volume | PIER Journals

Abstract

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 Google Scholar

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 Google Scholar

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 Google Scholar

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 Google Scholar

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 Google Scholar

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. Google Scholar

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 Google Scholar

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 Google Scholar

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 Google Scholar

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 Google Scholar

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.

13. Pendry, J. B., "Negative refraction makes a perfect lens," Phys. Rev. Lett., Vol. 85, 3966-3969, 2000.
doi:10.1103/PhysRevLett.85.3966 Google Scholar

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 Google Scholar

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 Google Scholar

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 Google Scholar

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. Google Scholar

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 Google Scholar

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 Google Scholar

20. Leonhardt, U., "Optical conformal mapping," Science, Vol. 312, No. 5781, 1777-1780, 2006.
doi:10.1126/science.1126493 Google Scholar

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 Google Scholar

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 Google Scholar

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 Google Scholar

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 Google Scholar

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 Google Scholar

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 Google Scholar

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 Google Scholar

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 Google Scholar

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 Google Scholar

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 Google Scholar

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. Google Scholar

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. Google Scholar

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 Google Scholar

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. Google Scholar

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. Google Scholar

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 Google Scholar

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 Google Scholar

Dr. Mariana Dalarsson

Dr. Martin Karl Norgren

Ms. Tatjana Asenov

Prof. Nebojsa Doncov