Controlling electric fields spatially by graded metamaterials: Implication on enhanced nonlinear optical responses," Opt. Commun., Vol. 282, 1028-1031, 2009. doi:10.1016/j.optcom.2008.11.028
26. Ramakrishna, S. A. and J. B. Pendry, "Spherical perfect lens: Solutions of Maxwell's equations for spherical geometry," Phys. Rev. B, Vol. 69, 115115, 2004. doi:10.1103/PhysRevB.69.115115
27. Smith, D. R., J. J. Mock, A. F. Starr, and D. Schurig, "A gradient index metamaterial," Phys. Rev. E, Vol. 71, 036609, 2005. doi:10.1103/PhysRevE.71.036609
28. Pinchuk, A. O. and G. C. Schatz, "Metamaterials with gradient negative index of refraction," J. Opt. Soc. Am. A, Vol. 24, A39-A44, 2007. doi:10.1364/JOSAA.24.000A39
29. Litchinitser, N. M., N. M., A. I. Maimistov, I. R. Gabitov, R. Z. Sagdeev, and V. M. Shalaev, "Metamaterials: Electromagnetic enhancement at zero-index transition," Opt. Lett., Vol. 33, 2350-2352, 2008. doi:10.1364/OL.33.002350
30. Dalarsson, M. and P. Tassin, "Analytical solution for wave propagation through a graded index interface between a right-handed and a left-handed material," Opt. Express, Vol. 17, No. 8, 6747-6752, 2009. doi:10.1364/OE.17.006747
31. Dalarsson, M., Z. Jaksic, and P. Tassin, "Exact analytical solution for oblique incidence on a graded index interface between a right-handed and a left-handed material," J. Optoel. Biomed. Mat., Vol. 1, 345-352, 2009.
32. Dalarsson, M., Z. Jaksic, and P. Tassin, "Structures containing left-handed metamaterials with refractive index gradient: Exact analytical versus numerical treatment," Microwave Rev., Vol. 15, 1-5, 2009.
33. Dalarsson, M., M. Norgren, and Z. Jak·sic, "Lossy gradient index metamaterial with sinusoidal periodicity of refractive index: Case of constant impedance throughout the structure," J. Nanophoton., Vol. 5, 051804, 2011. doi:10.1117/1.3590251
34. Dalarsson, M., M. Norgren, and Z. Jaksic, "Lossy wave propagation through a graded interface to a negative index material case of constant impedance," Microwave Rev., Vol. 17, 1-6, 2011.
35. Doncov, N., B. Milovanovic, T. Asenov, and J. Paul, "TLM modelling of left-handed metamaterials by using digital filtering techniques," Microwave Rev., Vol. 16, 2-7, 2010.
36. Paul, J., C. Christopoulos, and D. W. P. Thomas, "Generalized material models in TLM. Part I: Materials with frequency dependent properties," IEEE Trans. Antennas and Propag., Vol. 47, No. 10, 1528-1534, 1999. doi:10.1109/8.805895
37. Paul, J., C. Christopoulos, and D. W. P. Thomas, "Generalized material models in TLM. Part II: Materials with anisotropic properties," IEEE Trans. Antennas and Propag., Vol. 47, No. 10, 1535-1542, 1999. doi:10.1109/8.805896