Abstract

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

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  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).

2010 (1)

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]

2009 (10)

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]

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]

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]

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]

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]

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]

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]

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. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nature Photon. 3, 157-162 (2009).
[CrossRef]

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]

2008 (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]

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]

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]

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]

2004 (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]

1998 (1)

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

1972 (1)

P. B. Johnson and R. W. Christy, “Optical Constants of Noble Metals,” Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

1908 (1)

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

Adam, P.-M.

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]

Bartal, G.

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]

Biagioni, P.

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]

Burger, S.

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]

Burresi, M.

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]

Busch, K.

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]

Cabuz, A. I.

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]

Cassagne, D.

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]

Celebrano, M.

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]

Centeno, E.

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]

Cerullo, G.

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]

Chettiar, U. K.

Chipouline, 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]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical Constants of Noble Metals,” Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Decker, M.

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]

Dineen, C.

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]

Drachev, V. P.

Duo, L.

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]

Economou, E. N.

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]

Enkrich, C.

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]

Etrich, C.

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]

Felbacq, D.

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]

Feth, N.

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]

Finazzi, M.

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]

Fleischhauer, M.

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]

Foteinopoulou, S.

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]

Fu, L.

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]

Genov, D. A.

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]

Giessen, H.

N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nature Photon. 3, 157-162 (2009).
[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]

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]

Guizal, B.

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]

Guo, H.

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]

Heideman, R.

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]

Helgert, C.

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]

Husnik, M.

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]

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical Constants of Noble Metals,” Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

K¨astel, J.

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]

Kafesaki, M.

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]

Kaiser, S.

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]

Kampfrath, T.

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]

Kildishev, A. V.

Klein, M. W.

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]

Kley, E.-B.

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]

Konig, M.

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]

Koschny, T.

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]

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]

Kuipers, L.

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]

Langguth, L.

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]

Lederer, F.

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]

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]

Leinse, A.

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]

Linden, S.

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]

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]

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]

Liu, H.

N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nature Photon. 3, 157-162 (2009).
[CrossRef]

Liu, N.

N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nature Photon. 3, 157-162 (2009).
[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]

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]

Martin, O. J. F.

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

Menzel, C.

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]

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]

Mie, G.

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

Moloney, J. V.

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]

Niegemann, J.

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]

Oosten, D. V.

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]

Ozbay, E.

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]

Pertsch, T.

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]

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]

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]

Petschulat, J.

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]

Pfau, T.

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]

Piller, N. B.

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

Rockstuhl, C.

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]

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]

Savoini, M.

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]

Schoenmaker, H.

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]

Schweizer, H.

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]

Shalaev, V. M.

Shen, N.-H.

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]

Soukoulis, C. M.

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]

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]

T¨unnermann, A.

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]

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]

Tunnermann, A.

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]

Ulin-Avila, E.

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]

Valentine, J.

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]

Vynck, K.

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]

Wegener, M.

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]

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]

Weiss, T.

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]

Xiao, S.

Zavelani-Rossi, M.

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]

Zeng, Y.

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]

Zentgraf, T.

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]

Zhang, S.

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]

Zhang, X.

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]

Zhou, J.

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]

Zhu, S.

N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nature Photon. 3, 157-162 (2009).
[CrossRef]

Ann. Phys. (1)

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

Nature (1)

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. (2)

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. (2)

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. (1)

Phys. Rev. A (2)

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 (6)

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]

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]

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]

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]

Phys. Rev. E (1)

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

Phys. Rev. Lett. (1)

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 (2)

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 (8)

I. S. Gradstein and I. M. Ryshik, Tables of Series, Products and Integrals (Harry Deutsch, Frankfurt, 1981).

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]

A. Serdyukov, I. Semchenko, S. Tretyakov, A. Sihvola, Electromagnetics of bi-anisotropic materials: Theory and applications (Gordon and Breach, Amsterdam, 2001).

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).

. 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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Figures (4)

Fig. 1.
Fig. 1.

(Color online) The modulus of the magnetic field distribution (TM polarization) described by the eigenfunctions according to Eq. (5) for the first three expansion orders. We have plotted the eigenfunctions accounting for the cos() angular distributions only. The results for the sin() terms would result in a rotation by an angle of π/2m. By comparison with the field distributions of point multipoles, which are shown in the gray-scaled insets, it can already be anticipated that the eigenfunctions Ψ can be identified with multipole fields. Here m = 0 (a) corresponds to a magnetic dipole, m = 1 (b) is related to the electric dipole, while the electric quadrupole is associated with m = 2 (c).

Fig. 2.
Fig. 2.

(Color online) The illumination conditions, the orientation, and the definition of the geometrical parameters of the CW (a) and the SRR metaatom (b). The calculated Mie coefficients, related to the magnetic dipole (MD), the electric dipole (ED) and the electric quadrupole (EQ) for the CW (c) and the SRR (d) metaatom. Finally, the scattered magnetic fields for the magnetic (e) and the electric resonance (f) of the CW and the SRR (g,h) are shown, respectively. The gray-scaled insets show the exact multipole magnetic field distributions as in Fig. 1 to underline the similarities to the exact scattering field patterns.

Fig. 3.
Fig. 3.

(Color online) (a) The SRR geometry together with the three positions representing the selected origins for the multipole expansion. “A” is associated with the center of symmetry of the corresponding CW structure, while “B” and “C” represent additional positions out of the metaatoms center. The calculated Mie coefficients for the magnetic dipole (MD)(b),the electric dipole (ED) (c) and the electric quadrupole (EQ) (d). For completeness both electric dipole moments along the x (a + 1) and the y axis (a 1) are shown.

Fig. 4.
Fig. 4.

(Color online) The exact field patterns for the z component of the magnetic field for the magnetic dipole (a), the electric dipole (b), and the electric quadrupole (c) for carrier dynamics in the (x,y) plane. The two-dimensional field distributions below the three-dimensional ones show the three-dimensional fields for the respective multipole moment integrated along the z axis. The moduli of these two-dimensional fields correspond to the insets in Fig. 1(a–c) and precisely agree with ψm (r,ϕ), m∈{1,2,3}, respectively as shown in the derivation below.

Equations (55)

Equations on this page are rendered with MathJax. Learn more.

x , y 2 F z ( x , y ) + k ( ω ) 2 F z ( x , y ) = 0
1 R R ( R R F z ) + 1 R 2 ( R 2 ϕ 2 F z ) + k 2 F z = 0 .
F z = Σ m = Z m ( k R ) e im ϕ .
F z , s = Σ m = a m i m H m ( 1 ) ( k R ) e im ϕ ,
F z , s = Σ m = 0 [ a m + ψ m + ( R , ϕ , k ) + a m ψ m ( R , ϕ , k ) ] ,
ψ m + ( R , ϕ , k ) = i m H m ( 1 ) ( k R ) cos ( m ϕ ) ,
ψ m ( R , ϕ , k ) = i m + 1 H m ( 1 ) ( k R ) sin ( m ϕ ) ,
a m ± = ( a m ± a m ) ,
a m ± = 0 2 π d ϕ R 1 R 2 dRRF z ( R , ϕ ) ψ m ± ( R , ϕ , k ) 0 2 π d ϕ R 1 R 2 dRR ψ m ± ( R , ϕ , k ) 2 .
0 2 π d ϕ R 1 R 2 dRR ψ l ± ( R , ϕ , k ) ψ m ± ( R , ϕ , k ) 0 2 π d ϕ R 1 R 2 dRR ψ m ( R , ϕ , k ) 2
= R 1 R 2 dRRH l ( 1 ) ( R , k ) H m ( 1 ) ( R , k ) 0 2 π d ϕ R 1 R 2 dRR ψ m ( R , ϕ , k ) 2 0 2 π d ϕ [ sin ( l ϕ ) sin ( m ϕ ) + cos ( l ϕ ) cos ( m ϕ ) ] ,
= R 1 R 2 dRRH l ( 1 ) ( R , k ) H m ( 1 ) ( R , k ) R 1 R 2 dRR H m ( 1 ) ( kR ) 2 δ ml = δ ml .
A ( r ) = μ 0 4 π e ikr r d 3 r j ( r ) + μ 0 4 π e ikr r ( 1 r ik ) d 3 r j ( r ) ( n · r ) .
A ed ( r ) = i ω μ 0 4 π e ikr r p ( r ) ,
B ( r ) = × A ( r ) ,
B ed , z ( r ) = i ω μ 0 4 π e ikr r [ p y ( iky y r 2 ) p x ( ikx x r 2 ) ] .
B ed , z 2 D ( x , y ) = ω μ 0 k 4 [ cos ( ϕ ) p x sin ( ϕ ) p y ] H 1 ( 1 ) ( kR ) ,
= H 1 ( 1 ) ( kR ) [ a 1 + cos ( ϕ ) a 1 sin ( ϕ ) ] ,
a 1 ± ω μ 0 4 p x , y .
A ( r ) = μ 0 4 π e ikr r d 3 r j ( r ) + μ 0 4 π e ikr r ( 1 r ik ) d 3 r j ( r ) ( n · r ) .
A ed ( r ) = i ω μ 0 4 π e ikr r p ( r ) .
B ed , z ( r ) = x A y ( r ) y A x ( r ) ,
= i ω μ 0 4 π ( p y x p x y ) e ikr r .
B ed , z 2 D ( x , y ) = dzB ed , z ,
= i ω μ 0 4 π ( p y x p x y ) dz e ikr r .
dz e ik R 2 + z 2 R 2 + z 2 = i π H 0 ( 1 ) ( kR ) .
z H n ( 1 ) ( z ) = n z H n ( 1 ) ( z ) H n + 1 ( 1 ) ( z ) ,
X l H 0 ( 1 ) ( kR ) = H 1 ( 1 ) ( kR ) X l k R , X l { x , y } ,
B ed , z 2 D ( R , ϕ ) = ω μ 0 k 4 [ cos ( ϕ ) p x sin ( ϕ ) p y ] H 1 ( 1 ) ( k R ) .
F z , s = a 1 + ψ 1 + ( R , ϕ , k ) + a 1 ψ 1 ( R , ϕ , k ) ,
= [ a 1 + cos ( ϕ ) a 1 sin ( ϕ ) ] H 1 ( 1 ) ( kR ) .
a 1 + = ω μ 0 k 4 p x , a 1 = ω μ 0 k 4 p y .
A ( r ) = μ 0 4 π e ikr r ( 1 r ik ) d 3 r j ( r ) ( n · r )
= A md ( r ) + A eq ( r ) ,
A eq ( r ) = μ 0 4 π e ikr r ( 1 r ik ) d 3 r 1 2 [ ( n · r ) j ( r ) + ( n · j ( r ) ) r ] , Q
A md ( r ) = μ 0 4 π e ikr r ( 1 r ik ) [ 1 2 d 3 r r × j ( r ) ] m × n
= μ 0 4 π e ikr r ( 1 r ik ) m × n .
A md ( r ) = μ 0 4 π e ikr r 2 ( 1 r ik ) ( y e x + x e y ) m z .
X j e ikr r = X j e ikr r 2 ( 1 r ik ) , X j { x , y } ,
A md ( r ) = μ 0 m z 4 π ( e x y e y x ) e ikr r .
B md , z ( r ) = μ 0 m z 4 π ( 2 x 2 + 2 y 2 ) e ikr r .
B md , z 2 D ( x , y ) = i μ 0 μ ( ω ) m z 4 k 2 [ 2 kR H 1 ( 1 ) ( kR ) H 2 ( 1 ) ( kR ) ] ,
H 0 ( 1 , 2 ) ( z ) = 2 z H 1 ( 1 , 2 ) ( z ) H 2 ( 1 , 2 ) ( z ) ,
B md , z 2 D ( R , ϕ ) = i μ 0 m z k 2 4 H 0 ( 1 ) ( kR ) .
F x , s = a 0 H 0 ( 1 ) ( k R ) .
A eq ( r ) = i ω μ 0 8 π e ikr r ( 1 r ik ) d 3 r r ( n · r ) ρ ( r ) .
ρ ( r ) = Σ α = 1 N q α δ [ x x α ( t ) ] δ [ y y α ( t ) ] δ ( z ) .
d 3 r r ( n · r ) ρ ( r ) = Σ j = 1 3 e j Σ l = 1 3 d 3 r X j n l X l ρ ( r ) Q ̂ jl ( n ) e j .
Q ̂ ( n ) = Σ α = 1 N q α ( x α 2 n x x α y α n y 0 y α x α n x y α 2 n y 0 0 0 0 ) ( Q xx n x Q xy n y 0 Q xy n x Q yy n y 0 0 0 0 ) ,
A eq ( r ) = i ω μ 0 8 π [ e x ( Q xx x + Q xy y ) + e y ( Q yx x + Q yy y ) ] e ikr r .
B eq , z ( r ) = i ω μ 0 8 π [ Q yx ( 2 x 2 2 y 2 ) + Q yy 2 x y Q xx y x ] e ikr r .
B eq , z 2 D ( x , y ) = ω μ 0 8 [ Q xy ( 2 x 2 2 y 2 ) + ( Q yy Q xx ) 2 x y ] H 0 ( 1 ) ( kR ) ,
B eq , z 2 D ( R , ϕ ) = ω μ 0 k 2 8 { Q xy [ cos 2 ( ϕ ) sin 2 ( ϕ ) ] + ( Q yy Q xx ) cos ( ϕ ) sin ( ϕ ) } H 2 ( 1 ) ( kR ) .
B eq , z 2 D ( R , ϕ ) = ω μ 0 k 2 16 [ Q xy cos ( 2 ϕ ) + ( Q yy Q xx ) sin ( 2 ϕ ) ] H 2 ( 1 ) ( kR ) .
F z , s = [ a 2 + cos ( 2 ϕ ) + ia 2 sin ( 2 ϕ ) ] H 2 ( 1 ) ( kR ) .

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