After the prediction that strong enough optical activity may result in negative refraction and negative reflection, more and more artificial chiral metamaterials were designed and fabricated at difference frequency ranges from microwaves to optical waves. Therefore, a simple and robust method to retrieve the effective constitutive parameters for chiral metamaterials is urgently needed. Here, we analyze the wave propagation in chiral metamaterials and follow the regular retrieval procedure for ordinary metamaterials and apply it in chiral metamaterial slabs. Then based on the transfer matrix technique, the parameter retrieval is extended to treat samples with not only the substrate but also the top layers. After the parameter retrieval procedure, we take two examples to check our method and study how the substrate influences on the thin chiral metamaterials slabs. We find that the substrate may cause the homogeneous slab to be inhomogeneous, i.e. the reflections in forward and backward directions are different. However, the chiral metamaterial where the resonance element is embedded far away from the substrate is insensitive to the substrate.

© 2010 Optical Society of America

1. J. B. Pendry, “A chiral route to negative refraction,” Science 306, 1353–1355 (2004). [CrossRef] [PubMed]
2. S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, “Waves and energy in chiral nihility,” J. Electromagn. Waves Appl. 17, 695–706 (2003). [CrossRef]
3. C. Monzon, and D. W. Forester, “Negative refraction and focusing of circularly polarized waves in optically active media,” Phys. Rev. Lett. 95, 123904 (2005). [CrossRef] [PubMed]
4. S. Tretyakov, A. Sihvola, and L. Jylha, “Backward-wave regime and negative refraction in chiral composites,” Photonics Nanostruct. Fundam. Appl. 3, 107 (2005). [CrossRef]
5. V. Yannopapas, “Negative index of refraction in artificial chiral materials,” J. Phys. Condens. Matter 18, 6883–6890 (2006). [CrossRef]
6. V. M. Agranovich, Y. N. Gartstein, and A. A. Zakhidov, “Negative refraction in gyrotropic media,” Phys. Rev. B 73, 045114 (2006). [CrossRef]
7. C. Zhang, and T. J. Cui, “Negative reflections of electromagnetic waves in a strong chiral medium,” Appl. Phys. Lett. 91, 194101 (2007). [CrossRef]
8. J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. V. Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009). [CrossRef] [PubMed]
9. E. Plum, J. Zhou, J. Dong, V. A. Fedotov, Th. Koschny, C. M. Soukoulis, and N. I. Zheludev, “Metamaterial with negative index due to chirality,” Phys. Rev. B 79, 035407 (2009). [CrossRef]
10. J. Zhou, J. Dong, B. Wang, Th. Koschny, M. Kafesaki, and C. M. Soukoulis, “Negative refractive index due to chirality,” Phys. Rev. B 79, 121104 (2009). [CrossRef]
11. J. Dong, J. Zhou, Th. Koschny, and C. M. Soukoulis, “Bi-layer cross chiral structure with strong optical activity and negative refractive index,” Opt. Express 17, 14172–14179 (2009). [CrossRef] [PubMed]
12. E. Plum, V. A. Fedotov, and N. I. Zheludev, “Optical activity in extrinsically chiral metamaterial,” Appl. Phys. Lett. 93, 191911 (2008). [CrossRef]
13. E. Plum, X.-X. Liu, V. A. Fedotov, Y. Chen, D. P. Tsai, and N. I. Zheludev, “Metamaterials: optical activity without chirality,” Phys. Rev. Lett. 102, 113902 (2009). [CrossRef] [PubMed]
14. L. Jelinek, R. Marquěs, F. Mesa, and J. D. Baena, “Periodic arrangements of chiral scatterers providing negative refractive index bi-isotropic media,” Phys. Rev. B 77, 205110 (2008). [CrossRef]
15. B. Wang, J. Zhou, Th. Koschny, and C. M. Soukoulis, “Nonplanar chiral metamaterials with negative index,” Appl. Phys. Lett. 94, 151112 (2009). [CrossRef]
16. V. Yannopapas, “Circular dichroism in planar nonchiral plasmonic metamaterials,” Opt. Lett. 34, 632–634 (2009). [CrossRef] [PubMed]
17. S. Zhang, Y. S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett. 102, 023901 (2009). [CrossRef] [PubMed]
18. E. Plum, V. A. Fedotov, A. S. Schwanecke, Y. Chen, and N. I. Zheludev, “Giant optical gyrotropy due to electromagnetic coupling,” Appl. Phys. Lett. 90, 223113 (2007). [CrossRef]
19. M. Kuwata-Gonokami, N. Saito, Y. Ino, M. Kauranen, K. Jefimovs, T. Vallius, J. Turunen, and Y. Svirko, “Giant optical activity in quasi-two-dimensional planar nanostructures,” Phys. Rev. Lett. 95, 227401 (2005). [CrossRef] [PubMed]
20. M. Decker, R. Zhao, C. M. Soukoulis, S. Linden, and M. Wegener, “Twisted split-ring-resonator photonic metamaterial with huge optical activity,” Opt. Lett. 35, 1593–1595 (2010). [CrossRef] [PubMed]
21. M. Decker, M. Ruther, C. E. Kriegler, J. Zhou, C. M. Soukoulis, S. Linden, and M. Wegener, “Strong optical activity from twisted-cross photonic metamaterials,” Opt. Lett. 34, 1501–1503 (2009). [CrossRef] [PubMed]
22. M. Decker, M. W. Klein, M. Wegener, and S. Linden, “Circular dichroism of planar chiral magnetic metamaterials,” Opt. Lett. 32, 856–858 (2007). [CrossRef] [PubMed]
23. D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002). [CrossRef]
24. X. Chen, T. M. Grzegorczyk, B. I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70, 016608 (2004). [CrossRef]
25. Th. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. 93, 107402 (2004). [CrossRef] [PubMed]
26. Th. Koschny, P. Markoš, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, “Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials,” Phys. Rev. B 71, 245105 (2005). [CrossRef]
27. D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71, 036617 (2005). [CrossRef]
28. Z. Li, K. Aydin, and E. Ozbay, “Determination of the effective constitutive parameters of bianisotropic metamaterials from reflection and transmission coefficients,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79, 026610 (2009). [CrossRef]
29. B. Wang, J. Zhou, Th. Koschny, M. Kafesaki, and C. M. Soukoulis, “Chiral metamaterials: simulations and experiments,” J. Opt. A, Pure Appl. Opt. 11, 114003 (2009). [CrossRef]
30. D. H. Kwon, D. H. Werner, A. V. Kildishev, and V. M. Shalaev, “Material parameter retrieval procedure for general bi-isotropic metamaterials and its application to optical chiral negative-index metamaterial design,” Opt. Express 16, 11822–11829 (2008). [CrossRef] [PubMed]
31. I. V. Lindell,  et al., Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, Boston · London, 1994).
32. A. Serdyukov,  et al., Electromagnetics of Bi-anisotropic Materials: Theory and Applications (Gordon and Breach Science Publishers, Amsterdam, 2001).
33. Some people [see, for instance, A. Lakhtakia et al., “Reflection of plane waves at planar achiral-chiral interfaces: independence of the reflected polarization state from the incident polarization state,” J. Opt. Soc. Am. A 7, 1654 (1990).] use the Drude-Born-Fedorov relations: D = ε (B + β∇ ×E), B = μ(H + β∇ ×H), where β characterizes the strength of the chirality. They can be brought to the same form. The transformations between the parameters of the two systems are given in Ref. 26. [CrossRef]
34. CST MICROWAVE STUDIO (CST MWS) is a specialist tool for the 3D EM simulation of high frequency components, http://www.cst.com/Content/Products/MWS/Overview.aspx.
35. P. Markoš, and C. M. Soukoulis, Wave Propagation: From Electrons to Photonic crystals and Left-Handed Materials (Princeton University Press, Princeton, 2008).
36. R. Zhao, J. Zhou, Th. Koschny, E. N. Economou, and C. M. Soukoulis, “Repulsive Casimir force in chiral metamaterials,” Phys. Rev. Lett. 103, 103602 (2009). [CrossRef] [PubMed]
37. J. B. Pendry, A. J. Holden, and D. J. Robbins, “andW. J. Stewart,“ Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47, 2075 (1999).
38. E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937). [CrossRef]
39. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (2nd ed., Pergamon Press, Oxford, 1984), §104, p.362–367.

Appl. Phys. Lett.

• C. Zhang, and T. J. Cui, “Negative reflections of electromagnetic waves in a strong chiral medium,” Appl. Phys. Lett. 91, 194101 (2007). [CrossRef]
• E. Plum, V. A. Fedotov, and N. I. Zheludev, “Optical activity in extrinsically chiral metamaterial,” Appl. Phys. Lett. 93, 191911 (2008). [CrossRef]
• B. Wang, J. Zhou, Th. Koschny, and C. M. Soukoulis, “Nonplanar chiral metamaterials with negative index,” Appl. Phys. Lett. 94, 151112 (2009). [CrossRef]
• E. Plum, V. A. Fedotov, A. S. Schwanecke, Y. Chen, and N. I. Zheludev, “Giant optical gyrotropy due to electromagnetic coupling,” Appl. Phys. Lett. 90, 223113 (2007). [CrossRef]

IEEE Trans. Microw. Theory Tech.

• J. B. Pendry, A. J. Holden, and D. J. Robbins, “andW. J. Stewart,“ Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47, 2075 (1999).

J. Electromagn. Waves Appl.

• S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, “Waves and energy in chiral nihility,” J. Electromagn. Waves Appl. 17, 695–706 (2003). [CrossRef]

J. Opt. A, Pure Appl. Opt.

• B. Wang, J. Zhou, Th. Koschny, M. Kafesaki, and C. M. Soukoulis, “Chiral metamaterials: simulations and experiments,” J. Opt. A, Pure Appl. Opt. 11, 114003 (2009). [CrossRef]

J. Phys. Condens. Matter

• V. Yannopapas, “Negative index of refraction in artificial chiral materials,” J. Phys. Condens. Matter 18, 6883–6890 (2006). [CrossRef]

Opt. Express

• J. Dong, J. Zhou, Th. Koschny, and C. M. Soukoulis, “Bi-layer cross chiral structure with strong optical activity and negative refractive index,” Opt. Express 17, 14172–14179 (2009). [CrossRef] [PubMed]
• D. H. Kwon, D. H. Werner, A. V. Kildishev, and V. M. Shalaev, “Material parameter retrieval procedure for general bi-isotropic metamaterials and its application to optical chiral negative-index metamaterial design,” Opt. Express 16, 11822–11829 (2008). [CrossRef] [PubMed]

Opt. Lett.

• M. Decker, R. Zhao, C. M. Soukoulis, S. Linden, and M. Wegener, “Twisted split-ring-resonator photonic metamaterial with huge optical activity,” Opt. Lett. 35, 1593–1595 (2010). [CrossRef] [PubMed]
• M. Decker, M. Ruther, C. E. Kriegler, J. Zhou, C. M. Soukoulis, S. Linden, and M. Wegener, “Strong optical activity from twisted-cross photonic metamaterials,” Opt. Lett. 34, 1501–1503 (2009). [CrossRef] [PubMed]
• M. Decker, M. W. Klein, M. Wegener, and S. Linden, “Circular dichroism of planar chiral magnetic metamaterials,” Opt. Lett. 32, 856–858 (2007). [CrossRef] [PubMed]
• V. Yannopapas, “Circular dichroism in planar nonchiral plasmonic metamaterials,” Opt. Lett. 34, 632–634 (2009). [CrossRef] [PubMed]

Photonics Nanostruct. Fundam. Appl.

• S. Tretyakov, A. Sihvola, and L. Jylha, “Backward-wave regime and negative refraction in chiral composites,” Photonics Nanostruct. Fundam. Appl. 3, 107 (2005). [CrossRef]

Phys. Rev. B

• V. M. Agranovich, Y. N. Gartstein, and A. A. Zakhidov, “Negative refraction in gyrotropic media,” Phys. Rev. B 73, 045114 (2006). [CrossRef]
• L. Jelinek, R. Marquěs, F. Mesa, and J. D. Baena, “Periodic arrangements of chiral scatterers providing negative refractive index bi-isotropic media,” Phys. Rev. B 77, 205110 (2008). [CrossRef]
• E. Plum, J. Zhou, J. Dong, V. A. Fedotov, Th. Koschny, C. M. Soukoulis, and N. I. Zheludev, “Metamaterial with negative index due to chirality,” Phys. Rev. B 79, 035407 (2009). [CrossRef]
• J. Zhou, J. Dong, B. Wang, Th. Koschny, M. Kafesaki, and C. M. Soukoulis, “Negative refractive index due to chirality,” Phys. Rev. B 79, 121104 (2009). [CrossRef]
• D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002). [CrossRef]
• Th. Koschny, P. Markoš, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, “Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials,” Phys. Rev. B 71, 245105 (2005). [CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

• D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71, 036617 (2005). [CrossRef]
• Z. Li, K. Aydin, and E. Ozbay, “Determination of the effective constitutive parameters of bianisotropic metamaterials from reflection and transmission coefficients,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79, 026610 (2009). [CrossRef]
• X. Chen, T. M. Grzegorczyk, B. I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70, 016608 (2004). [CrossRef]

Phys. Rev. Lett.

• Th. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. 93, 107402 (2004). [CrossRef] [PubMed]
• M. Kuwata-Gonokami, N. Saito, Y. Ino, M. Kauranen, K. Jefimovs, T. Vallius, J. Turunen, and Y. Svirko, “Giant optical activity in quasi-two-dimensional planar nanostructures,” Phys. Rev. Lett. 95, 227401 (2005). [CrossRef] [PubMed]
• R. Zhao, J. Zhou, Th. Koschny, E. N. Economou, and C. M. Soukoulis, “Repulsive Casimir force in chiral metamaterials,” Phys. Rev. Lett. 103, 103602 (2009). [CrossRef] [PubMed]
• E. Plum, X.-X. Liu, V. A. Fedotov, Y. Chen, D. P. Tsai, and N. I. Zheludev, “Metamaterials: optical activity without chirality,” Phys. Rev. Lett. 102, 113902 (2009). [CrossRef] [PubMed]
• S. Zhang, Y. S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett. 102, 023901 (2009). [CrossRef] [PubMed]
• C. Monzon, and D. W. Forester, “Negative refraction and focusing of circularly polarized waves in optically active media,” Phys. Rev. Lett. 95, 123904 (2005). [CrossRef] [PubMed]

Rev. Mod. Phys.

• E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937). [CrossRef]

Science

• J. B. Pendry, “A chiral route to negative refraction,” Science 306, 1353–1355 (2004). [CrossRef] [PubMed]
• J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. V. Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009). [CrossRef] [PubMed]

Other

• L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (2nd ed., Pergamon Press, Oxford, 1984), §104, p.362–367.
• I. V. Lindell,  et al., Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, Boston · London, 1994).
• A. Serdyukov,  et al., Electromagnetics of Bi-anisotropic Materials: Theory and Applications (Gordon and Breach Science Publishers, Amsterdam, 2001).
• Some people [see, for instance, A. Lakhtakia et al., “Reflection of plane waves at planar achiral-chiral interfaces: independence of the reflected polarization state from the incident polarization state,” J. Opt. Soc. Am. A 7, 1654 (1990).] use the Drude-Born-Fedorov relations: D = ε (B + β∇ ×E), B = μ(H + β∇ ×H), where β characterizes the strength of the chirality. They can be brought to the same form. The transformations between the parameters of the two systems are given in Ref. 26. [CrossRef]
• CST MICROWAVE STUDIO (CST MWS) is a specialist tool for the 3D EM simulation of high frequency components, http://www.cst.com/Content/Products/MWS/Overview.aspx.
• P. Markoš, and C. M. Soukoulis, Wave Propagation: From Electrons to Photonic crystals and Left-Handed Materials (Princeton University Press, Princeton, 2008).

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