Abstract

The current and field distribution in a 2D metamaterial consisting of resonant elements in a hexagonal arrangement are found assuming magnetic interaction between the elements. The dispersion equation of magnetoinductive (MI) waves is derived with the aid of the direct and reciprocal lattice familiar from solid state theory. A continuous model for the current variation in the elements is introduced leading to the familiar wave equation in the form of a second order differential equation. The current distributions are shown to exhibit a series of spatial resonances for rectangular, circular and hexagonal boundaries. The axial and radial components of the resulting magnetic field are compared with previously obtained experimental results on a Swiss Roll metamaterial with hexagonal boundaries. Experimental and theoretical results are also compared for the near field image of an object in the shape of the letter M followed by a more general discussion of imaging. It is concluded that a theoretical formulation based on the propagation of MI waves can correctly describe the experimental results.

©2005 Optical Society of America

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  1. R. E. Collin , Field Theory of Guided Waves ( Oxford University Press, Oxford , 1991 ).
  2. E. Yablonovitch , “ Inhibited spontaneous emission in solid-state physics and electronics ,” Phys. Rev. Lett.   58 , 2059 – 2062 ( 1987 ).
    [Crossref] [PubMed]
  3. S. John , “ Strong localization of photons in certain disordered dielectric superlattices ,” Phys. Rev. Lett.   58 , 2486 – 2489 ( 1987 ).
    [Crossref] [PubMed]
  4. J. B. Pendry , A. J. Holden , D. J. Robbins , and W. J. Stewart , “ Magnetism from conductors and enhanced nonlinear phenomena ,” IEEE Trans. Microwave Theory Tech.   47 , 2075 – 2084 ( 1999 ).
    [Crossref]
  5. J. B. Pendry , “ Negative refraction makes a perfect lens ,” Phys. Rev. Lett.   85 , 3966 – 3969 ( 2000 ).
    [Crossref] [PubMed]
  6. D. O. S. Melville and R. J. Blaikie , “ Super-resolution imaging through a planar silver layer ,” Opt. Express   13 , 2127 – 2134 ( 2005 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-6-2127 .
    [Crossref] [PubMed]
  7. N. Fang , H. Lee , C. Sun , and X. Zhang , “ Sub-diffraction-limited optical imaging with a silver superlens ,” Science   308 , 534 – 537 ( 2005 ).
    [Crossref] [PubMed]
  8. M. C. K. Wiltshire , J. B. Pendry , I. R. Young , D. J. Larkman , D. J. Gilderdale , and J. V. Hajnal , “ Microstructured magnetic materials for RF flux guides in Magnetic Resonance Imaging ,” Science   291 , 849 – 851 ( 2001 ).
    [Crossref] [PubMed]
  9. M. C. K. Wiltshire , J. V. Hajnal , J. B. Pendry , D. J. Edwards , and C. J. Stevens , “ Metamaterial endoscope for magnetic field transfer: near field imaging with magnetic wires ,” Opt. Express   11 , 709 – 715 ( 2003 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-709 .
    [Crossref] [PubMed]
  10. M. J. Freire and R. Marques , “ Planar magnetoinductive lens for three-dimensional subwavelength imaging ,” Appl. Phys. Lett.   86 , 182505 -1–3 ( 2005 ).
    [Crossref]
  11. E. Shamonina , V. A. Kalinin , K. H. Ringhofer , and L. Solymar , “ Magneto-inductive waveguide ,” Electron. Lett.   38 , 371 – 373 ( 2002 ).
    [Crossref]
  12. E. Shamonina , V.A. Kalinin , K. H. Ringhofer , and L. Solymar , “ Magnetoinductive waves in one, two, and three dimensions ,” J. Appl. Phys.   92 : 6252 – 6261 ( 2002 ).
    [Crossref]
  13. M. C. K. Wiltshire , E. Shamonina , I. R. Young , and L. Solymar , “ Dispersion characteristics of magneto-inductive waves: comparison between theory and experiment ,” Electron. Lett.   39 , 215 – 217 ( 2003 ).
    [Crossref]
  14. M. C. K. Wiltshire , E. Shamonina , I. R. Young , and L. Solymar , “ Experimental and theoretical study of magneto-inductive waves supported by one-dimensional arrays of “swiss rolls” ,” J. Appl. Phys.   95 , 4488 – 4493 ( 2004 ).
    [Crossref]
  15. R. Marques , F. Mesa , J. Martel , and F. Medina , “ Comparative analysis of edge- and broadside-coupled split ring resonators for metamaterial design - Theory and experiments ,” IEEE Trans. Antennas Prop.   51 , 2572 – 2581 ( 2003 ).
    [Crossref]
  16. J. D. Baena , R. Marques , F. Medina , and J. Martel , “ Artificial magnetic metamaterial design by using spiral resonators ,” Phys. Rev. B   69 , 014402 -1–5 ( 2004 ).
    [Crossref]
  17. M. Shamonin , E. Shamonina , V. Kalinin , and L. Solymar , “ Properties of a metamaterial element: analytical solutions and numerical simulations for a singly split double ring ,” J. Appl. Phys.   95 , 3778 – 3784 ( 2004 ).
    [Crossref]
  18. F. W. Grover , Inductance Calculations: Working Formulas and Tables ( Instrument Society of America, Research Triangle Park, N.C. , 1981 ).
  19. M. C. K. Wiltshire , J. V. Hajnal , J. B. Pendry , and D. J. Edwards , “ RF field transmission through Swiss Rolls - an anisotropic magnetic metamaterial ,” in Proceedings of the 27th ESA Antenna Technology Workshop on Innovative Periodic Antennas: Electromagnetic Bandgap, Left-handed Materials, Fractal and Frequency Selective Surfaces , Santiago de Compostela, Spain , 9 – 11 March, 2004 .
  20. D. R. Smith , D. Schurig , J. J. Mock , P. Kolinko , and P. Rye , “ Partial focusing of radiation by a slab of indefinite media ,” Appl. Phys. Lett.   84 , 2244 – 2246 ( 2004 ).
    [Crossref]
  21. S. Maslovski , S. Tretyakov , and P. Alitalo , “ Near-field enhancement and imaging in double planar polariton-resonant structures ,” J. Appl. Phys.   96 , 1293 – 1300 ( 2004 ).
    [Crossref]
  22. I.-H. Lin , C. Caloz , and T. Itoh , “ Near-field focusing by a nonuniform leaky-wave interface ,” Microw. Opt. Techn. Lett.   44 , 416 – 418 ( 2005 ).
    [Crossref]
  23. R. Zengerle , “ Light-propagation in singly and doubly periodic planar wave-guides ,” J. Mod. Opt.   34 , 1589 – 1617 ( 1987 ).
    [Crossref]
  24. P. A. Belov , C. R. Simovski , and P. Ikonen , “ Canalization of subwavelength images by electromagnetic crystals ,” Phys. Rev.   71 , 193105 -1–4 ( 2005 ).
    [Crossref]
  25. T. J. Yen , W. J. Padilla , N. Fang , D. C. Vier , D. R. Smith , J. B. Pendry , D. N. Basov , and X. Zhang , “ Terahertz magnetic response from artificial materials ,” Science   303 , 1494 – 1496 ( 2004 ).
    [Crossref] [PubMed]
  26. S. Linden , C. Enkrich , M. Wegener , J. F. Zhou , T. Koschny , and C. M. Soukoulis , “ Magnetic response of metamaterials at 100 terahertz ,” Science   306 , 1351 – 1353 ( 2004 ).
    [Crossref] [PubMed]

2005 (5)

M. J. Freire and R. Marques , “ Planar magnetoinductive lens for three-dimensional subwavelength imaging ,” Appl. Phys. Lett.   86 , 182505 -1–3 ( 2005 ).
[Crossref]

P. A. Belov , C. R. Simovski , and P. Ikonen , “ Canalization of subwavelength images by electromagnetic crystals ,” Phys. Rev.   71 , 193105 -1–4 ( 2005 ).
[Crossref]

I.-H. Lin , C. Caloz , and T. Itoh , “ Near-field focusing by a nonuniform leaky-wave interface ,” Microw. Opt. Techn. Lett.   44 , 416 – 418 ( 2005 ).
[Crossref]

N. Fang , H. Lee , C. Sun , and X. Zhang , “ Sub-diffraction-limited optical imaging with a silver superlens ,” Science   308 , 534 – 537 ( 2005 ).
[Crossref] [PubMed]

D. O. S. Melville and R. J. Blaikie , “ Super-resolution imaging through a planar silver layer ,” Opt. Express   13 , 2127 – 2134 ( 2005 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-6-2127 .
[Crossref] [PubMed]

2004 (7)

T. J. Yen , W. J. Padilla , N. Fang , D. C. Vier , D. R. Smith , J. B. Pendry , D. N. Basov , and X. Zhang , “ Terahertz magnetic response from artificial materials ,” Science   303 , 1494 – 1496 ( 2004 ).
[Crossref] [PubMed]

S. Linden , C. Enkrich , M. Wegener , J. F. Zhou , T. Koschny , and C. M. Soukoulis , “ Magnetic response of metamaterials at 100 terahertz ,” Science   306 , 1351 – 1353 ( 2004 ).
[Crossref] [PubMed]

M. C. K. Wiltshire , E. Shamonina , I. R. Young , and L. Solymar , “ Experimental and theoretical study of magneto-inductive waves supported by one-dimensional arrays of “swiss rolls” ,” J. Appl. Phys.   95 , 4488 – 4493 ( 2004 ).
[Crossref]

J. D. Baena , R. Marques , F. Medina , and J. Martel , “ Artificial magnetic metamaterial design by using spiral resonators ,” Phys. Rev. B   69 , 014402 -1–5 ( 2004 ).
[Crossref]

M. Shamonin , E. Shamonina , V. Kalinin , and L. Solymar , “ Properties of a metamaterial element: analytical solutions and numerical simulations for a singly split double ring ,” J. Appl. Phys.   95 , 3778 – 3784 ( 2004 ).
[Crossref]

D. R. Smith , D. Schurig , J. J. Mock , P. Kolinko , and P. Rye , “ Partial focusing of radiation by a slab of indefinite media ,” Appl. Phys. Lett.   84 , 2244 – 2246 ( 2004 ).
[Crossref]

S. Maslovski , S. Tretyakov , and P. Alitalo , “ Near-field enhancement and imaging in double planar polariton-resonant structures ,” J. Appl. Phys.   96 , 1293 – 1300 ( 2004 ).
[Crossref]

2003 (3)

R. Marques , F. Mesa , J. Martel , and F. Medina , “ Comparative analysis of edge- and broadside-coupled split ring resonators for metamaterial design - Theory and experiments ,” IEEE Trans. Antennas Prop.   51 , 2572 – 2581 ( 2003 ).
[Crossref]

M. C. K. Wiltshire , E. Shamonina , I. R. Young , and L. Solymar , “ Dispersion characteristics of magneto-inductive waves: comparison between theory and experiment ,” Electron. Lett.   39 , 215 – 217 ( 2003 ).
[Crossref]

M. C. K. Wiltshire , J. V. Hajnal , J. B. Pendry , D. J. Edwards , and C. J. Stevens , “ Metamaterial endoscope for magnetic field transfer: near field imaging with magnetic wires ,” Opt. Express   11 , 709 – 715 ( 2003 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-709 .
[Crossref] [PubMed]

2002 (2)

E. Shamonina , V. A. Kalinin , K. H. Ringhofer , and L. Solymar , “ Magneto-inductive waveguide ,” Electron. Lett.   38 , 371 – 373 ( 2002 ).
[Crossref]

E. Shamonina , V.A. Kalinin , K. H. Ringhofer , and L. Solymar , “ Magnetoinductive waves in one, two, and three dimensions ,” J. Appl. Phys.   92 : 6252 – 6261 ( 2002 ).
[Crossref]

2001 (1)

M. C. K. Wiltshire , J. B. Pendry , I. R. Young , D. J. Larkman , D. J. Gilderdale , and J. V. Hajnal , “ Microstructured magnetic materials for RF flux guides in Magnetic Resonance Imaging ,” Science   291 , 849 – 851 ( 2001 ).
[Crossref] [PubMed]

2000 (1)

J. B. Pendry , “ Negative refraction makes a perfect lens ,” Phys. Rev. Lett.   85 , 3966 – 3969 ( 2000 ).
[Crossref] [PubMed]

1999 (1)

J. B. Pendry , A. J. Holden , D. J. Robbins , and W. J. Stewart , “ Magnetism from conductors and enhanced nonlinear phenomena ,” IEEE Trans. Microwave Theory Tech.   47 , 2075 – 2084 ( 1999 ).
[Crossref]

1987 (3)

R. Zengerle , “ Light-propagation in singly and doubly periodic planar wave-guides ,” J. Mod. Opt.   34 , 1589 – 1617 ( 1987 ).
[Crossref]

E. Yablonovitch , “ Inhibited spontaneous emission in solid-state physics and electronics ,” Phys. Rev. Lett.   58 , 2059 – 2062 ( 1987 ).
[Crossref] [PubMed]

S. John , “ Strong localization of photons in certain disordered dielectric superlattices ,” Phys. Rev. Lett.   58 , 2486 – 2489 ( 1987 ).
[Crossref] [PubMed]

Alitalo, P.

S. Maslovski , S. Tretyakov , and P. Alitalo , “ Near-field enhancement and imaging in double planar polariton-resonant structures ,” J. Appl. Phys.   96 , 1293 – 1300 ( 2004 ).
[Crossref]

Baena, J. D.

J. D. Baena , R. Marques , F. Medina , and J. Martel , “ Artificial magnetic metamaterial design by using spiral resonators ,” Phys. Rev. B   69 , 014402 -1–5 ( 2004 ).
[Crossref]

Basov, D. N.

T. J. Yen , W. J. Padilla , N. Fang , D. C. Vier , D. R. Smith , J. B. Pendry , D. N. Basov , and X. Zhang , “ Terahertz magnetic response from artificial materials ,” Science   303 , 1494 – 1496 ( 2004 ).
[Crossref] [PubMed]

Belov, P. A.

P. A. Belov , C. R. Simovski , and P. Ikonen , “ Canalization of subwavelength images by electromagnetic crystals ,” Phys. Rev.   71 , 193105 -1–4 ( 2005 ).
[Crossref]

Blaikie, R. J.

Caloz, C.

I.-H. Lin , C. Caloz , and T. Itoh , “ Near-field focusing by a nonuniform leaky-wave interface ,” Microw. Opt. Techn. Lett.   44 , 416 – 418 ( 2005 ).
[Crossref]

Collin, R. E.

R. E. Collin , Field Theory of Guided Waves ( Oxford University Press, Oxford , 1991 ).

Edwards, D. J.

M. C. K. Wiltshire , J. V. Hajnal , J. B. Pendry , D. J. Edwards , and C. J. Stevens , “ Metamaterial endoscope for magnetic field transfer: near field imaging with magnetic wires ,” Opt. Express   11 , 709 – 715 ( 2003 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-709 .
[Crossref] [PubMed]

M. C. K. Wiltshire , J. V. Hajnal , J. B. Pendry , and D. J. Edwards , “ RF field transmission through Swiss Rolls - an anisotropic magnetic metamaterial ,” in Proceedings of the 27th ESA Antenna Technology Workshop on Innovative Periodic Antennas: Electromagnetic Bandgap, Left-handed Materials, Fractal and Frequency Selective Surfaces , Santiago de Compostela, Spain , 9 – 11 March, 2004 .

Enkrich, C.

S. Linden , C. Enkrich , M. Wegener , J. F. Zhou , T. Koschny , and C. M. Soukoulis , “ Magnetic response of metamaterials at 100 terahertz ,” Science   306 , 1351 – 1353 ( 2004 ).
[Crossref] [PubMed]

Fang, N.

N. Fang , H. Lee , C. Sun , and X. Zhang , “ Sub-diffraction-limited optical imaging with a silver superlens ,” Science   308 , 534 – 537 ( 2005 ).
[Crossref] [PubMed]

T. J. Yen , W. J. Padilla , N. Fang , D. C. Vier , D. R. Smith , J. B. Pendry , D. N. Basov , and X. Zhang , “ Terahertz magnetic response from artificial materials ,” Science   303 , 1494 – 1496 ( 2004 ).
[Crossref] [PubMed]

Freire, M. J.

M. J. Freire and R. Marques , “ Planar magnetoinductive lens for three-dimensional subwavelength imaging ,” Appl. Phys. Lett.   86 , 182505 -1–3 ( 2005 ).
[Crossref]

Gilderdale, D. J.

M. C. K. Wiltshire , J. B. Pendry , I. R. Young , D. J. Larkman , D. J. Gilderdale , and J. V. Hajnal , “ Microstructured magnetic materials for RF flux guides in Magnetic Resonance Imaging ,” Science   291 , 849 – 851 ( 2001 ).
[Crossref] [PubMed]

Grover, F. W.

F. W. Grover , Inductance Calculations: Working Formulas and Tables ( Instrument Society of America, Research Triangle Park, N.C. , 1981 ).

Hajnal, J. V.

M. C. K. Wiltshire , J. V. Hajnal , J. B. Pendry , D. J. Edwards , and C. J. Stevens , “ Metamaterial endoscope for magnetic field transfer: near field imaging with magnetic wires ,” Opt. Express   11 , 709 – 715 ( 2003 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-709 .
[Crossref] [PubMed]

M. C. K. Wiltshire , J. B. Pendry , I. R. Young , D. J. Larkman , D. J. Gilderdale , and J. V. Hajnal , “ Microstructured magnetic materials for RF flux guides in Magnetic Resonance Imaging ,” Science   291 , 849 – 851 ( 2001 ).
[Crossref] [PubMed]

M. C. K. Wiltshire , J. V. Hajnal , J. B. Pendry , and D. J. Edwards , “ RF field transmission through Swiss Rolls - an anisotropic magnetic metamaterial ,” in Proceedings of the 27th ESA Antenna Technology Workshop on Innovative Periodic Antennas: Electromagnetic Bandgap, Left-handed Materials, Fractal and Frequency Selective Surfaces , Santiago de Compostela, Spain , 9 – 11 March, 2004 .

Holden, A. J.

J. B. Pendry , A. J. Holden , D. J. Robbins , and W. J. Stewart , “ Magnetism from conductors and enhanced nonlinear phenomena ,” IEEE Trans. Microwave Theory Tech.   47 , 2075 – 2084 ( 1999 ).
[Crossref]

Ikonen, P.

P. A. Belov , C. R. Simovski , and P. Ikonen , “ Canalization of subwavelength images by electromagnetic crystals ,” Phys. Rev.   71 , 193105 -1–4 ( 2005 ).
[Crossref]

Itoh, T.

I.-H. Lin , C. Caloz , and T. Itoh , “ Near-field focusing by a nonuniform leaky-wave interface ,” Microw. Opt. Techn. Lett.   44 , 416 – 418 ( 2005 ).
[Crossref]

John, S.

S. John , “ Strong localization of photons in certain disordered dielectric superlattices ,” Phys. Rev. Lett.   58 , 2486 – 2489 ( 1987 ).
[Crossref] [PubMed]

Kalinin, V.

M. Shamonin , E. Shamonina , V. Kalinin , and L. Solymar , “ Properties of a metamaterial element: analytical solutions and numerical simulations for a singly split double ring ,” J. Appl. Phys.   95 , 3778 – 3784 ( 2004 ).
[Crossref]

Kalinin, V. A.

E. Shamonina , V. A. Kalinin , K. H. Ringhofer , and L. Solymar , “ Magneto-inductive waveguide ,” Electron. Lett.   38 , 371 – 373 ( 2002 ).
[Crossref]

Kalinin, V.A.

E. Shamonina , V.A. Kalinin , K. H. Ringhofer , and L. Solymar , “ Magnetoinductive waves in one, two, and three dimensions ,” J. Appl. Phys.   92 : 6252 – 6261 ( 2002 ).
[Crossref]

Kolinko, P.

D. R. Smith , D. Schurig , J. J. Mock , P. Kolinko , and P. Rye , “ Partial focusing of radiation by a slab of indefinite media ,” Appl. Phys. Lett.   84 , 2244 – 2246 ( 2004 ).
[Crossref]

Koschny, T.

S. Linden , C. Enkrich , M. Wegener , J. F. Zhou , T. Koschny , and C. M. Soukoulis , “ Magnetic response of metamaterials at 100 terahertz ,” Science   306 , 1351 – 1353 ( 2004 ).
[Crossref] [PubMed]

Larkman, D. J.

M. C. K. Wiltshire , J. B. Pendry , I. R. Young , D. J. Larkman , D. J. Gilderdale , and J. V. Hajnal , “ Microstructured magnetic materials for RF flux guides in Magnetic Resonance Imaging ,” Science   291 , 849 – 851 ( 2001 ).
[Crossref] [PubMed]

Lee, H.

N. Fang , H. Lee , C. Sun , and X. Zhang , “ Sub-diffraction-limited optical imaging with a silver superlens ,” Science   308 , 534 – 537 ( 2005 ).
[Crossref] [PubMed]

Lin, I.-H.

I.-H. Lin , C. Caloz , and T. Itoh , “ Near-field focusing by a nonuniform leaky-wave interface ,” Microw. Opt. Techn. Lett.   44 , 416 – 418 ( 2005 ).
[Crossref]

Linden, S.

S. Linden , C. Enkrich , M. Wegener , J. F. Zhou , T. Koschny , and C. M. Soukoulis , “ Magnetic response of metamaterials at 100 terahertz ,” Science   306 , 1351 – 1353 ( 2004 ).
[Crossref] [PubMed]

Marques, R.

M. J. Freire and R. Marques , “ Planar magnetoinductive lens for three-dimensional subwavelength imaging ,” Appl. Phys. Lett.   86 , 182505 -1–3 ( 2005 ).
[Crossref]

J. D. Baena , R. Marques , F. Medina , and J. Martel , “ Artificial magnetic metamaterial design by using spiral resonators ,” Phys. Rev. B   69 , 014402 -1–5 ( 2004 ).
[Crossref]

R. Marques , F. Mesa , J. Martel , and F. Medina , “ Comparative analysis of edge- and broadside-coupled split ring resonators for metamaterial design - Theory and experiments ,” IEEE Trans. Antennas Prop.   51 , 2572 – 2581 ( 2003 ).
[Crossref]

Martel, J.

J. D. Baena , R. Marques , F. Medina , and J. Martel , “ Artificial magnetic metamaterial design by using spiral resonators ,” Phys. Rev. B   69 , 014402 -1–5 ( 2004 ).
[Crossref]

R. Marques , F. Mesa , J. Martel , and F. Medina , “ Comparative analysis of edge- and broadside-coupled split ring resonators for metamaterial design - Theory and experiments ,” IEEE Trans. Antennas Prop.   51 , 2572 – 2581 ( 2003 ).
[Crossref]

Maslovski, S.

S. Maslovski , S. Tretyakov , and P. Alitalo , “ Near-field enhancement and imaging in double planar polariton-resonant structures ,” J. Appl. Phys.   96 , 1293 – 1300 ( 2004 ).
[Crossref]

Medina, F.

J. D. Baena , R. Marques , F. Medina , and J. Martel , “ Artificial magnetic metamaterial design by using spiral resonators ,” Phys. Rev. B   69 , 014402 -1–5 ( 2004 ).
[Crossref]

R. Marques , F. Mesa , J. Martel , and F. Medina , “ Comparative analysis of edge- and broadside-coupled split ring resonators for metamaterial design - Theory and experiments ,” IEEE Trans. Antennas Prop.   51 , 2572 – 2581 ( 2003 ).
[Crossref]

Melville, D. O. S.

Mesa, F.

R. Marques , F. Mesa , J. Martel , and F. Medina , “ Comparative analysis of edge- and broadside-coupled split ring resonators for metamaterial design - Theory and experiments ,” IEEE Trans. Antennas Prop.   51 , 2572 – 2581 ( 2003 ).
[Crossref]

Mock, J. J.

D. R. Smith , D. Schurig , J. J. Mock , P. Kolinko , and P. Rye , “ Partial focusing of radiation by a slab of indefinite media ,” Appl. Phys. Lett.   84 , 2244 – 2246 ( 2004 ).
[Crossref]

Padilla, W. J.

T. J. Yen , W. J. Padilla , N. Fang , D. C. Vier , D. R. Smith , J. B. Pendry , D. N. Basov , and X. Zhang , “ Terahertz magnetic response from artificial materials ,” Science   303 , 1494 – 1496 ( 2004 ).
[Crossref] [PubMed]

Pendry, J. B.

T. J. Yen , W. J. Padilla , N. Fang , D. C. Vier , D. R. Smith , J. B. Pendry , D. N. Basov , and X. Zhang , “ Terahertz magnetic response from artificial materials ,” Science   303 , 1494 – 1496 ( 2004 ).
[Crossref] [PubMed]

M. C. K. Wiltshire , J. V. Hajnal , J. B. Pendry , D. J. Edwards , and C. J. Stevens , “ Metamaterial endoscope for magnetic field transfer: near field imaging with magnetic wires ,” Opt. Express   11 , 709 – 715 ( 2003 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-709 .
[Crossref] [PubMed]

M. C. K. Wiltshire , J. B. Pendry , I. R. Young , D. J. Larkman , D. J. Gilderdale , and J. V. Hajnal , “ Microstructured magnetic materials for RF flux guides in Magnetic Resonance Imaging ,” Science   291 , 849 – 851 ( 2001 ).
[Crossref] [PubMed]

J. B. Pendry , “ Negative refraction makes a perfect lens ,” Phys. Rev. Lett.   85 , 3966 – 3969 ( 2000 ).
[Crossref] [PubMed]

J. B. Pendry , A. J. Holden , D. J. Robbins , and W. J. Stewart , “ Magnetism from conductors and enhanced nonlinear phenomena ,” IEEE Trans. Microwave Theory Tech.   47 , 2075 – 2084 ( 1999 ).
[Crossref]

M. C. K. Wiltshire , J. V. Hajnal , J. B. Pendry , and D. J. Edwards , “ RF field transmission through Swiss Rolls - an anisotropic magnetic metamaterial ,” in Proceedings of the 27th ESA Antenna Technology Workshop on Innovative Periodic Antennas: Electromagnetic Bandgap, Left-handed Materials, Fractal and Frequency Selective Surfaces , Santiago de Compostela, Spain , 9 – 11 March, 2004 .

Ringhofer, K. H.

E. Shamonina , V. A. Kalinin , K. H. Ringhofer , and L. Solymar , “ Magneto-inductive waveguide ,” Electron. Lett.   38 , 371 – 373 ( 2002 ).
[Crossref]

E. Shamonina , V.A. Kalinin , K. H. Ringhofer , and L. Solymar , “ Magnetoinductive waves in one, two, and three dimensions ,” J. Appl. Phys.   92 : 6252 – 6261 ( 2002 ).
[Crossref]

Robbins, D. J.

J. B. Pendry , A. J. Holden , D. J. Robbins , and W. J. Stewart , “ Magnetism from conductors and enhanced nonlinear phenomena ,” IEEE Trans. Microwave Theory Tech.   47 , 2075 – 2084 ( 1999 ).
[Crossref]

Rye, P.

D. R. Smith , D. Schurig , J. J. Mock , P. Kolinko , and P. Rye , “ Partial focusing of radiation by a slab of indefinite media ,” Appl. Phys. Lett.   84 , 2244 – 2246 ( 2004 ).
[Crossref]

Schurig, D.

D. R. Smith , D. Schurig , J. J. Mock , P. Kolinko , and P. Rye , “ Partial focusing of radiation by a slab of indefinite media ,” Appl. Phys. Lett.   84 , 2244 – 2246 ( 2004 ).
[Crossref]

Shamonin, M.

M. Shamonin , E. Shamonina , V. Kalinin , and L. Solymar , “ Properties of a metamaterial element: analytical solutions and numerical simulations for a singly split double ring ,” J. Appl. Phys.   95 , 3778 – 3784 ( 2004 ).
[Crossref]

Shamonina, E.

M. Shamonin , E. Shamonina , V. Kalinin , and L. Solymar , “ Properties of a metamaterial element: analytical solutions and numerical simulations for a singly split double ring ,” J. Appl. Phys.   95 , 3778 – 3784 ( 2004 ).
[Crossref]

M. C. K. Wiltshire , E. Shamonina , I. R. Young , and L. Solymar , “ Experimental and theoretical study of magneto-inductive waves supported by one-dimensional arrays of “swiss rolls” ,” J. Appl. Phys.   95 , 4488 – 4493 ( 2004 ).
[Crossref]

M. C. K. Wiltshire , E. Shamonina , I. R. Young , and L. Solymar , “ Dispersion characteristics of magneto-inductive waves: comparison between theory and experiment ,” Electron. Lett.   39 , 215 – 217 ( 2003 ).
[Crossref]

E. Shamonina , V.A. Kalinin , K. H. Ringhofer , and L. Solymar , “ Magnetoinductive waves in one, two, and three dimensions ,” J. Appl. Phys.   92 : 6252 – 6261 ( 2002 ).
[Crossref]

E. Shamonina , V. A. Kalinin , K. H. Ringhofer , and L. Solymar , “ Magneto-inductive waveguide ,” Electron. Lett.   38 , 371 – 373 ( 2002 ).
[Crossref]

Simovski, C. R.

P. A. Belov , C. R. Simovski , and P. Ikonen , “ Canalization of subwavelength images by electromagnetic crystals ,” Phys. Rev.   71 , 193105 -1–4 ( 2005 ).
[Crossref]

Smith, D. R.

T. J. Yen , W. J. Padilla , N. Fang , D. C. Vier , D. R. Smith , J. B. Pendry , D. N. Basov , and X. Zhang , “ Terahertz magnetic response from artificial materials ,” Science   303 , 1494 – 1496 ( 2004 ).
[Crossref] [PubMed]

D. R. Smith , D. Schurig , J. J. Mock , P. Kolinko , and P. Rye , “ Partial focusing of radiation by a slab of indefinite media ,” Appl. Phys. Lett.   84 , 2244 – 2246 ( 2004 ).
[Crossref]

Solymar, L.

M. Shamonin , E. Shamonina , V. Kalinin , and L. Solymar , “ Properties of a metamaterial element: analytical solutions and numerical simulations for a singly split double ring ,” J. Appl. Phys.   95 , 3778 – 3784 ( 2004 ).
[Crossref]

M. C. K. Wiltshire , E. Shamonina , I. R. Young , and L. Solymar , “ Experimental and theoretical study of magneto-inductive waves supported by one-dimensional arrays of “swiss rolls” ,” J. Appl. Phys.   95 , 4488 – 4493 ( 2004 ).
[Crossref]

M. C. K. Wiltshire , E. Shamonina , I. R. Young , and L. Solymar , “ Dispersion characteristics of magneto-inductive waves: comparison between theory and experiment ,” Electron. Lett.   39 , 215 – 217 ( 2003 ).
[Crossref]

E. Shamonina , V. A. Kalinin , K. H. Ringhofer , and L. Solymar , “ Magneto-inductive waveguide ,” Electron. Lett.   38 , 371 – 373 ( 2002 ).
[Crossref]

E. Shamonina , V.A. Kalinin , K. H. Ringhofer , and L. Solymar , “ Magnetoinductive waves in one, two, and three dimensions ,” J. Appl. Phys.   92 : 6252 – 6261 ( 2002 ).
[Crossref]

Soukoulis, C. M.

S. Linden , C. Enkrich , M. Wegener , J. F. Zhou , T. Koschny , and C. M. Soukoulis , “ Magnetic response of metamaterials at 100 terahertz ,” Science   306 , 1351 – 1353 ( 2004 ).
[Crossref] [PubMed]

Stevens, C. J.

Stewart, W. J.

J. B. Pendry , A. J. Holden , D. J. Robbins , and W. J. Stewart , “ Magnetism from conductors and enhanced nonlinear phenomena ,” IEEE Trans. Microwave Theory Tech.   47 , 2075 – 2084 ( 1999 ).
[Crossref]

Sun, C.

N. Fang , H. Lee , C. Sun , and X. Zhang , “ Sub-diffraction-limited optical imaging with a silver superlens ,” Science   308 , 534 – 537 ( 2005 ).
[Crossref] [PubMed]

Tretyakov, S.

S. Maslovski , S. Tretyakov , and P. Alitalo , “ Near-field enhancement and imaging in double planar polariton-resonant structures ,” J. Appl. Phys.   96 , 1293 – 1300 ( 2004 ).
[Crossref]

Vier, D. C.

T. J. Yen , W. J. Padilla , N. Fang , D. C. Vier , D. R. Smith , J. B. Pendry , D. N. Basov , and X. Zhang , “ Terahertz magnetic response from artificial materials ,” Science   303 , 1494 – 1496 ( 2004 ).
[Crossref] [PubMed]

Wegener, M.

S. Linden , C. Enkrich , M. Wegener , J. F. Zhou , T. Koschny , and C. M. Soukoulis , “ Magnetic response of metamaterials at 100 terahertz ,” Science   306 , 1351 – 1353 ( 2004 ).
[Crossref] [PubMed]

Wiltshire, M. C. K.

M. C. K. Wiltshire , E. Shamonina , I. R. Young , and L. Solymar , “ Experimental and theoretical study of magneto-inductive waves supported by one-dimensional arrays of “swiss rolls” ,” J. Appl. Phys.   95 , 4488 – 4493 ( 2004 ).
[Crossref]

M. C. K. Wiltshire , E. Shamonina , I. R. Young , and L. Solymar , “ Dispersion characteristics of magneto-inductive waves: comparison between theory and experiment ,” Electron. Lett.   39 , 215 – 217 ( 2003 ).
[Crossref]

M. C. K. Wiltshire , J. V. Hajnal , J. B. Pendry , D. J. Edwards , and C. J. Stevens , “ Metamaterial endoscope for magnetic field transfer: near field imaging with magnetic wires ,” Opt. Express   11 , 709 – 715 ( 2003 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-709 .
[Crossref] [PubMed]

M. C. K. Wiltshire , J. B. Pendry , I. R. Young , D. J. Larkman , D. J. Gilderdale , and J. V. Hajnal , “ Microstructured magnetic materials for RF flux guides in Magnetic Resonance Imaging ,” Science   291 , 849 – 851 ( 2001 ).
[Crossref] [PubMed]

M. C. K. Wiltshire , J. V. Hajnal , J. B. Pendry , and D. J. Edwards , “ RF field transmission through Swiss Rolls - an anisotropic magnetic metamaterial ,” in Proceedings of the 27th ESA Antenna Technology Workshop on Innovative Periodic Antennas: Electromagnetic Bandgap, Left-handed Materials, Fractal and Frequency Selective Surfaces , Santiago de Compostela, Spain , 9 – 11 March, 2004 .

Yablonovitch, E.

E. Yablonovitch , “ Inhibited spontaneous emission in solid-state physics and electronics ,” Phys. Rev. Lett.   58 , 2059 – 2062 ( 1987 ).
[Crossref] [PubMed]

Yen, T. J.

T. J. Yen , W. J. Padilla , N. Fang , D. C. Vier , D. R. Smith , J. B. Pendry , D. N. Basov , and X. Zhang , “ Terahertz magnetic response from artificial materials ,” Science   303 , 1494 – 1496 ( 2004 ).
[Crossref] [PubMed]

Young, I. R.

M. C. K. Wiltshire , E. Shamonina , I. R. Young , and L. Solymar , “ Experimental and theoretical study of magneto-inductive waves supported by one-dimensional arrays of “swiss rolls” ,” J. Appl. Phys.   95 , 4488 – 4493 ( 2004 ).
[Crossref]

M. C. K. Wiltshire , E. Shamonina , I. R. Young , and L. Solymar , “ Dispersion characteristics of magneto-inductive waves: comparison between theory and experiment ,” Electron. Lett.   39 , 215 – 217 ( 2003 ).
[Crossref]

M. C. K. Wiltshire , J. B. Pendry , I. R. Young , D. J. Larkman , D. J. Gilderdale , and J. V. Hajnal , “ Microstructured magnetic materials for RF flux guides in Magnetic Resonance Imaging ,” Science   291 , 849 – 851 ( 2001 ).
[Crossref] [PubMed]

Zengerle, R.

R. Zengerle , “ Light-propagation in singly and doubly periodic planar wave-guides ,” J. Mod. Opt.   34 , 1589 – 1617 ( 1987 ).
[Crossref]

Zhang, X.

N. Fang , H. Lee , C. Sun , and X. Zhang , “ Sub-diffraction-limited optical imaging with a silver superlens ,” Science   308 , 534 – 537 ( 2005 ).
[Crossref] [PubMed]

T. J. Yen , W. J. Padilla , N. Fang , D. C. Vier , D. R. Smith , J. B. Pendry , D. N. Basov , and X. Zhang , “ Terahertz magnetic response from artificial materials ,” Science   303 , 1494 – 1496 ( 2004 ).
[Crossref] [PubMed]

Zhou, J. F.

S. Linden , C. Enkrich , M. Wegener , J. F. Zhou , T. Koschny , and C. M. Soukoulis , “ Magnetic response of metamaterials at 100 terahertz ,” Science   306 , 1351 – 1353 ( 2004 ).
[Crossref] [PubMed]

Appl. Phys. Lett. (2)

M. J. Freire and R. Marques , “ Planar magnetoinductive lens for three-dimensional subwavelength imaging ,” Appl. Phys. Lett.   86 , 182505 -1–3 ( 2005 ).
[Crossref]

D. R. Smith , D. Schurig , J. J. Mock , P. Kolinko , and P. Rye , “ Partial focusing of radiation by a slab of indefinite media ,” Appl. Phys. Lett.   84 , 2244 – 2246 ( 2004 ).
[Crossref]

Electron. Lett. (2)

E. Shamonina , V. A. Kalinin , K. H. Ringhofer , and L. Solymar , “ Magneto-inductive waveguide ,” Electron. Lett.   38 , 371 – 373 ( 2002 ).
[Crossref]

M. C. K. Wiltshire , E. Shamonina , I. R. Young , and L. Solymar , “ Dispersion characteristics of magneto-inductive waves: comparison between theory and experiment ,” Electron. Lett.   39 , 215 – 217 ( 2003 ).
[Crossref]

IEEE Trans. Antennas Prop. (1)

R. Marques , F. Mesa , J. Martel , and F. Medina , “ Comparative analysis of edge- and broadside-coupled split ring resonators for metamaterial design - Theory and experiments ,” IEEE Trans. Antennas Prop.   51 , 2572 – 2581 ( 2003 ).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

J. B. Pendry , A. J. Holden , D. J. Robbins , and W. J. Stewart , “ Magnetism from conductors and enhanced nonlinear phenomena ,” IEEE Trans. Microwave Theory Tech.   47 , 2075 – 2084 ( 1999 ).
[Crossref]

J. Appl. Phys. (4)

M. Shamonin , E. Shamonina , V. Kalinin , and L. Solymar , “ Properties of a metamaterial element: analytical solutions and numerical simulations for a singly split double ring ,” J. Appl. Phys.   95 , 3778 – 3784 ( 2004 ).
[Crossref]

M. C. K. Wiltshire , E. Shamonina , I. R. Young , and L. Solymar , “ Experimental and theoretical study of magneto-inductive waves supported by one-dimensional arrays of “swiss rolls” ,” J. Appl. Phys.   95 , 4488 – 4493 ( 2004 ).
[Crossref]

E. Shamonina , V.A. Kalinin , K. H. Ringhofer , and L. Solymar , “ Magnetoinductive waves in one, two, and three dimensions ,” J. Appl. Phys.   92 : 6252 – 6261 ( 2002 ).
[Crossref]

S. Maslovski , S. Tretyakov , and P. Alitalo , “ Near-field enhancement and imaging in double planar polariton-resonant structures ,” J. Appl. Phys.   96 , 1293 – 1300 ( 2004 ).
[Crossref]

J. Mod. Opt. (1)

R. Zengerle , “ Light-propagation in singly and doubly periodic planar wave-guides ,” J. Mod. Opt.   34 , 1589 – 1617 ( 1987 ).
[Crossref]

Microw. Opt. Techn. Lett. (1)

I.-H. Lin , C. Caloz , and T. Itoh , “ Near-field focusing by a nonuniform leaky-wave interface ,” Microw. Opt. Techn. Lett.   44 , 416 – 418 ( 2005 ).
[Crossref]

Opt. Express (2)

Phys. Rev. (1)

P. A. Belov , C. R. Simovski , and P. Ikonen , “ Canalization of subwavelength images by electromagnetic crystals ,” Phys. Rev.   71 , 193105 -1–4 ( 2005 ).
[Crossref]

Phys. Rev. B (1)

J. D. Baena , R. Marques , F. Medina , and J. Martel , “ Artificial magnetic metamaterial design by using spiral resonators ,” Phys. Rev. B   69 , 014402 -1–5 ( 2004 ).
[Crossref]

Phys. Rev. Lett. (3)

E. Yablonovitch , “ Inhibited spontaneous emission in solid-state physics and electronics ,” Phys. Rev. Lett.   58 , 2059 – 2062 ( 1987 ).
[Crossref] [PubMed]

S. John , “ Strong localization of photons in certain disordered dielectric superlattices ,” Phys. Rev. Lett.   58 , 2486 – 2489 ( 1987 ).
[Crossref] [PubMed]

J. B. Pendry , “ Negative refraction makes a perfect lens ,” Phys. Rev. Lett.   85 , 3966 – 3969 ( 2000 ).
[Crossref] [PubMed]

Science (4)

N. Fang , H. Lee , C. Sun , and X. Zhang , “ Sub-diffraction-limited optical imaging with a silver superlens ,” Science   308 , 534 – 537 ( 2005 ).
[Crossref] [PubMed]

M. C. K. Wiltshire , J. B. Pendry , I. R. Young , D. J. Larkman , D. J. Gilderdale , and J. V. Hajnal , “ Microstructured magnetic materials for RF flux guides in Magnetic Resonance Imaging ,” Science   291 , 849 – 851 ( 2001 ).
[Crossref] [PubMed]

T. J. Yen , W. J. Padilla , N. Fang , D. C. Vier , D. R. Smith , J. B. Pendry , D. N. Basov , and X. Zhang , “ Terahertz magnetic response from artificial materials ,” Science   303 , 1494 – 1496 ( 2004 ).
[Crossref] [PubMed]

S. Linden , C. Enkrich , M. Wegener , J. F. Zhou , T. Koschny , and C. M. Soukoulis , “ Magnetic response of metamaterials at 100 terahertz ,” Science   306 , 1351 – 1353 ( 2004 ).
[Crossref] [PubMed]

Other (3)

R. E. Collin , Field Theory of Guided Waves ( Oxford University Press, Oxford , 1991 ).

F. W. Grover , Inductance Calculations: Working Formulas and Tables ( Instrument Society of America, Research Triangle Park, N.C. , 1981 ).

M. C. K. Wiltshire , J. V. Hajnal , J. B. Pendry , and D. J. Edwards , “ RF field transmission through Swiss Rolls - an anisotropic magnetic metamaterial ,” in Proceedings of the 27th ESA Antenna Technology Workshop on Innovative Periodic Antennas: Electromagnetic Bandgap, Left-handed Materials, Fractal and Frequency Selective Surfaces , Santiago de Compostela, Spain , 9 – 11 March, 2004 .

Supplementary Material (4)

» Media 1: AVI (2046 KB)     
» Media 2: AVI (2249 KB)     
» Media 3: AVI (2549 KB)     
» Media 4: AVI (2095 KB)     

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Figures (7)

Fig. 1.
Fig. 1. (a) Lattice of resonant elements with hexagonal arrangement. (b) Direct vectors of the hexagonal lattice (d 1,d 2) and vectors of the reciprocal lattice (b 1,b 2).

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Fig. 2.
Fig. 2. 2D dispersion equation of MI waves for a hexagonal metamaterial showing the contour lines of frequency as a function of kxd and kyd. The boundaries of the first Brillouin zone are shown by bold lines.

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Fig. 3.
Fig. 3. Current distribution for circular boundary conditions at various frequencies (a–c). (2.05 MB) Movie of current distribution in the frequency range from 1.5 to 0.7 ω 0 (72 frames).

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Fig. 4.
Fig. 4. Current distribution for rectangular boundary conditions at various frequencies (a–d). (2.25 MB). Movie of current distribution in the frequency range from 1.5 to 0.7 ω 0 (77 frames).

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Fig. 5.
Fig. 5. Magnetic field distribution at four resonant frequencies: normal component (a–d) and tangential component (e–h). (2.1 MB, 2.55MB) Corresponding movies in the frequency range from 1.5 to 0.7 ω 0 (86 frames).

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Fig. 6.
Fig. 6. Resonant frequencies of a hexagonal metamaterial. Experimental values are taken from Ref.[19].

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Fig. 7.
Fig. 7. The magnetic field pattern (normal component) at ω= 0.98ω 0 (a) and ω=1.01ω0 (b).

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Equations (19)

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Z I n , m + jωM ( I n , m − 1 + I n , m + 1 + I n − 1 , m + I n + 1 , m + I n − 1 , m + 1 + I n + 1 , m − 1 ) = 0
r n , m = n d 1 + m d 2
d 1 = i x d , d 2 = ( i x cos ⁡ 60 ° + i y sin ⁡ 60 ° ) d
d 1 ∙ b 1 = 1 , d 1 ∙ b 2 = 0 ,
d 2 ∙ b 1 = 0 , d 2 ∙ b 2 = 1 .
I n , m = I 0,0 exp ( − j k ∙ r n , m ) .
k = 2 π ( f 1 b 1 + f 2 b 2 )
ω ω 0 = { 1 + κ [ cos ⁡ ( 2 π f 1 ) + cos ⁡ ( 2 π f 2 ) + cos ⁡ ( 2 π f 1 − 2 π f 2 ) ] } − 1 2
I = Z − 1 V .
I ( ν + Δ ν , μ + Δ μ ) = { 1 + Δ ν ∂ ∂ ν + Δ μ ∂ ∂ μ +
+ 1 2 [ ( Δ ν ) 2 ∂ 2 ∂ ν 2 + 2 Δ ν Δ μ ∂ 2 ∂ ν ∂ μ + ( Δ μ ) 2 ∂ 2 ∂ μ 2 ] } I ( ν , μ )
∂ 2 I ∂ ν 2 − ∂ 2 I ∂ ν ∂ μ + ∂ 2 I ∂ μ 2 + 1 2 d 2 ( 6 + Z jωM ) I = 0
∂ 2 ∂ ν 2 − ∂ 2 ∂ ν ∂ μ + ∂ 2 ∂ μ 2 = 3 4 ( ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 )
∂ 2 I ∂ x 2 + ∂ 2 I ∂ y 2 + k 2 I = 0
k 2 = 4 d 2 [ 1 + 1 3 κ ( 1 − ω 2 ω 0 2 ) ]
I = I 0 sin ⁡ ( k x x ) sin ⁡ ( k y y )
I = I 0 J 0 ( kr )
kR = ρ i
k x L x = p x π and k y L y = p y π

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