We introduce a technique to decompose the scattered near field of two-dimensional arbitrary metaatoms into its multipole contributions. To this end we expand the scattered field upon plane wave illumination into cylindrical harmonics as known from Mie’s theory. By relating these cylindrical harmonics to the field radiated by Cartesian multipoles, the contribution of the lowest order electric and magnetic multipoles can be identified. Revealing these multipoles is essential for the design of metamaterials because they largely determine the character of light propagation. In particular, having this information at hand it is straightforward to distinguish between effects that result either from the arrangement of the metaatoms or from their particular design.

© 2010 Optical Society of America

1. S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic Response of Metamaterials at 100 Terahertz,” Science 306, 1351-1352 (2004). [CrossRef] [PubMed]
2. N.-H. Shen, S. Foteinopoulou, M. Kafesaki, T. Koschny, E. Ozbay, E. N. Economou, and C. M. Soukoulis, “Compact planar far-field superlens based on anisotropic left-handed metamaterials,” Phys. Rev. B 80, 115123 (2009). [CrossRef]
3. S. Xiao, U. K. Chettiar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Yellow-light negative-index metamaterials,” Opt. Lett. 34, 3478-3480 (2009). [CrossRef] [PubMed]
4. J. Valentine, 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 455, 376-380 (2008). [CrossRef] [PubMed]
5. N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nature Mat. 7, 31-37 (2008). [CrossRef]
6. D. J. Cho, F. Wang, X. Zhang, and Y. R. Shen, “Contribution of the electric quadrupole resonance in optical metamaterials,” Phys. Rev. B 78, 121101(R) (2008). [CrossRef]
7. J. Petschulat, C. Menzel, A. Chipouline, C. Rockstuhl, A. Tunnermann, F. Lederer, and T. Pertsch, “Multipole approach to metamaterials,” Phys. Rev. A 78, 043811 (2008). [CrossRef]
8. K. Vynck, D. Felbacq, E. Centeno, A. I. Cabuz, D. Cassagne, and B. Guizal, “All-Dielectric Rod-Type Metamaterials at Optical Frequencies,” Phys. Rev. Lett. 102, 133901 (2009). [CrossRef] [PubMed]
9. M. Burresi, D. V. Oosten, T. Kampfrath, H. Schoenmaker, R. Heideman, A. Leinse, and L. Kuipers, “Probing the Magnetic Field of Light at Optical Frequencies,” Science 326, 550-553 (2009). [CrossRef] [PubMed]
10. M. Decker, S. Burger, S. Linden, andM.Wegener, “Magnetization waves in split-ring-resonator arrays: Evidence for retardation effects,” Phys. Rev. B 80, 193102 (2009). [CrossRef]
11. N. Papasimakis, V. A. Fedotov, Y. H. Fu, D. P. Tsai, and N. I. Zheludev, “Coherent and incoherent metamaterials and order-disorder transitions,” Phys. Rev. B 80, 041102(R) (2009). [CrossRef]
12. N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nature Photon. 3, 157-162 (2009). [CrossRef]
13. J. Petschulat, A. Chipouline, A. T¨unnermann, T. Pertsch, C. Menzel, C. Rockstuhl, and F. Lederer, “Multipole nonlinearity of metamaterials,” Phys. Rev. A 80, 063828 (2009). [CrossRef]
14. Y. Zeng, C. Dineen, and J. V. Moloney, “Magnetic dipole moments in single and coupled split-ring resonators,” Phys. Rev. B 81, 075116 (2010). [CrossRef]
15. A. Serdyukov, I. Semchenko, S. Tretyakov, A. Sihvola, Electromagnetics of bi-anisotropic materials: Theory and applications (Gordon and Breach, Amsterdam, 2001).
16. C. Helgert, C. Rockstuhl, C. Etrich, C. Menzel, E.-B. Kley, A. T¨unnermann, F. Lederer, and T. Pertsch, “Effective properties of amorphous metamaterials,” Phys. Rev. B 79, 233107 (2009). [CrossRef]
17. G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 25, 377-445 (1908). [CrossRef]
18. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
19. R. E. Raab and O. L. D. Lange, Multipole Theory in Electromagnetism (Clarendon, Oxford, 2005).
20. N. Liu, L. Langguth, T. Weiss, J. K¨astel, M. Fleischhauer, T. Pfau and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nature Mat. 8, 758-762 (2009). [CrossRef]
21. P. B. Johnson and R. W. Christy, “Optical Constants of Noble Metals,” Phys. Rev. B 6, 4370-4379 (1972). [CrossRef]
22. . We applied the commercial product COMSOL. (www.comsol.com)
23. M. Husnik, M. W. Klein, N. Feth, M. Konig, J. Niegemann, K. Busch, S. Linden, and M. Wegener, “Absolute extinction cross-section of individual magnetic split-ring resonators,” Nature Photon. 2, 614-617 (2008). [CrossRef]
24. M. Celebrano, M. Savoini, P. Biagioni, M. Zavelani-Rossi, P.-M. Adam, L. Duo, G. Cerullo, and M. Finazzi, “Retrieving the complex polarizability of single plasmonic nanoresonators,” Phys. Rev. B 80,153407 (2009). [CrossRef]
25. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
26. O. J. F. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E 58, 3909-3915 (1998). [CrossRef]
27. I. S. Gradstein and I. M. Ryshik, Tables of Series, Products and Integrals (Harry Deutsch, Frankfurt, 1981).

Ann. Phys.

• G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 25, 377-445 (1908). [CrossRef]

Nature

• J. Valentine, 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 455, 376-380 (2008). [CrossRef] [PubMed]

Nature Mat.

• N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nature Mat. 7, 31-37 (2008). [CrossRef]
• N. Liu, L. Langguth, T. Weiss, J. K¨astel, M. Fleischhauer, T. Pfau and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nature Mat. 8, 758-762 (2009). [CrossRef]

Nature Photon.

• M. Husnik, M. W. Klein, N. Feth, M. Konig, J. Niegemann, K. Busch, S. Linden, and M. Wegener, “Absolute extinction cross-section of individual magnetic split-ring resonators,” Nature Photon. 2, 614-617 (2008). [CrossRef]
• N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nature Photon. 3, 157-162 (2009). [CrossRef]

Opt. Lett.

• S. Xiao, U. K. Chettiar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Yellow-light negative-index metamaterials,” Opt. Lett. 34, 3478-3480 (2009). [CrossRef] [PubMed]

Phys. Rev. A

• J. Petschulat, A. Chipouline, A. T¨unnermann, T. Pertsch, C. Menzel, C. Rockstuhl, and F. Lederer, “Multipole nonlinearity of metamaterials,” Phys. Rev. A 80, 063828 (2009). [CrossRef]
• J. Petschulat, C. Menzel, A. Chipouline, C. Rockstuhl, A. Tunnermann, F. Lederer, and T. Pertsch, “Multipole approach to metamaterials,” Phys. Rev. A 78, 043811 (2008). [CrossRef]

Phys. Rev. B

• M. Decker, S. Burger, S. Linden, andM.Wegener, “Magnetization waves in split-ring-resonator arrays: Evidence for retardation effects,” Phys. Rev. B 80, 193102 (2009). [CrossRef]
• N.-H. Shen, S. Foteinopoulou, M. Kafesaki, T. Koschny, E. Ozbay, E. N. Economou, and C. M. Soukoulis, “Compact planar far-field superlens based on anisotropic left-handed metamaterials,” Phys. Rev. B 80, 115123 (2009). [CrossRef]
• Y. Zeng, C. Dineen, and J. V. Moloney, “Magnetic dipole moments in single and coupled split-ring resonators,” Phys. Rev. B 81, 075116 (2010). [CrossRef]
• C. Helgert, C. Rockstuhl, C. Etrich, C. Menzel, E.-B. Kley, A. T¨unnermann, F. Lederer, and T. Pertsch, “Effective properties of amorphous metamaterials,” Phys. Rev. B 79, 233107 (2009). [CrossRef]
• M. Celebrano, M. Savoini, P. Biagioni, M. Zavelani-Rossi, P.-M. Adam, L. Duo, G. Cerullo, and M. Finazzi, “Retrieving the complex polarizability of single plasmonic nanoresonators,” Phys. Rev. B 80,153407 (2009). [CrossRef]
• P. B. Johnson and R. W. Christy, “Optical Constants of Noble Metals,” Phys. Rev. B 6, 4370-4379 (1972). [CrossRef]

Phys. Rev. E

• O. J. F. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E 58, 3909-3915 (1998). [CrossRef]

Phys. Rev. Lett.

• K. Vynck, D. Felbacq, E. Centeno, A. I. Cabuz, D. Cassagne, and B. Guizal, “All-Dielectric Rod-Type Metamaterials at Optical Frequencies,” Phys. Rev. Lett. 102, 133901 (2009). [CrossRef] [PubMed]

Science

• M. Burresi, D. V. Oosten, T. Kampfrath, H. Schoenmaker, R. Heideman, A. Leinse, and L. Kuipers, “Probing the Magnetic Field of Light at Optical Frequencies,” Science 326, 550-553 (2009). [CrossRef] [PubMed]
• S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic Response of Metamaterials at 100 Terahertz,” Science 306, 1351-1352 (2004). [CrossRef] [PubMed]

Other

• D. J. Cho, F. Wang, X. Zhang, and Y. R. Shen, “Contribution of the electric quadrupole resonance in optical metamaterials,” Phys. Rev. B 78, 121101(R) (2008). [CrossRef]
• N. Papasimakis, V. A. Fedotov, Y. H. Fu, D. P. Tsai, and N. I. Zheludev, “Coherent and incoherent metamaterials and order-disorder transitions,” Phys. Rev. B 80, 041102(R) (2009). [CrossRef]
• J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
• R. E. Raab and O. L. D. Lange, Multipole Theory in Electromagnetism (Clarendon, Oxford, 2005).
• A. Serdyukov, I. Semchenko, S. Tretyakov, A. Sihvola, Electromagnetics of bi-anisotropic materials: Theory and applications (Gordon and Breach, Amsterdam, 2001).
• I. S. Gradstein and I. M. Ryshik, Tables of Series, Products and Integrals (Harry Deutsch, Frankfurt, 1981).
• . We applied the commercial product COMSOL. (www.comsol.com)
• C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

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